The On-Line Encyclopedia of Integer Sequences, Recent Additions

This is a section of the main database for the On-Line Encyclopedia of Integer Sequences.

It shows the most recently added sequences in reverse chronological order.

For more information see the following pages:
( www.research.att.com/~njas/sequences/ then )

    Seis.html:          Welcome
    index.html:         Lookup 
    demo1.html:         Demos 
    Sindx.html:         Index 
    WebCam.html:        WebCam
    Submit.html:        Contribute new sequence or comment 
    eishelp1.html:      Internal format 
    eishelp2.html:      Beautified format
    transforms.html:    Transforms 
    Spuzzle.html:       Puzzles 
    Shot.html:          Hot 
    classic.html:       Classics
    ol.html:            Superseeker 
    pages.html:         More pages 

Maintained  by:         N. J. A. Sloane (njas@research.att.com), 
    home page:          www.research.att.com/~njas/
    

The WebCam at www.research.att.com/~njas/sequences/WebCam.html
is another way to browse the recent additions.

[If the database has just been resorted into lexicographic order,
the present file will be empty, but the WebCam will still work.]

(start)

%I A078946
%S A078946 5,17,227,1277,1607,3527,3917,4637,4787,27737,38447,39227,44267,71327,
%T A078946 97367,99707,113147,122027,122387,124337,165707,183497,187127,191447,
%U A078946 197957,198827,275447,290657,312197,317957,347057,349397,416387,418337
%N A078946 Primes p such that the differences between the 5 consecutive primes starting with p are (2,4,6,2).
%C A078946 Equivalently, p, p+2, p+6, p+12 and p+14 are consecutive primes.
%D A078946 G. E. Andrews, MacMahon's prime numbers of measurement, Amer. Math. Monthly, 82 (1975), 922-923.
%D A078946 R. K. Guy, Unsolved Problems in Number Theory, E30.
%D A078946 P. A. MacMahon, The prime numbers of measurement on a scale, Proc. Camb. Phil. Soc. 21 (1923), 651-654; reprinted in Coll. Papers I, pp. 797-800. [Michel Lagneau (mn.lagneau2(AT)orange.fr), Feb 03 2010]
%H A078946 G. E. Andrews, <a href="http://www.jstor.org/stable/2318498">MacMahon's prime numbers of measurement</a>, Amer. Math. Monthly, 82 (1975), 922-923. [Michel Lagneau (mn.lagneau2(AT)orange.fr), Feb 03 2010]
%H A078946 T. Forbes, <a href="http://mathworld.wolfram.com/PrimeTriplet.html">Prime k-tuplets</a>
%H A078946 Eric Weisstein's World of Mathematics,<a href="http://mathworld.wolfram.com/PrimeTriplet.html"> Link to a section of The World of Mathematics</a>.
%e A078946 227 is in the sequence since 227, 229, 233, 239 and 241 are consecutive primes.
%e A078946 The two first sequences are : (5,7,11,17,19), (17,19,23,29,31) [Michel Lagneau (mn.lagneau2(AT)orange.fr), Feb 03 2010]
%p A078946 for n from 1 by 2 to 110000 do; if isprime(n) and isprime(n+2) and isprime(n+6) and isprime(n+12) and isprime(n + 14) then print(n) else fi;od; # Michel Lagneau (mn.lagneau2(AT)orange.fr), Feb 03 2010
%Y A078946 Cf. A001223, A078866, A078867, A078946-A078971, A022006, A022007.
%Y A078946 Cf. A172454, A073648, A098412 [Michel Lagneau (mn.lagneau2(AT)orange.fr), Feb 03 2010]
%K A078946 nonn,new
%O A078946 1,1
%A A078946 Labos E. (labos(AT)ana.sote.hu), Dec 19 2002
%E A078946 Edited by Dean Hickerson (dean.hickerson(AT)yahoo.com), Dec 20 2002
%E A078946 5 inserted by Michel Lagneau (mn.lagneau2(AT)orange.fr), Feb 03 2010

%I A134648
%S A134648 0,1,90,44730,56586600,154700988750,807998767676100,7373018003758407000,
%T A134648 109829050417159537464000,2532230252503738514963235000,
%U A134648 86574740102712303011539719750000,4237239732072431006302896746240010000
%N A134648 Number of 2n X n (0,1)-matrices with row sums 2 and column sums 4.
%H A134648 R. H. Hardin, <a href="b134648.txt">Table of n, a(n) for n=1..49</a>
%F A134648 a(m,n) = 24^{-m} Sum_{alpha = 0 ..m} Sum_{beta = 0 .. m-alpha } \frac{ (-1)^{(m-alpha -beta )}3^{alpha }6^{(m-alpha -beta )}m!n!(4beta +2(m-alpha -beta ))!}{alpha !beta !(m-alpha -beta )!(2beta +(m-alpha-beta ))!2^{2beta +(m-alpha -beta )}}
%e A134648 Number of 4 X 2: 1
%e A134648 Number of 6 X 3: 90
%e A134648 Number of 8 X 4: 44730
%e A134648 Number of 10 X 5: 5658660
%Y A134648 Cf. A132202, A134646, A000681, A000986, A134645, A139670, etc.
%K A134648 nonn
%O A134648 1,3
%A A134648 Shanzhen Gao (sgao2(AT)fau.edu), Nov 05 2007
%E A134648 a(7) onwards from R. H. Hardin (rhhardin(AT)att.net), Oct 18 2009

%I A134646
%S A134646 0,2,31,1344,111920,16214000,3758757240,1310799454720,655551508577280,
%T A134646 452647176631372800,418399785559398720000,504669505260741099417600,
%U A134646 777461035821119354357452800,1501959201213688265322501427200
%N A134646 Number of n X n (0,1,2)-matrices with every row sum 3 and column sum 3.
%D A134646 Zhonghua Tan and Shanzhen Gao, Counting (0,1,2)-Matrices, submitted.
%H A134646 R. H. Hardin, <a href="b134646.txt">Table of n, a(n) for n=1..99</a>
%F A134646 a(n) = 6^{-n} Sum_{alpha = 0 .. n} Sum_{beta = 0 .. n-alpha } \frac{(-4)^{(n-alpha -beta )}3^{beta }(n!)^{2}(beta +3alpha )!}{(alpha !)^{2}beta !(n-alpha -beta )!6^{alpha }}
%e A134646 a(2) = 2:
%e A134646 21 12
%e A134646 12 21
%Y A134646 Cf. A000681, A134645.
%K A134646 nonn,easy,new
%O A134646 1,2
%A A134646 Shanzhen Gao (sgao2(AT)fau.edu), Nov 05 2007
%E A134646 Definition corrected and a(7) and a(8) found (by direct enumeration) by R. H. Hardin, Oct 18 2009
%E A134646 a(9) - a(99) from R. H. Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A173058
%S A173058 8,512,
%T A173058 1056589062271330492704679569833033213037694652072243044255921418053347805113449718948834511775314375789348789986514257357764695119005371074501077956925879153816773367998010168337463035352852882106048465816422376808296056585503123477676793797534072952979077161795475996672
%N A173058 Leyland numbers (Cubes), a^b+b^a, a and b > 1.
%e A173058 2^3=8, 8^3=512,
%e A173058 101851798816724304313422284420468908052573419683296812531807022467719064988166\
%e A173058 8353091698688^3=1056...6672
%t A173058 f[a_,b_]:=a^b+b^a; Select[Union[Flatten[Table[f[a,b],{a,2,150},{b,2,150}]]],IntegerQ[(#1)^(1/3)]&]
%Y A173058 Cf. A076980, A173054, A173055, A173056
%K A173058 nonn,new,bref
%O A173058 1,1
%A A173058 Vladimir Orlovsky (4vladimir(AT)gmail.com), Feb 08 2010

%I A173057
%S A173057 2,5,10,17,40,69,100,137,190,249,320,393,472,705,944,1237,1548,1861,
%T A173057 2178,2551,2930,3523,4122,4841,5574,6313,7110,9443,11782,14175,16574,
%U A173057 19513,22632,25769,29502,33241,37034,40831,46770,53963,61294,68627
%N A173057 Partial sums of A024770.
%C A173057 Partial sums of right-truncatable primes, primes whose every prefix is prime (in decimal representation). The sequence has 83 terms. The subsequence of prime partial sums of right-truncatable primes begins: 2, 5, 17, 137, 1237, 1861, 2551, 199483. What is the largest value in the subsubsequence of right-truncatable prime partial sums of right-truncatable primes?
%e A173057 a(50) = 2 + 3 + 5 + 7 + 23 + 29 + 31 + 37 + 53 + 59 + 71 + 73 + 79 + 233 + 239 + 293 + 311 + 313 + 317 + 373 + 379 + 593 + 599 + 719 + 733 + 739 + 797 + 2333 + 2339 + 2393 + 2399 + 2939 + 3119 + 3137 + 3733 + 3739 + 3793 + 3797 + 5939 + 7193 + 7331 + 7333 + 7393 + 23333 + 23339 + 23399 + 23993 + 29399 + 31193 + 31379.
%Y A173057 Cf. A000040, A024770, A033664, A024785 (left-trucatable primes), A032437, A020994, A052023, A052024, A052025, A050986, A050987, A069866.
%K A173057 base,fini,nonn,new
%O A173057 1,1
%A A173057 Jonathan Vos Post (jvospost3(AT)gmail.com), Feb 08 2010

%I A173056
%S A173056 8,17,54,57,177,368,2169,2530,6250,8361,94932,131361,178478,524649,
%T A173056 1596520,1647086,8389137,48989176,129145076,536871753,1162268326,
%U A173056 1221074418,1996813914,2147484609,94143190994,96951758924,137438954841
%N A173056 Numbers of the form a^b+b^a, a and b are primes.
%C A173056 2^2+2^2=8, 2^3+3^2=17, 3^3+3^3=54, 2^5+5^2=57, 2^7+7^2=177, 3^5+5^3=368,..
%t A173056 f[a_,b_]:=Prime[a]^Prime[b]+Prime[b]^Prime[a]; Take[Union[Flatten[Table[f[a,b],{a,1,60},{b,1,60}]]],50]
%Y A173056 Cf. A076980, A173054, A173055
%K A173056 nonn,new
%O A173056 1,1
%A A173056 Vladimir Orlovsky (4vladimir(AT)gmail.com), Feb 08 2010

%I A173055
%S A173055 368,2530,94932,178478,1596520,48989176,129145076,1162268326,1221074418,
%T A173055 1996813914,94143190994,96951758924,762940872982,19073488804224,
%U A173055 36314872537968,68630377389272,232630924325880,617673396313738
%N A173055 Numbers of the form a^b+b^a, a and b are odd primes, b > a.
%C A173055 3^5 + 5^3 = 368, 3^7 + 7^3 = 2530, 5^7 + 7^5 = 94932,..
%t A173055 f[a_,b_]:=Prime[a]^Prime[b]+Prime[b]^Prime[a]; Take[Union[Flatten[Table[f[a,b],{a,2,60},{b,a+1,60}]]],40]
%Y A173055 Cf. A076980, A173054
%K A173055 nonn,new
%O A173055 1,1
%A A173055 Vladimir Orlovsky (4vladimir(AT)gmail.com), Feb 08 2010

%I A173054
%S A173054 17,32,57,100,145,177,320,368,593,945,1124,1649,2169,2530,4240,5392,
%T A173054 7073,8361,16580,18785,20412,23401,32993,60049,65792,69632,94932,131361,
%U A173054 178478,262468,268705,397585,423393,524649,533169,1048976,1058576
%N A173054 Numbers of the form a^b+b^a, a > 1, b > a.
%C A173054 2^3+3^2=17,..
%t A173054 f[a_,b_]:=a^b+b^a; Take[Union[Flatten[Table[f[a,b],{a,2,50},{b,a+1,50}]]],80]
%Y A173054 Cf. A076980
%K A173054 nonn,new
%O A173054 1,1
%A A173054 Vladimir Orlovsky (4vladimir(AT)gmail.com), Feb 08 2010

%I A173053
%S A173053 0,1,3,118,942,25690
%N A173053 Numbers n such that a(n)=2^(2*n)+2*n+1 is a prime
%C A173053 So a(n) = 2^(x-1)+x at x=2n+1 and these x are in A061422. We got six odd x in A061422 which leads to known solutions n=0, 1, 3, 118, 942, 25690, from Richard Mathar (mathar@strw.leidenuniv.nl)
%F A173053 a(n)=2^(2*n)+2*n+1
%e A173053 For n=0, a(0)=1+1=2; n=1, a(1)=2^2+3; n=3, a(n)=2^6+7=71; 2^236+237=110427941548649020598956093796432407239217743554726184882600387580788973; 2^1884+1885= 1382012053811764656063603814922461897232169149133334820257746561118767804680043393092721633850612925251668622490076521344323973994295501605289654085673382070337356164068219336016728992335814914911412187690428571398333020743770166324877584497363683636171472658647851335511884201240262003030383530995329899916429574621923842877572383659283690961568644353925008051273579630011760909888042372159186092413046670909688409283414159958649645833376606067462959428860119024655756469887393606641633377838666492170994665643096356310676185469174583336345878025903816130428525348701; 2^51380+51381 (Number more than 10000 digits).
%Y A173053 Cf. A061422
%K A173053 nonn,new
%O A173053 0,3
%A A173053 Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Feb 08 2010

%I A173052
%S A173052 1,3,16,53,160,273,410,1423,2460,3539,4776,6143,7522,17601,27724,37860,
%T A173052 47999,58236,68515,78882,89261,101640,115319,215598,315977,417214,
%U A173052 519561,621940,725619,849098,1850335,2852682,3855061,4858740,5871089
%N A173052 Partial sums of A072857.
%C A173052 Partial sums of primeval numbers. Primeval number: a prime which "contains" more primes in it than any preceding number. Here "contains" means may be constructed from a subset of its digits. E.g., 1379 contains 3, 7, 13, 17, 19, 31, 37, 71, 73, 79, 97, 137, 139, 173, 179, 193, 197, 317, 379, 397, 719, 739, 937, 971, 1973, 3719, 3917, 7193, 9137, 9173, and 9371. The subsequence of prime partial sums of primeval numbers begins: 3, 53, 1423, 3539, 6143, 89261, 115319, 315977. What is the smallest primeval prime partial sums of primeval numbers, i.e. the intersection of this sequence with A119535?
%H A173052 Chris K. Caldwell AND G. L. Honaker, Jr., <a href="http://primes.utm.edu/curios/includes/glossary.pdf">A BRIEF PRIME CURIOS! GLOSSARY</a>.
%F A173052 a(n) = SUM[i=1..n] A072857(i) = SUM[i=1..n] {numbers that set a record for the number of distinct primes that can be obtained by permuting some subset of their digits}.
%e A173052 a(36) = 1 + 2 + 13 + 37 + 107 + 113 + 137 + 1013 + 1037 + 1079 + 1237 + 1367 + 1379 + 10079 + 10123 + 10136 + 10139 + 10237 + 10279 + 10367 + 10379 + 12379 + 13679 + 100279 + 100379 + 101237 + 102347 + 102379 + 103679 + 123479 + 1001237 + 1002347 + 1002379 + 1003679 + 1012349 + 1012379.
%Y A173052 Cf. A000040, A072857, A039993, A075053, A076497, A076449, A119535 (prime subsequence).
%K A173052 base,easy,nonn,new
%O A173052 1,2
%A A173052 Jonathan Vos Post (jvospost3(AT)gmail.com), Feb 08 2010

%I A173051
%S A173051 10123457689,20246923478,30370389375,40493875054,50617360823,
%T A173051 60740857680,70864405549,80987954228,91111523175,101235101824
%N A173051 Partial sums of A050288.
%C A173051 Partial sums of (base 10) Pandigital primes. Note that almost all primes are pandigital. a(59) is (after the first value) the first prime in this sequence. What is the smallest pandigital prime partial sum of (base 10) pandigital primes? In other bases?
%F A173051 a(n) = SUM[i=1..n] A050288(i) = SUM[i=1..n] {p is prime and p, base 10, has all 10 digits in its decimal representation, digits may appear multiple times}.
%e A173051 The least prime after a(1) is a(59) = 10123457689 + 10123465789 + 10123465897 + 10123485679 + 10123485769 + 10123496857 + 10123547869 + 10123548679 + 10123568947 + 10123578649 + 10123586947 + 10123598467 + 10123654789 + 10123684759 + 10123685749 + 10123694857 + 10123746859 + 10123784569 + 10123846597 + 10123849657 + 10123854679 + 10123876549 + 10123945687 + 10123956487 + 10123965847 + 10123984657 + 10124356789 + 10124358697 + 10124365879 + 10124365987 + 10124369587 + 10124378569 + 10124385967 + 10124389567 + 10124395867 + 10124398657 + 10124536789 + 10124538769 + 10124563789 + 10124563879 + 10124563987 + 10124568793 + 10124576893 + 10124578693 + 10124579863 + 10124583967 + 10124586397 + 10124589637 + 10124593867 + 10124596873 + 10124597683 + 10124635879 + 10124635897 + 10124638759 + 10124659873 + 10124673859 + 10124678953 + 10124683759 + 10124685379 = 597325496783 is prime.
%Y A173051 Cf. A000040, A050288, A050290.
%K A173051 base,easy,nonn,new
%O A173051 1,1
%A A173051 Jonathan Vos Post (jvospost3(AT)gmail.com), Feb 08 2010

%I A172963
%S A172963 1,670,133721236,2043207067941480,1047367983257881640491925,
%T A172963 10336768663024665454036757208361400,
%U A172963 1313075006655438005305387554231256477491331800
%N A172963 Number of 2*n X n 0..9 arrays with row sums 9 and column sums 18
%H A172963 R. H. Hardin, <a href="b172963.txt">Table of n, a(n) for n=1..10</a>
%K A172963 nonn,new
%O A172963 1,2
%A A172963 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172962
%S A172962 1,3,55,10147,22069251,602351808741,215717608046511873,
%T A172962 1046591482728407939338275,70417932475495769964322670258947,
%U A172962 66879971495995556019851723767661130580273
%N A172962 Number of n X n 0..9 arrays with row sums n and column sums n
%H A172962 R. H. Hardin, <a href="b172962.txt">Table of n, a(n) for n=1..11</a>
%K A172962 nonn,new
%O A172962 1,2
%A A172962 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172961
%S A172961 1,3139,19546975,171475190227,1846780916614531,22800663583664571781,
%T A172961 310323674631037864285609,4543765474410728783088589075,
%U A172961 70417932475495769964322670258947,1142003787585101049672484911221507209
%N A172961 Number of n X 9 0..9 arrays with row sums 9 and column sums n
%H A172961 R. H. Hardin, <a href="b172961.txt">Table of n, a(n) for n=1..24</a>
%K A172961 nonn,new
%O A172961 1,2
%A A172961 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172960
%S A172960 1,8953,176703337,5158851352489,190172119451839801,
%T A172960 8184214105554904614241,393277602707575435993011793,
%U A172960 20524231145279587298876714943337,1142003787585101049672484911221507209
%N A172960 Number of n X 10 0..9 arrays with row sums 10 and column sums n
%H A172960 R. H. Hardin, <a href="b172960.txt">Table of n, a(n) for n=1..18</a>
%K A172960 nonn,new
%O A172960 1,2
%A A172960 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172959
%S A172959 0,9,2124,5010670,79190239681,6289730555976945,2022961772806402753444,
%T A172959 2233030862952577841545024528,7423987282347762864682376427695952,
%U A172959 66879971495995556019851723767661130580273
%N A172959 Number of n X n 0..9 arrays with row sums 10 and column sums 10
%H A172959 R. H. Hardin, <a href="b172959.txt">Table of n, a(n) for n=1..14</a>
%K A172959 nonn,new
%O A172959 1,2
%A A172959 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172958
%S A172958 1,10,1540,2309384,20856798285,842286559093240,123538613356253145400,
%T A172958 56272722406349235035916800,70417932475495769964322670258947,
%U A172958 219302242655748448462474951981229489200
%N A172958 Number of n X n 0..9 arrays with row sums 9 and column sums 9
%H A172958 R. H. Hardin, <a href="b172958.txt">Table of n, a(n) for n=1..15</a>
%K A172958 nonn,new
%O A172958 1,2
%A A172958 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172957
%S A172957 0,8,2667,9939400,268482337210,40250607905812176,26993049504287277951750,
%T A172957 68257203489300320501394173184,567530705296541288110503654350454672,
%U A172957 13875282554259983875399222296482616642748800
%N A172957 Number of n X n 0..9 arrays with row sums 11 and column sums 11
%H A172957 R. H. Hardin, <a href="b172957.txt">Table of n, a(n) for n=1..12</a>
%K A172957 nonn,new
%O A172957 1,2
%A A172957 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172956
%S A172956 0,7,3079,17909911,814185714530,223679092210809116,
%T A172956 300437743830090052243120,1659993098152833453274748097748,
%U A172956 32763575419677710012600143229118604368
%N A172956 Number of n X n 0..9 arrays with row sums 12 and column sums 12
%H A172956 R. H. Hardin, <a href="b172956.txt">Table of n, a(n) for n=1..11</a>
%K A172956 nonn,new
%O A172956 1,2
%A A172956 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172955
%S A172955 0,6,3300,29325760,2208879837845,1087071687451329840,
%T A172955 2832943476223117064033156,32889801403868602076336080073600,
%U A172955 1473836226725275907157399237234998604948
%N A172955 Number of n X n 0..9 arrays with row sums 13 and column sums 13
%H A172955 R. H. Hardin, <a href="b172955.txt">Table of n, a(n) for n=1..11</a>
%K A172955 nonn,new
%O A172955 1,2
%A A172955 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172954
%S A172954 0,5,3300,43803256,5371024286115,4641887910957574500,
%T A172954 22868384331806129843900090,540079347936742523341917036638800,
%U A172954 52908783231949497453582944166083684222304
%N A172954 Number of n X n 0..9 arrays with row sums 14 and column sums 14
%H A172954 R. H. Hardin, <a href="b172954.txt">Table of n, a(n) for n=1..10</a>
%K A172954 nonn,new
%O A172954 1,2
%A A172954 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172953
%S A172953 0,4,3079,59930080,11745341072931,17491353277138361640,
%T A172953 159265228322047609940846213,7444077333816868635568524618503936,
%U A172953 1543634978788620301563132446180065043318640
%N A172953 Number of n X n 0..9 arrays with row sums 15 and column sums 15
%H A172953 R. H. Hardin, <a href="b172953.txt">Table of n, a(n) for n=1..10</a>
%K A172953 nonn,new
%O A172953 1,2
%A A172953 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172952
%S A172952 0,344,107624896,1883878552356976,1012681356319237050478275,
%T A172952 10180416476430663944707156952314700,
%U A172952 1303360851994578257391361738660273438063376250
%N A172952 Number of 2*n X n 0..8 arrays with row sums 9 and column sums 18
%H A172952 R. H. Hardin, <a href="b172952.txt">Table of n, a(n) for n=1..10</a>
%K A172952 nonn,new
%O A172952 1,2
%A A172952 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172951
%S A172951 1,489,49207203,274963523715301,38828651093351514263025,
%T A172951 81979160780389846116516969784320,
%U A172951 1773632104073646452662520542376150560342605
%N A172951 Number of 2*n X n 0..8 arrays with row sums 8 and column sums 16
%H A172951 R. H. Hardin, <a href="b172951.txt">Table of n, a(n) for n=1..11</a>
%K A172951 nonn,new
%O A172951 1,2
%A A172951 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172950
%S A172950 1,3,55,10147,22069251,602351808741,215717608046511873,
%T A172950 1046591482728407939338275,70413374705157290394475959841507,
%U A172950 66845086415203272975671039831846565573553
%N A172950 Number of n X n 0..8 arrays with row sums n and column sums n
%H A172950 R. H. Hardin, <a href="b172950.txt">Table of n, a(n) for n=1..12</a>
%K A172950 nonn,new
%O A172950 1,2
%A A172950 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172949
%S A172949 1,73789,14823083089,4855940642591941,2124709987334672961301,
%T A172949 1125711921752126203903372141,683259412052443643145910905655009,
%U A172949 459028914923930661875625466306150410469
%N A172949 Number of n X 12 0..8 arrays with row sums 12 and column sums n
%H A172949 R. H. Hardin, <a href="b172949.txt">Table of n, a(n) for n=1..12</a>
%K A172949 nonn,new
%O A172949 1,2
%A A172949 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172948
%S A172948 1,8953,176703337,5158851352489,190172119451839801,
%T A172948 8184214105554904614241,393277602707575435993011793,
%U A172948 20524231145279587298876714943337,1141793131372748920690525916472692809
%N A172948 Number of n X 10 0..8 arrays with row sums 10 and column sums n
%H A172948 R. H. Hardin, <a href="b172948.txt">Table of n, a(n) for n=1..18</a>
%K A172948 nonn,new
%O A172948 1,2
%A A172948 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172947
%S A172947 1,1107,2186815,5796870115,18343212299091,65338547748658101,
%T A172947 253310893747894263177,1046591482728407939338275,
%U A172947 4543765474410728783088589075,20524231145279587298876714943337
%N A172947 Number of n X 8 0..8 arrays with row sums 8 and column sums n
%H A172947 R. H. Hardin, <a href="b172947.txt">Table of n, a(n) for n=1..35</a>
%K A172947 nonn,new
%O A172947 1,2
%A A172947 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172946
%S A172946 1,25653,1612210777,157366288875709,19947450199250403501,
%T A172946 3006271871518328730143421,512421117125922437799936404913,
%U A172946 95761990178886923105405778950731677
%N A172946 Number of n X 11 0..8 arrays with row sums 11 and column sums n
%H A172946 R. H. Hardin, <a href="b172946.txt">Table of n, a(n) for n=1..14</a>
%K A172946 nonn,new
%O A172946 1,2
%A A172946 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172945
%S A172945 1,3139,19546975,171475190227,1846780916614531,22800663583664571781,
%T A172945 310323674631037864285609,4543765474410728783088589075,
%U A172945 70413374705157290394475959841507,1141793131372748920690525916472692809
%N A172945 Number of n X 9 0..8 arrays with row sums 9 and column sums n
%H A172945 R. H. Hardin, <a href="b172945.txt">Table of n, a(n) for n=1..24</a>
%K A172945 nonn,new
%O A172945 1,2
%A A172945 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172944
%S A172944 1,9,1035,981541,4855258305,94459713879600,5945968652327831925,
%T A172944 1046591482728407939338275,459761347800901006933211075259,
%U A172944 460129061613105910803354676484897475
%N A172944 Number of n X n 0..8 arrays with row sums 8 and column sums 8
%H A172944 R. H. Hardin, <a href="b172944.txt">Table of n, a(n) for n=1..17</a>
%K A172944 nonn,new
%O A172944 1,2
%A A172944 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172943
%S A172943 0,6,2100,8777176,252302863410,39004916064327576,26552617137288925164846,
%T A172943 67648499181095959493255910336,564659092427723249985264571037981712,
%U A172943 13834003902493629391621853438501279412753600
%N A172943 Number of n X n 0..8 arrays with row sums 11 and column sums 11
%H A172943 R. H. Hardin, <a href="b172943.txt">Table of n, a(n) for n=1..12</a>
%K A172943 nonn,new
%O A172943 1,2
%A A172943 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172942
%S A172942 0,5,2191,14424679,715535002730,207412191964163486,
%T A172942 287389284114499903096180,1616438584669598518787107631636,
%U A172942 32234252378649647001051217798280395344
%N A172942 Number of n X n 0..8 arrays with row sums 12 and column sums 12
%H A172942 R. H. Hardin, <a href="b172942.txt">Table of n, a(n) for n=1..12</a>
%K A172942 nonn,new
%O A172942 1,2
%A A172942 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172941
%S A172941 0,4,2100,21224392,1776676349670,943257597506859240,
%T A172941 2584986583431322279831286,30994847539761067648293062000000,
%U A172941 1417657385775947212537904178152631764244
%N A172941 Number of n X n 0..8 arrays with row sums 13 and column sums 13
%H A172941 R. H. Hardin, <a href="b172941.txt">Table of n, a(n) for n=1..11</a>
%K A172941 nonn,new
%O A172941 1,2
%A A172941 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172940
%S A172940 0,3,1842,28083496,3882567048795,3691124596208508930,
%T A172940 19502304786439261984306250,483784356286505032648292371201968,
%U A172940 49022230603053508588871953902168912724896
%N A172940 Number of n X n 0..8 arrays with row sums 14 and column sums 14
%H A172940 R. H. Hardin, <a href="b172940.txt">Table of n, a(n) for n=1..10</a>
%K A172940 nonn,new
%O A172940 1,2
%A A172940 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172939
%S A172939 0,2,1462,33427744,7500881437251,12504843842710113540,
%T A172939 124430557151683165279097411,6218376942668389682021820614125312,
%U A172939 1354372742296729064606689624633970992039248
%N A172939 Number of n X n 0..8 arrays with row sums 15 and column sums 15
%H A172939 R. H. Hardin, <a href="b172939.txt">Table of n, a(n) for n=1..10</a>
%K A172939 nonn,new
%O A172939 1,2
%A A172939 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172938
%S A172938 0,8,1462,2285392,20799366165,841536749932180,123497359693624468650,
%T A172938 56264817255407588985669120,70413374705157290394475959841507,
%U A172938 219295201090394751786316169884571962500
%N A172938 Number of n X n 0..8 arrays with row sums 9 and column sums 9
%H A172938 R. H. Hardin, <a href="b172938.txt">Table of n, a(n) for n=1..15</a>
%K A172938 nonn,new
%O A172938 1,2
%A A172938 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172937
%S A172937 0,7,1842,4753822,77598074001,6236610887770425,2015220250384997352592,
%T A172937 2228880487828542717259248816,7416866626501160507667175518765840,
%U A172937 66845086415203272975671039831846565573553
%N A172937 Number of n X n 0..8 arrays with row sums 10 and column sums 10
%H A172937 R. H. Hardin, <a href="b172937.txt">Table of n, a(n) for n=1..14</a>
%K A172937 nonn,new
%O A172937 1,2
%A A172937 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172936
%S A172936 1,18152,513799394688,3453235602818369547520,
%T A172936 1835784677576057001102381224194900,
%U A172936 37246115504386074787217299933684238287194514560
%N A172936 Number of 3*n X n 0..7 arrays with row sums 7 and column sums 21
%H A172936 R. H. Hardin, <a href="b172936.txt">Table of n, a(n) for n=1..10</a>
%K A172936 nonn,new
%O A172936 1,2
%A A172936 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172935
%S A172935 0,231,37537218,245905976350021,36903579347747029033800,
%T A172935 79946751614323391528510957685750,
%U A172935 1750260314638524131612128230641517272720595
%N A172935 Number of 2*n X n 0..7 arrays with row sums 8 and column sums 16
%H A172935 R. H. Hardin, <a href="b172935.txt">Table of n, a(n) for n=1..11</a>
%K A172935 nonn,new
%O A172935 1,2
%A A172935 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172934
%S A172934 1,344,16205400,29920962972240,1030178236624445762100,
%T A172934 405285856727517597572011434240,1284495236763845536447959729080655425337,
%U A172934 25229644523536541011926273204266840907955788193920
%N A172934 Number of 2*n X n 0..7 arrays with row sums 7 and column sums 14
%H A172934 R. H. Hardin, <a href="b172934.txt">Table of n, a(n) for n=1..12</a>
%K A172934 nonn,new
%O A172934 1,2
%A A172934 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172933
%S A172933 0,146,62923906,1389194436231112,851595316175807779373475,
%T A172933 9204278755664744827525498829547800,
%U A172933 1226956821626286798384954346260110391913131030
%N A172933 Number of 2*n X n 0..7 arrays with row sums 9 and column sums 18
%H A172933 R. H. Hardin, <a href="b172933.txt">Table of n, a(n) for n=1..10</a>
%K A172933 nonn,new
%O A172933 1,2
%A A172933 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172932
%S A172932 1,3,55,10147,22069251,602351808741,215717608046511873,
%T A172932 1046211088756248825569475,70241609479301663107223809203427,
%U A172932 66348852357002987700605359091266362631153
%N A172932 Number of n X n 0..7 arrays with row sums n and column sums n
%H A172932 R. H. Hardin, <a href="b172932.txt">Table of n, a(n) for n=1..12</a>
%K A172932 nonn,new
%O A172932 1,2
%A A172932 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172931
%S A172931 1,393,248137,200148649,187403959401,193953937375521,215717608046511873,
%T A172931 253310893747894263177,310323674631037864285609,
%U A172931 393277602707575435993011793,512421117125922437799936404913
%N A172931 Number of n X 7 0..7 arrays with row sums 7 and column sums n
%H A172931 R. H. Hardin, <a href="b172931.txt">Table of n, a(n) for n=1..45</a>
%K A172931 nonn,new
%O A172931 1,2
%A A172931 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172930
%S A172930 1,8953,176703337,5158851352489,190172119451839801,
%T A172930 8184214105554904614241,393277602707575435993011793,
%U A172930 20489936895192142861722091906537,1136916899291276422087603683231884809
%N A172930 Number of n X 10 0..7 arrays with row sums 10 and column sums n
%H A172930 R. H. Hardin, <a href="b172930.txt">Table of n, a(n) for n=1..18</a>
%K A172930 nonn,new
%O A172930 1,2
%A A172930 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172929
%S A172929 1,3139,19546975,171475190227,1846780916614531,22800663583664571781,
%T A172929 310323674631037864285609,4539465445191675494919391795,
%U A172929 70241609479301663107223809203427,1136916899291276422087603683231884809
%N A172929 Number of n X 9 0..7 arrays with row sums 9 and column sums n
%H A172929 R. H. Hardin, <a href="b172929.txt">Table of n, a(n) for n=1..24</a>
%K A172929 nonn,new
%O A172929 1,2
%A A172929 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172928
%S A172928 1,1107,2186815,5796870115,18343212299091,65338547748658101,
%T A172928 253310893747894263177,1046211088756248825569475,
%U A172928 4539465445191675494919391795,20489936895192142861722091906537
%N A172928 Number of n X 8 0..7 arrays with row sums 8 and column sums n
%H A172928 R. H. Hardin, <a href="b172928.txt">Table of n, a(n) for n=1..34</a>
%K A172928 nonn,new
%O A172928 1,2
%A A172928 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172927
%S A172927 1,73789,14823083089,4855940642591941,2124709987334672961301,
%T A172927 1125711921752126203903372141,683259412052443643145910905655009,
%U A172927 457473931605348347085693323651070310789
%N A172927 Number of n X 12 0..7 arrays with row sums 12 and column sums n
%H A172927 R. H. Hardin, <a href="b172927.txt">Table of n, a(n) for n=1..13</a>
%K A172927 nonn,new
%O A172927 1,2
%A A172927 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172926
%S A172926 1,25653,1612210777,157366288875709,19947450199250403501,
%T A172926 3006271871518328730143421,512421117125922437799936404913,
%U A172926 95523201836591926809187126531656477
%N A172926 Number of n X 11 0..7 arrays with row sums 11 and column sums n
%H A172926 R. H. Hardin, <a href="b172926.txt">Table of n, a(n) for n=1..16</a>
%K A172926 nonn,new
%O A172926 1,2
%A A172926 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172925
%S A172925 0,4,1356,6487624,207015210610,34361552147620896,24478936797552315587502,
%T A172925 64168030353697742360750556800,545286936782342675064752807847170544,
%U A172925 13512485901432412441378747065599223927264000
%N A172925 Number of n X n 0..7 arrays with row sums 11 and column sums 11
%H A172925 R. H. Hardin, <a href="b172925.txt">Table of n, a(n) for n=1..12</a>
%K A172925 nonn,new
%O A172925 1,2
%A A172925 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172924
%S A172924 0,3,1216,9231031,517724571330,165121260897866936,
%T A172924 245134923025161071850640,1446656016533971629592879758484,
%U A172924 29816381062270870630530816568741792176
%N A172924 Number of n X n 0..7 arrays with row sums 12 and column sums 12
%H A172924 R. H. Hardin, <a href="b172924.txt">Table of n, a(n) for n=1..12</a>
%K A172924 nonn,new
%O A172924 1,2
%A A172924 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172923
%S A172923 0,6,1216,2113288,20095347085,828045453260560,122498884159587560730,
%T A172923 56021411242071401516696320,70241609479301663107223809203427,
%U A172923 218979482042361137186381999071773029400
%N A172923 Number of n X n 0..7 arrays with row sums 9 and column sums 9
%H A172923 R. H. Hardin, <a href="b172923.txt">Table of n, a(n) for n=1..15</a>
%K A172923 nonn,new
%O A172923 1,2
%A A172923 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172922
%S A172922 0,1,666,12441192,2028016023915,2213060559783689880,
%T A172922 13095078470636262271880150,355581688897614695233992602967856,
%U A172922 38666510837044677496604059187075890364736
%N A172922 Number of n X n 0..7 arrays with row sums 14 and column sums 14
%H A172922 R. H. Hardin, <a href="b172922.txt">Table of n, a(n) for n=1..10</a>
%K A172922 nonn,new
%O A172922 1,3
%A A172922 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172921
%S A172921 0,0,406,11492512,3178005285051,6245350445865892320,
%T A172921 71187872386623811677450179,3978462276964182243369639664709376,
%U A172921 948836768023263746262225246333504275312592
%N A172921 Number of n X n 0..7 arrays with row sums 15 and column sums 15
%H A172921 R. H. Hardin, <a href="b172921.txt">Table of n, a(n) for n=1..10</a>
%K A172921 nonn,new
%O A172921 1,3
%A A172921 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172920
%S A172920 0,5,1356,3978958,70252384001,5897868149018025,1951415739647747836330,
%T A172920 2186630273077730637194339120,7330158006601231605920532966798192,
%U A172920 66348852357002987700605359091266362631153
%N A172920 Number of n X n 0..7 arrays with row sums 10 and column sums 10
%H A172920 R. H. Hardin, <a href="b172920.txt">Table of n, a(n) for n=1..15</a>
%K A172920 nonn,new
%O A172920 1,2
%A A172920 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172919
%S A172919 1,8,666,381424,976395820,8575979362560,215717608046511873,
%T A172919 13590707419428422843904,1933216160887575268614599040,
%U A172919 571506744082785127089569166384000
%N A172919 Number of n X n 0..7 arrays with row sums 7 and column sums 7
%H A172919 R. H. Hardin, <a href="b172919.txt">Table of n, a(n) for n=1..20</a>
%K A172919 nonn,new
%O A172919 1,2
%A A172919 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172918
%S A172918 0,7,966,965557,4830921860,94285363835070,5941344401501213145,
%T A172918 1046211088756248825569475,459676589298753554382247541064,
%U A172918 460083089716507694000345247855463725
%N A172918 Number of n X n 0..7 arrays with row sums 8 and column sums 8
%H A172918 R. H. Hardin, <a href="b172918.txt">Table of n, a(n) for n=1..18</a>
%K A172918 nonn,new
%O A172918 1,2
%A A172918 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172917
%S A172917 122975608,114601242382721619224,255860848198497899739373566171520,
%T A172917 835840537229783152562450810757651648205244120,
%U A172917 3380280640943155648574196396675785374012700247929920799808
%N A172917 Number of 4*n X 8 0..6 arrays with row sums 6 and column sums 3*n
%H A172917 R. H. Hardin, <a href="b172917.txt">Table of n, a(n) for n=1..11</a>
%K A172917 nonn,new
%O A172917 1,1
%A A172917 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172916
%S A172916 352128,811889208811800,4801114002903931560293544,
%T A172916 43212752077206858567543807299695080,
%U A172916 493557244824104842248683942590327045986131328
%N A172916 Number of 3*n X 9 0..6 arrays with row sums 6 and column sums 2*n
%H A172916 R. H. Hardin, <a href="b172916.txt">Table of n, a(n) for n=1..13</a>
%K A172916 nonn,new
%O A172916 1,1
%A A172916 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172915
%S A172915 0,4332,280069725072,2575828082701033916800,
%T A172915 1572889460074298922779258762727000,
%U A172915 34135169535125235488192553066649784793796595520
%N A172915 Number of 3*n X n 0..6 arrays with row sums 7 and column sums 21
%H A172915 R. H. Hardin, <a href="b172915.txt">Table of n, a(n) for n=1..10</a>
%K A172915 nonn,new
%O A172915 1,2
%A A172915 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172914
%S A172914 1,9331,69327885583,69404300198350892236,3373513537183933018216388397350,
%T A172914 4081159676302022080974044453167358453805921,
%U A172914 76587973156243192496223417009664279130517333252040973460
%N A172914 Number of 3*n X n 0..6 arrays with row sums 6 and column sums 18
%H A172914 R. H. Hardin, <a href="b172914.txt">Table of n, a(n) for n=1..11</a>
%K A172914 nonn,new
%O A172914 1,2
%A A172914 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172913
%S A172913 3432,492872166072,276686670102097569600,292602598138566972319466403000,
%T A172913 446297806870359768599238974918072797632,
%U A172913 864293171071670188384186291883505606721028274624
%N A172913 Number of 2*n X 14 0..6 arrays with row sums 7 and column sums n
%H A172913 R. H. Hardin, <a href="b172913.txt">Table of n, a(n) for n=1..12</a>
%K A172913 nonn,new
%O A172913 1,1
%A A172913 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172912
%S A172912 924,6152037276,127676780106968400,4509741556140302851191900,
%T A172912 216845410132724601463488385879824,
%U A172912 12753041531161741070937921930120292019856
%N A172912 Number of 2*n X 12 0..6 arrays with row sums 6 and column sums n
%H A172912 R. H. Hardin, <a href="b172912.txt">Table of n, a(n) for n=1..16</a>
%K A172912 nonn,new
%O A172912 1,1
%A A172912 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172911
%S A172911 0,146,11455470,25512834226600,950699986010003704500,
%T A172911 388015827514187438302900020000,1252647814330243252910452100864057894337,
%U A172911 24848247984277498423162949237290870891470475838400
%N A172911 Number of 2*n X n 0..6 arrays with row sums 7 and column sums 14
%H A172911 R. H. Hardin, <a href="b172911.txt">Table of n, a(n) for n=1..12</a>
%K A172911 nonn,new
%O A172911 1,2
%A A172911 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172910
%S A172910 0,44,22119391,649096595660632,496395338691326078075775,
%T A172910 6287689969637354023974752834007500,
%U A172910 936126190440040230109711933212013162356155460
%N A172910 Number of 2*n X n 0..6 arrays with row sums 9 and column sums 18
%H A172910 R. H. Hardin, <a href="b172910.txt">Table of n, a(n) for n=1..10</a>
%K A172910 nonn,new
%O A172910 1,2
%A A172910 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172909
%S A172909 0,85,18852138,160367559138277,28295985978361118312100,
%T A172909 67572411262488084551685074297100,
%U A172909 1569272515289638694593745151597761337133425
%N A172909 Number of 2*n X n 0..6 arrays with row sums 8 and column sums 16
%H A172909 R. H. Hardin, <a href="b172909.txt">Table of n, a(n) for n=1..11</a>
%K A172909 nonn,new
%O A172909 1,2
%A A172909 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172908
%S A172908 1,231,4645936,2503134855620,18123941520898218600,
%T A172908 1125711921752126203903372141,436763455430178705879474096982721048,
%U A172908 835840537229783152562450810757651648205244120
%N A172908 Number of 2*n X n 0..6 arrays with row sums 6 and column sums 12
%H A172908 R. H. Hardin, <a href="b172908.txt">Table of n, a(n) for n=1..15</a>
%K A172908 nonn,new
%O A172908 1,2
%A A172908 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172907
%S A172907 1,3,55,10147,22069251,602351808741,215298243454580553,
%T A172907 1035075803117199670120995,68263787608482538080643047338467,
%U A172907 62721314697296961151225820176430830979953
%N A172907 Number of n X n 0..6 arrays with row sums n and column sums n
%H A172907 R. H. Hardin, <a href="b172907.txt">Table of n, a(n) for n=1..12</a>
%K A172907 nonn,new
%O A172907 1,2
%A A172907 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172906
%S A172906 1,73789,14823083089,4855940642591941,2124709987334672961301,
%T A172906 1125711921752126203903372141,671206979825282639695814241561409,
%U A172906 439431084223425427245984332541081989989
%N A172906 Number of n X 12 0..6 arrays with row sums 12 and column sums n
%H A172906 R. H. Hardin, <a href="b172906.txt">Table of n, a(n) for n=1..14</a>
%K A172906 nonn,new
%O A172906 1,2
%A A172906 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172905
%S A172905 1,1107,2186815,5796870115,18343212299091,65338547748658101,
%T A172905 252150480931735527177,1035075803117199670120995,
%U A172905 4460486602857545788513340755,19987118442361472471477122541737
%N A172905 Number of n X 8 0..6 arrays with row sums 8 and column sums n
%H A172905 R. H. Hardin, <a href="b172905.txt">Table of n, a(n) for n=1..35</a>
%K A172905 nonn,new
%O A172905 1,2
%A A172905 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172904
%S A172904 1,141,28681,7100821,1985311701,602351808741,193953937375521,
%T A172904 65338547748658101,22800663583664571781,8184214105554904614241,
%U A172904 3006271871518328730143421,1125711921752126203903372141
%N A172904 Number of n X 6 0..6 arrays with row sums 6 and column sums n
%H A172904 R. H. Hardin, <a href="b172904.txt">Table of n, a(n) for n=1..61</a>
%K A172904 nonn,new
%O A172904 1,2
%A A172904 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172903
%S A172903 1,393,248137,200148649,187403959401,193953937375521,215298243454580553,
%T A172903 252150480931735527177,307972094130678784570009,
%U A172903 389031356288581849014042193,505162253803288598888223487713
%N A172903 Number of n X 7 0..6 arrays with row sums 7 and column sums n
%H A172903 R. H. Hardin, <a href="b172903.txt">Table of n, a(n) for n=1..45</a>
%K A172903 nonn,new
%O A172903 1,2
%A A172903 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172902
%S A172902 1,25653,1612210777,157366288875709,19947450199250403501,
%T A172902 3006271871518328730143421,505162253803288598888223487713,
%U A172902 92474463058795212569743127797930077
%N A172902 Number of n X 11 0..6 arrays with row sums 11 and column sums n
%H A172902 R. H. Hardin, <a href="b172902.txt">Table of n, a(n) for n=1..16</a>
%K A172902 nonn,new
%O A172902 1,2
%A A172902 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172901
%S A172901 1,3139,19546975,171475190227,1846780916614531,22800663583664571781,
%T A172901 307972094130678784570009,4460486602857545788513340755,
%U A172901 68263787608482538080643047338467,1092146417007065643218131873660038409
%N A172901 Number of n X 9 0..6 arrays with row sums 9 and column sums n
%H A172901 R. H. Hardin, <a href="b172901.txt">Table of n, a(n) for n=1..24</a>
%K A172901 nonn,new
%O A172901 1,2
%A A172901 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172900
%S A172900 1,8953,176703337,5158851352489,190172119451839801,
%T A172900 8184214105554904614241,389031356288581849014042193,
%U A172900 19987118442361472471477122541737,1092146417007065643218131873660038409
%N A172900 Number of n X 10 0..6 arrays with row sums 10 and column sums n
%H A172900 R. H. Hardin, <a href="b172900.txt">Table of n, a(n) for n=1..18</a>
%K A172900 nonn,new
%O A172900 1,2
%A A172900 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172899
%S A172899 0,6,606,371272,966989100,8540999195040,215298243454580553,
%T A172899 13576914921321831895872,1932115872363940267546247712,
%U A172899 571313477490432486300617373331200
%N A172899 Number of n X n 0..6 arrays with row sums 7 and column sums 7
%H A172899 R. H. Hardin, <a href="b172899.txt">Table of n, a(n) for n=1..20</a>
%K A172899 nonn,new
%O A172899 1,2
%A A172899 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172898
%S A172898 0,0,231,3416992,411376189470,292963407932973960,
%T A172898 1019341412751580730961408,14800896794108577720470005943168,
%U A172898 787822511525554259096702212708356345684
%N A172898 Number of n X n 0..6 arrays with row sums 13 and column sums 13
%H A172898 R. H. Hardin, <a href="b172898.txt">Table of n, a(n) for n=1..11</a>
%K A172898 nonn,new
%O A172898 1,3
%A A172898 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172897
%S A172897 0,0,120,2571486,549507732155,741887829992249010,
%T A172897 5238384476572540280509985,165111855382438587451340513833160,
%U A172897 20360140686173139397061732442741625077240
%N A172897 Number of n X n 0..6 arrays with row sums 14 and column sums 14
%H A172897 R. H. Hardin, <a href="b172897.txt">Table of n, a(n) for n=1..10</a>
%K A172897 nonn,new
%O A172897 1,3
%A A172897 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172896
%S A172896 0,0,55,1624984,604695015651,1519010563560867720,
%T A172896 21289875032822659308693914,1419681395223483661783817488296832,
%U A172896 393979033195135438178790605724141071905152
%N A172896 Number of n X n 0..6 arrays with row sums 15 and column sums 15
%H A172896 R. H. Hardin, <a href="b172896.txt">Table of n, a(n) for n=1..10</a>
%K A172896 nonn,new
%O A172896 1,3
%A A172896 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172895
%S A172895 0,3,756,2571486,51386071001,4735169155116045,1673127629798953193698,
%T A172895 1961572542256259206207069360,6782830842618515149125681541406256,
%U A172895 62721314697296961151225820176430830979953
%N A172895 Number of n X n 0..6 arrays with row sums 10 and column sums 10
%H A172895 R. H. Hardin, <a href="b172895.txt">Table of n, a(n) for n=1..15</a>
%K A172895 nonn,new
%O A172895 1,2
%A A172895 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172894
%S A172894 1,7,406,132724,164176640,602351808741,5562418293759978,
%T A172894 114601242382721619224,4801114002903931560293544,
%U A172894 379957050243738294456427057200,53499490664883668314419655562118240
%N A172894 Number of n X n 0..6 arrays with row sums 6 and column sums 6
%H A172894 R. H. Hardin, <a href="b172894.txt">Table of n, a(n) for n=1..25</a>
%K A172894 nonn,new
%O A172894 1,2
%A A172894 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172893
%S A172893 0,4,811,1624984,17046289085,747054974484580,114730784954807798280,
%T A172893 53679014250694594378460480,68263787608482538080643047338467,
%U A172893 214735064836225772744899504151062116600
%N A172893 Number of n X n 0..6 arrays with row sums 9 and column sums 9
%H A172893 R. H. Hardin, <a href="b172893.txt">Table of n, a(n) for n=1..15</a>
%K A172893 nonn,new
%O A172893 1,2
%A A172893 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172892
%S A172892 0,2,606,3416992,125879845460,23548106318659596,18419713779316213098798,
%T A172892 51819721687533499412779736320,464092875294121232699449367220361200,
%U A172892 11955340654858543651865928739816679891692800
%N A172892 Number of n X n 0..6 arrays with row sums 11 and column sums 11
%H A172892 R. H. Hardin, <a href="b172892.txt">Table of n, a(n) for n=1..12</a>
%K A172892 nonn,new
%O A172892 1,2
%A A172892 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172891
%S A172891 0,1,406,3789991,252171345610,92945759551815326,155517962859038458507570,
%T A172891 1010575567197244536303853333716,22475445868427318276367515097088853904,
%U A172891 1540307034615560553585359584559295771487536000
%N A172891 Number of n X n 0..6 arrays with row sums 12 and column sums 12
%H A172891 R. H. Hardin, <a href="b172891.txt">Table of n, a(n) for n=1..12</a>
%K A172891 nonn,new
%O A172891 1,3
%A A172891 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172890
%S A172890 0,5,756,855877,4545739180,91288388141100,5834634337385578935,
%T A172890 1035075803117199670120995,456649509237912722080809820584,
%U A172890 458136022051795651814655857322653175
%N A172890 Number of n X n 0..6 arrays with row sums 8 and column sums 8
%H A172890 R. H. Hardin, <a href="b172890.txt">Table of n, a(n) for n=1..17</a>
%K A172890 nonn,new
%O A172890 1,2
%A A172890 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172889
%S A172889 30767936616,1522567006061525982469296,
%T A172889 145638518631283768165426487679968270016,
%U A172889 18430464510875205633785528020178855385292893289980336
%N A172889 Number of 6*n X 6 0..5 arrays with row sums 5 and column sums 5*n
%H A172889 R. H. Hardin, <a href="b172889.txt">Table of n, a(n) for n=1..11</a>
%K A172889 nonn,new
%O A172889 1,1
%A A172889 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172888
%S A172888 122975608,114246189025868653880,251497324742297754394272933901120,
%T A172888 808114910894339803643339208801511167291450040,
%U A172888 3211690737914923068885811116444743739077811551329906417728
%N A172888 Number of 4*n X 8 0..5 arrays with row sums 6 and column sums 3*n
%H A172888 R. H. Hardin, <a href="b172888.txt">Table of n, a(n) for n=1..11</a>
%K A172888 nonn,new
%O A172888 1,1
%A A172888 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172887
%S A172887 1,135954,49244182264431,5312943290737316385142992,
%T A172887 49886018232605740875666730594683213081,
%U A172887 18430464510875205633785528020178855385292893289980336
%N A172887 Number of 4*n X n 0..5 arrays with row sums 5 and column sums 20
%H A172887 R. H. Hardin, <a href="b172887.txt">Table of n, a(n) for n=1..11</a>
%K A172887 nonn,new
%O A172887 1,2
%A A172887 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172886
%S A172886 352128,811889208811800,4791845694620286764862960,
%T A172886 42869833252266149325697583980525320,
%U A172886 486080087657753670049657485797288505931441728
%N A172886 Number of 3*n X 9 0..5 arrays with row sums 6 and column sums 2*n
%H A172886 R. H. Hardin, <a href="b172886.txt">Table of n, a(n) for n=1..13</a>
%K A172886 nonn,new
%O A172886 1,1
%A A172886 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172885
%S A172885 756756,19292117692187340,2275745987915517509338826880,
%T A172885 540777843742195976921274441959885997900,
%U A172885 192083384534505302178334348591748127157718391470256
%N A172885 Number of 3*n X 15 0..5 arrays with row sums 5 and column sums n
%H A172885 R. H. Hardin, <a href="b172885.txt">Table of n, a(n) for n=1..13</a>
%K A172885 nonn,new
%O A172885 1,1
%A A172885 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172884
%S A172884 1,4332,7020205512,822065583733092864,2756256275031328927730303391,
%T A172884 145638518631283768165426487679968270016,
%U A172884 79865167852275876121290622323639890979486114456768
%N A172884 Number of 3*n X n 0..5 arrays with row sums 5 and column sums 15
%H A172884 R. H. Hardin, <a href="b172884.txt">Table of n, a(n) for n=1..13</a>
%K A172884 nonn,new
%O A172884 1,2
%A A172884 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172883
%S A172883 0,580,61236085104,889348003628262655840,
%T A172883 747407928835934560435386462917600,
%U A172883 20140279098377603754458043886598358319686833280
%N A172883 Number of 3*n X n 0..5 arrays with row sums 7 and column sums 21
%H A172883 R. H. Hardin, <a href="b172883.txt">Table of n, a(n) for n=1..10</a>
%K A172883 nonn,new
%O A172883 1,2
%A A172883 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172882
%S A172882 0,1751,31175004775,45086287276225676316,2621840984501107521339241754700,
%T A172882 3490160725561441439459874178515370729198161,
%U A172882 69170157963114769801393728393183054793115323713429835710
%N A172882 Number of 3*n X n 0..5 arrays with row sums 6 and column sums 18
%H A172882 R. H. Hardin, <a href="b172882.txt">Table of n, a(n) for n=1..11</a>
%K A172882 nonn,new
%O A172882 1,2
%A A172882 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172881
%S A172881 3432,492872166072,276686670102097569600,292602598138566972319466403000,
%T A172881 446297806870359768599238974918072797632,
%U A172881 862926669194086047399157525873929691696067254464
%N A172881 Number of 2*n X 14 0..5 arrays with row sums 7 and column sums n
%H A172881 R. H. Hardin, <a href="b172881.txt">Table of n, a(n) for n=1..12</a>
%K A172881 nonn,new
%O A172881 1,1
%A A172881 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172880
%S A172880 924,6152037276,127676780106968400,4509741556140302851191900,
%T A172880 216845410132724601463488385879824,
%U A172880 12745340913298249545448510285014313578576
%N A172880 Number of 2*n X 12 0..5 arrays with row sums 6 and column sums n
%H A172880 R. H. Hardin, <a href="b172880.txt">Table of n, a(n) for n=1..17</a>
%K A172880 nonn,new
%O A172880 1,1
%A A172880 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172879
%S A172879 252,79796556,62810128195920,76034933358200884620,
%T A172879 118285830126660123474844752,216845410132724601463488385879824,
%U A172879 446297806870359768599238974918072797632
%N A172879 Number of 2*n X 10 0..5 arrays with row sums 5 and column sums n
%H A172879 R. H. Hardin, <a href="b172879.txt">Table of n, a(n) for n=1..22</a>
%K A172879 nonn,new
%O A172879 1,1
%A A172879 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172878
%S A172878 0,44,4573530,13632458329840,619930160682301188900,
%T A172878 288573946007975323336605745440,1016267487290196803817219026865112113177,
%U A172878 21370280923598082571921850540899933826060972530560
%N A172878 Number of 2*n X n 0..5 arrays with row sums 7 and column sums 14
%H A172878 R. H. Hardin, <a href="b172878.txt">Table of n, a(n) for n=1..13</a>
%K A172878 nonn,new
%O A172878 1,2
%A A172878 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172877
%S A172877 1,146,1114623,149961161616,190172119451839801,1522567006061525982469296,
%T A172877 58163545802347204143878939431446,
%U A172877 8621638450661720405052664706050051958784
%N A172877 Number of 2*n X n 0..5 arrays with row sums 5 and column sums 10
%H A172877 R. H. Hardin, <a href="b172877.txt">Table of n, a(n) for n=1..18</a>
%K A172877 nonn,new
%O A172877 1,2
%A A172877 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172876
%S A172876 0,6,2931721,121455149938696,126089364994137051749175,
%T A172876 2069759825203620045218768697951200,
%U A172876 381163169870068345716673283360625263636254050
%N A172876 Number of 2*n X n 0..5 arrays with row sums 9 and column sums 18
%H A172876 R. H. Hardin, <a href="b172876.txt">Table of n, a(n) for n=1..10</a>
%K A172876 nonn,new
%O A172876 1,2
%A A172876 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172875
%S A172875 0,19,4573530,53051137724677,12135602542619786290200,
%T A172875 35453583475103188963340339100930,
%U A172875 958392436155577046885078330426780564873175
%N A172875 Number of 2*n X n 0..5 arrays with row sums 8 and column sums 16
%H A172875 R. H. Hardin, <a href="b172875.txt">Table of n, a(n) for n=1..11</a>
%K A172875 nonn,new
%O A172875 1,2
%A A172875 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172874
%S A172874 0,85,2931721,1972609390276,15850794059115529500,
%T A172874 1038651765223403011577901181,415055613581699421523189700844639043,
%U A172874 808114910894339803643339208801511167291450040
%N A172874 Number of 2*n X n 0..5 arrays with row sums 6 and column sums 12
%H A172874 R. H. Hardin, <a href="b172874.txt">Table of n, a(n) for n=1..15</a>
%K A172874 nonn,new
%O A172874 1,2
%A A172874 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172873
%S A172873 1,3,55,10147,22069251,596500235301,206239836523724193,
%T A172873 936518400276375286684035,57040154877904164444585301792867,
%U A172873 47374533605081761722921471083862749119153
%N A172873 Number of n X n 0..5 arrays with row sums n and column sums n
%H A172873 R. H. Hardin, <a href="b172873.txt">Table of n, a(n) for n=1..12</a>
%K A172873 nonn,new
%O A172873 1,2
%A A172873 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172872
%S A172872 1,393,248137,200148649,187403959401,189988969424241,206239836523724193,
%T A172872 236069185705118888697,281689709632342945209529,
%U A172872 347532030376658655480001393,440641949482293338531582699793
%N A172872 Number of n X 7 0..5 arrays with row sums 7 and column sums n
%H A172872 R. H. Hardin, <a href="b172872.txt">Table of n, a(n) for n=1..45</a>
%K A172872 nonn,new
%O A172872 1,2
%A A172872 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172871
%S A172871 1,3139,19546975,171475190227,1846780916614531,21822522304624196821,
%T A172871 281689709632342945209529,3899377047452140922854977715,
%U A172871 57040154877904164444585301792867,872202510697313574048780502647781609
%N A172871 Number of n X 9 0..5 arrays with row sums 9 and column sums n
%H A172871 R. H. Hardin, <a href="b172871.txt">Table of n, a(n) for n=1..24</a>
%K A172871 nonn,new
%O A172871 1,2
%A A172871 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172870
%S A172870 1,73789,14823083089,4855940642591941,2124709987334672961301,
%T A172870 1038651765223403011577901181,571578325753594187243824601539009,
%U A172870 346070048320872370739353554842532631189
%N A172870 Number of n X 12 0..5 arrays with row sums 12 and column sums n
%H A172870 R. H. Hardin, <a href="b172870.txt">Table of n, a(n) for n=1..14</a>
%K A172870 nonn,new
%O A172870 1,2
%A A172870 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172869
%S A172869 1,25653,1612210777,157366288875709,19947450199250403501,
%T A172869 2808322218896473982264541,440641949482293338531582699793,
%U A172869 75420658619178832623793205612642877
%N A172869 Number of n X 11 0..5 arrays with row sums 11 and column sums n
%H A172869 R. H. Hardin, <a href="b172869.txt">Table of n, a(n) for n=1..16</a>
%K A172869 nonn,new
%O A172869 1,2
%A A172869 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172868
%S A172868 1,51,3391,261331,22069251,1985311701,187403959401,18343212299091,
%T A172868 1846780916614531,190172119451839801,19947450199250403501,
%U A172868 2124709987334672961301,229273855680211513815541
%N A172868 Number of n X 5 0..5 arrays with row sums 5 and column sums n
%H A172868 R. H. Hardin, <a href="b172868.txt">Table of n, a(n) for n=1..95</a>
%K A172868 nonn,new
%O A172868 1,2
%A A172868 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172867
%S A172867 1,8953,176703337,5158851352489,190172119451839801,
%T A172867 7739410228312751312641,347532030376658655480001393,
%U A172867 16878669938423495235752712550537,872202510697313574048780502647781609
%N A172867 Number of n X 10 0..5 arrays with row sums 10 and column sums n
%H A172867 R. H. Hardin, <a href="b172867.txt">Table of n, a(n) for n=1..18</a>
%K A172867 nonn,new
%O A172867 1,2
%A A172867 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172866
%S A172866 1,1107,2186815,5796870115,18343212299091,63276181484771781,
%T A172866 236069185705118888697,936518400276375286684035,
%U A172866 3899377047452140922854977715,16878669938423495235752712550537
%N A172866 Number of n X 8 0..5 arrays with row sums 8 and column sums n
%H A172866 R. H. Hardin, <a href="b172866.txt">Table of n, a(n) for n=1..35</a>
%K A172866 nonn,new
%O A172866 1,2
%A A172866 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172865
%S A172865 1,141,28681,7100821,1985311701,596500235301,189988969424241,
%T A172865 63276181484771781,21822522304624196821,7739410228312751312641,
%U A172865 2808322218896473982264541,1038651765223403011577901181
%N A172865 Number of n X 6 0..5 arrays with row sums 6 and column sums n
%H A172865 R. H. Hardin, <a href="b172865.txt">Table of n, a(n) for n=1..61</a>
%K A172865 nonn,new
%O A172865 1,2
%A A172865 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172864
%S A172864 0,0,6,126660,39205553940,72443102322181770,669339995480640046936115,
%T A172864 26684517894223197491871690098160,
%U A172864 4047051071271585783777915818898222938640
%N A172864 Number of n X n 0..5 arrays with row sums 14 and column sums 14
%H A172864 R. H. Hardin, <a href="b172864.txt">Table of n, a(n) for n=1..10</a>
%K A172864 nonn,new
%O A172864 1,3
%A A172864 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172863
%S A172863 0,0,1,40176,22989265901,83308687622580480,1590632917730889408984804,
%T A172863 138992936515606925094500549847168,
%U A172863 49028617232539794829106747972829351457296
%N A172863 Number of n X n 0..5 arrays with row sums 15 and column sums 15
%H A172863 R. H. Hardin, <a href="b172863.txt">Table of n, a(n) for n=1..10</a>
%K A172863 nonn,new
%O A172863 1,4
%A A172863 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172862
%S A172862 1,6,231,40176,22069251,30767936616,94161778046406,569304690994400256,
%T A172862 6274236760589024662176,118285830126660123474844752,
%U A172862 3623440212198461411381072575512,172850452498398420310370097345242112
%N A172862 Number of n X n 0..5 arrays with row sums 5 and column sums 5
%H A172862 R. H. Hardin, <a href="b172862.txt">Table of n, a(n) for n=1..35</a>
%K A172862 nonn,new
%O A172862 1,2
%A A172862 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172861
%S A172861 0,1,231,965838,22989265901,2461655754351495,985771420457603052511,
%T A172861 1279614273812450904341423760,4801060151437579142777056341136944,
%U A172861 47374533605081761722921471083862749119153
%N A172861 Number of n X n 0..5 arrays with row sums 10 and column sums 10
%H A172861 R. H. Hardin, <a href="b172861.txt">Table of n, a(n) for n=1..15</a>
%K A172861 nonn,new
%O A172861 1,3
%A A172861 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172860
%S A172860 0,2,355,838792,10177326885,502857653609080,84763676021354613990,
%T A172860 42550449824987801797296000,57040154877904164444585301792867,
%U A172860 186651541822247767681950912578484269400
%N A172860 Number of n X n 0..5 arrays with row sums 9 and column sums 9
%H A172860 R. H. Hardin, <a href="b172860.txt">Table of n, a(n) for n=1..15</a>
%K A172860 nonn,new
%O A172860 1,2
%A A172860 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172859
%S A172859 0,3,432,571189,3422583180,74885351406570,5074214421358248075,
%T A172859 936518400276375286684035,424497177203869984956631226568,
%U A172859 433946937800667955296084195110940225
%N A172859 Number of n X n 0..5 arrays with row sums 8 and column sums 8
%H A172859 R. H. Hardin, <a href="b172859.txt">Table of n, a(n) for n=1..17</a>
%K A172859 nonn,new
%O A172859 1,2
%A A172859 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172858
%S A172858 0,4,432,305872,863754620,7977812965920,206239836523724193,
%T A172858 13199605245131506373760,1895476867367724956193779520,
%U A172858 563701189834142639263745251516800
%N A172858 Number of n X n 0..5 arrays with row sums 7 and column sums 7
%H A172858 R. H. Hardin, <a href="b172858.txt">Table of n, a(n) for n=1..20</a>
%K A172858 nonn,new
%O A172858 1,2
%A A172858 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172857
%S A172857 0,0,120,838792,39205553940,8867410986416760,8131811998479859238220,
%T A172857 26150377893243567040474379136,261894259231474231217142097747097616,
%U A172857 7402668628365416446775191706258616053932800
%N A172857 Number of n X n 0..5 arrays with row sums 11 and column sums 11
%H A172857 R. H. Hardin, <a href="b172857.txt">Table of n, a(n) for n=1..12</a>
%K A172857 nonn,new
%O A172857 1,3
%A A172857 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172856
%S A172856 0,0,55,571189,50984927470,23739529252540151,48305357301322263405109,
%T A172856 370911274700550803678566199652,9515813289536524717883338564939648176,
%U A172856 736766457419317377244512506283142581954316500
%N A172856 Number of n X n 0..5 arrays with row sums 12 and column sums 12
%H A172856 R. H. Hardin, <a href="b172856.txt">Table of n, a(n) for n=1..12</a>
%K A172856 nonn,new
%O A172856 1,3
%A A172856 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172855
%S A172855 0,5,355,126660,160936020,596500235301,5533046895002949,
%T A172855 114246189025868653880,4791845694620286764862960,
%U A172855 379477402498522892976988816800,53453544889838303139764031973905360
%N A172855 Number of n X n 0..5 arrays with row sums 6 and column sums 6
%H A172855 R. H. Hardin, <a href="b172855.txt">Table of n, a(n) for n=1..27</a>
%K A172855 nonn,new
%O A172855 1,2
%A A172855 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172854
%S A172854 0,0,21,305872,50984927470,47703798786066840,209367461452656000222414,
%T A172854 3715952487541217998469935646848,235540792783974946951038854447958540420,
%U A172854 48023201408596246269169102263301826304599932800
%N A172854 Number of n X n 0..5 arrays with row sums 13 and column sums 13
%H A172854 R. H. Hardin, <a href="b172854.txt">Table of n, a(n) for n=1..11</a>
%K A172854 nonn,new
%O A172854 1,3
%A A172854 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172853
%S A172853 936670590450,2344764719918663269508531250,
%T A172853 12954552338378953331770652180864486570952000,
%U A172853 100143576421221755324646806706234908029792932524731851041250
%N A172853 Number of 7*n X 7 0..4 arrays with row sums 4 and column sums 4*n
%H A172853 R. H. Hardin, <a href="b172853.txt">Table of n, a(n) for n=1..10</a>
%K A172853 nonn,new
%O A172853 1,1
%A A172853 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172852
%S A172852 29990997180,1295886527534643220837980,
%T A172852 107592348627283647736871768193003754560,
%U A172852 11800125905631135878642235825767415740742008020764060
%N A172852 Number of 6*n X 6 0..4 arrays with row sums 5 and column sums 5*n
%H A172852 R. H. Hardin, <a href="b172852.txt">Table of n, a(n) for n=1..11</a>
%K A172852 nonn,new
%O A172852 1,1
%A A172852 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172851
%S A172851 2224955,1020771217185075,784651415725030722439025,
%T A172851 752392911462866118274927841209875,
%U A172851 816183539809026724414261163240882996876580
%N A172851 Number of 5*n X 5 0..4 arrays with row sums 4 and column sums 4*n
%H A172851 R. H. Hardin, <a href="b172851.txt">Table of n, a(n) for n=1..19</a>
%K A172851 nonn,new
%O A172851 1,1
%A A172851 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172850
%S A172850 261331,76034933358200884620,540777843742195976921274441959885997900,
%T A172850 3037037527560979348033392392364047201967869489244978518088000,
%U A172850 2513219607356377151857448686432978558773726758522740963015057806546618777717045360000
%N A172850 Number of 5*n X 4*n 0..4 arrays with row sums 4 and column sums 5
%H A172850 R. H. Hardin, <a href="b172850.txt">Table of n, a(n) for n=1..10</a>
%K A172850 nonn,new
%O A172850 1,1
%A A172850 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172849
%S A172849 1328792850,14356628851597700179050,472456616741846499933573760930072500,
%T A172849 25272101855153630387493389888938604484045485426250,
%U A172849 1775919804991679659362869123694921059736265812420785710667676600
%N A172849 Number of 5*n X 10 0..4 arrays with row sums 4 and column sums 2*n
%H A172849 R. H. Hardin, <a href="b172849.txt">Table of n, a(n) for n=1..11</a>
%K A172849 nonn,new
%O A172849 1,1
%A A172849 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172848
%S A172848 381,5158851352489,953508835712685968036205885,
%T A172848 112733074361221363983325781513913693180776500,
%U A172848 1775919804991679659362869123694921059736265812420785710667676600
%N A172848 Number of 5*n X 2*n 0..4 arrays with row sums 4 and column sums 10
%H A172848 R. H. Hardin, <a href="b172848.txt">Table of n, a(n) for n=1..10</a>
%K A172848 nonn,new
%O A172848 1,1
%A A172848 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172847
%S A172847 305540235000,54918587341306311174536985000,
%T A172847 85504723470329847456379126545730424791071000000,
%U A172847 351499515037067154764556950151761640777013401021265237589794665000
%N A172847 Number of 5*n X 20 0..4 arrays with row sums 4 and column sums n
%H A172847 R. H. Hardin, <a href="b172847.txt">Table of n, a(n) for n=1..10</a>
%K A172847 nonn,new
%O A172847 1,1
%A A172847 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172846
%S A172846 1,856945,3366607202713755,5184059089729557142445800537,
%T A172846 816183539809026724414261163240882996876580,
%U A172846 5606794155185465322404088275221975704760242518879730852405
%N A172846 Number of 5*n X n 0..4 arrays with row sums 4 and column sums 20
%H A172846 R. H. Hardin, <a href="b172846.txt">Table of n, a(n) for n=1..10</a>
%K A172846 nonn,new
%O A172846 1,2
%A A172846 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172845
%S A172845 122975608,105468145309870490840,204422478568944191589314127357280,
%T A172845 576542290820108436100865963035873493286055960,
%U A172845 2008712019540133157801970175616166646400717506140422115648
%N A172845 Number of 4*n X 8 0..4 arrays with row sums 6 and column sums 3*n
%H A172845 R. H. Hardin, <a href="b172845.txt">Table of n, a(n) for n=1..11</a>
%K A172845 nonn,new
%O A172845 1,1
%A A172845 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172844
%S A172844 63063000,163081394186253543000,2253836332035499736062301563800000,
%T A172844 66583272259816204746406627604055923436126087000,
%U A172844 3037037527560979348033392392364047201967869489244978518088000
%N A172844 Number of 4*n X 16 0..4 arrays with row sums 4 and column sums n
%H A172844 R. H. Hardin, <a href="b172844.txt">Table of n, a(n) for n=1..12</a>
%K A172844 nonn,new
%O A172844 1,1
%A A172844 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172843
%S A172843 0,8092,10307646260703,2017878173399604368853000,
%T A172843 26274389782658999258821623642836586801,
%U A172843 11800125905631135878642235825767415740742008020764060
%N A172843 Number of 4*n X n 0..4 arrays with row sums 5 and column sums 20
%H A172843 R. H. Hardin, <a href="b172843.txt">Table of n, a(n) for n=1..11</a>
%K A172843 nonn,new
%O A172843 1,2
%A A172843 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172842
%S A172842 1,38165,1239277723200,4792275283921293214435,
%T A172842 752392911462866118274927841209875,
%U A172842 2424849638741414909681878324689804399805160625
%N A172842 Number of 4*n X n 0..4 arrays with row sums 4 and column sums 16
%H A172842 R. H. Hardin, <a href="b172842.txt">Table of n, a(n) for n=1..12</a>
%K A172842 nonn,new
%O A172842 1,2
%A A172842 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172841
%S A172841 2385,1047649905,811889208811800,816961921232733507825,
%T A172841 953508835712685968036205885,1224549137633779037361672006525000,
%U A172841 1681239341019391732613433550580491453200
%N A172841 Number of 3*n X 6 0..4 arrays with row sums 4 and column sums 2*n
%H A172841 R. H. Hardin, <a href="b172841.txt">Table of n, a(n) for n=1..28</a>
%K A172841 nonn,new
%O A172841 1,1
%A A172841 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172840
%S A172840 352128,811889208811800,4512934309457492170923216,
%T A172840 37736268336701361777964258280263320,
%U A172840 398985427051117716339042202464595044584127768
%N A172840 Number of 3*n X 9 0..4 arrays with row sums 6 and column sums 2*n
%H A172840 R. H. Hardin, <a href="b172840.txt">Table of n, a(n) for n=1..13</a>
%K A172840 nonn,new
%O A172840 1,1
%A A172840 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172839
%S A172839 19,7100821,811889208811800,4509741556140302851191900,
%T A172839 472456616741846499933573760930072500,
%U A172839 525065279695190719776277892214398176478661270000
%N A172839 Number of 3*n X 2*n 0..4 arrays with row sums 4 and column sums 6
%H A172839 R. H. Hardin, <a href="b172839.txt">Table of n, a(n) for n=1..16</a>
%K A172839 nonn,new
%O A172839 1,1
%A A172839 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172838
%S A172838 34650,3536978063850,1182802402030354182000,
%T A172838 683327637694741065563262206250,540777843742195976921274441959885997900,
%U A172838 525065279695190719776277892214398176478661270000
%N A172838 Number of 3*n X 12 0..4 arrays with row sums 4 and column sums n
%H A172838 R. H. Hardin, <a href="b172838.txt">Table of n, a(n) for n=1..17</a>
%K A172838 nonn,new
%O A172838 1,1
%A A172838 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172837
%S A172837 756756,19292117692187340,2275745987915517509338826880,
%T A172837 540777843742195976921274441959885997900,
%U A172837 191792366099321483346227837613264957393284852811696
%N A172837 Number of 3*n X 15 0..4 arrays with row sums 5 and column sums n
%H A172837 R. H. Hardin, <a href="b172837.txt">Table of n, a(n) for n=1..13</a>
%K A172837 nonn,new
%O A172837 1,1
%A A172837 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172836
%S A172836 0,141,3757921291,8871900691867826316,753400574499912868356844716700,
%T A172836 1329096037020824944632051261867911008990401,
%U A172836 32446537190781931300574314797091618240587813777700345470
%N A172836 Number of 3*n X n 0..4 arrays with row sums 6 and column sums 18
%H A172836 R. H. Hardin, <a href="b172836.txt">Table of n, a(n) for n=1..11</a>
%K A172836 nonn,new
%O A172836 1,2
%A A172836 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172835
%S A172835 0,580,2332105236,419542137630783552,1775125806066054844658927631,
%T A172835 107592348627283647736871768193003754560,
%U A172835 64299439861739255917710165174342650682005907347040
%N A172835 Number of 3*n X n 0..4 arrays with row sums 5 and column sums 15
%H A172835 R. H. Hardin, <a href="b172835.txt">Table of n, a(n) for n=1..13</a>
%K A172835 nonn,new
%O A172835 1,2
%A A172835 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172834
%S A172834 0,20,2332105236,55821470965449838000,73884729561168010198433552532700,
%T A172834 2932027248013823224822937070664663188676388640,
%U A172834 2229074240908452234163017626536657006251051388006192816445889
%N A172834 Number of 3*n X n 0..4 arrays with row sums 7 and column sums 21
%H A172834 R. H. Hardin, <a href="b172834.txt">Table of n, a(n) for n=1..10</a>
%K A172834 nonn,new
%O A172834 1,2
%A A172834 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172833
%S A172833 1,1751,484582350,4855940642591941,784651415725030722439025,
%T A172833 1224549137633779037361672006525000,
%U A172833 12954552338378953331770652180864486570952000
%N A172833 Number of 3*n X n 0..4 arrays with row sums 4 and column sums 12
%H A172833 R. H. Hardin, <a href="b172833.txt">Table of n, a(n) for n=1..17</a>
%K A172833 nonn,new
%O A172833 1,2
%A A172833 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172832
%S A172832 3432,492872166072,276686670102097569600,292602598138566972319466403000,
%T A172832 437263276341891154808563746635206959552,
%U A172832 823612105010876272181767605679102344847556205504
%N A172832 Number of 2*n X 14 0..4 arrays with row sums 7 and column sums n
%H A172832 R. H. Hardin, <a href="b172832.txt">Table of n, a(n) for n=1..12</a>
%K A172832 nonn,new
%O A172832 1,1
%A A172832 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172831
%S A172831 252,79796556,62810128195920,76034933358200884620,
%T A172831 117660704031419990309224272,214168486397890522888429407832464,
%U A172831 437263276341891154808563746635206959552
%N A172831 Number of 2*n X 10 0..4 arrays with row sums 5 and column sums n
%H A172831 R. H. Hardin, <a href="b172831.txt">Table of n, a(n) for n=1..22</a>
%K A172831 nonn,new
%O A172831 1,1
%A A172831 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172830
%S A172830 924,6152037276,127676780106968400,4509741556140302851191900,
%T A172830 214168486397890522888429407832464,
%U A172830 12385351313597469698382588100678714628496
%N A172830 Number of 2*n X 12 0..4 arrays with row sums 6 and column sums n
%H A172830 R. H. Hardin, <a href="b172830.txt">Table of n, a(n) for n=1..16</a>
%K A172830 nonn,new
%O A172830 1,1
%A A172830 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172829
%S A172829 70,1093050,33833709700,1455918295922650,76034933358200884620,
%T A172829 4509741556140302851191900,292602598138566972319466403000,
%U A172829 20294029043417487085670761524956250
%N A172829 Number of 2*n X 8 0..4 arrays with row sums 4 and column sums n
%H A172829 R. H. Hardin, <a href="b172829.txt">Table of n, a(n) for n=1..31</a>
%K A172829 nonn,new
%O A172829 1,1
%A A172829 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172828
%S A172828 0,44,587295,102738568968,149563116897254161,1295886527534643220837980,
%T A172828 51948810321889380622545174356004,
%U A172828 7942329340789530344493069550687020249600
%N A172828 Number of 2*n X n 0..4 arrays with row sums 5 and column sums 10
%H A172828 R. H. Hardin, <a href="b172828.txt">Table of n, a(n) for n=1..18</a>
%K A172828 nonn,new
%O A172828 1,2
%A A172828 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172827
%S A172827 0,0,28681,2449665403360,4267335124853033521800,
%T A172827 106107338700855492388229350341800,
%U A172827 27866376170914402863794530315047399586447050
%N A172827 Number of 2*n X n 0..4 arrays with row sums 9 and column sums 18
%H A172827 R. H. Hardin, <a href="b172827.txt">Table of n, a(n) for n=1..10</a>
%K A172827 nonn,new
%O A172827 1,3
%A A172827 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172826
%S A172826 1,85,210960,5796870115,1020771217185075,816961921232733507825,
%T A172826 2344764719918663269508531250,20294029043417487085670761524956250,
%U A172826 464104041410086779771666773956560859328250
%N A172826 Number of 2*n X n 0..4 arrays with row sums 4 and column sums 8
%H A172826 R. H. Hardin, <a href="b172826.txt">Table of n, a(n) for n=1..24</a>
%K A172826 nonn,new
%O A172826 1,2
%A A172826 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172825
%S A172825 0,1,210960,3763475372581,1240457049015641715900,
%T A172825 4983454435498876364888734279560,
%U A172825 177385566361246999857025320669145541704035
%N A172825 Number of 2*n X n 0..4 arrays with row sums 8 and column sums 16
%H A172825 R. H. Hardin, <a href="b172825.txt">Table of n, a(n) for n=1..11</a>
%K A172825 nonn,new
%O A172825 1,3
%A A172825 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172824
%S A172824 0,19,805231,744168838420,7608443835495995100,
%T A172824 594239211348093910990313341,269741028364409749133570268559118335,
%U A172824 576542290820108436100865963035873493286055960
%N A172824 Number of 2*n X n 0..4 arrays with row sums 6 and column sums 12
%H A172824 R. H. Hardin, <a href="b172824.txt">Table of n, a(n) for n=1..15</a>
%K A172824 nonn,new
%O A172824 1,2
%A A172824 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172823
%S A172823 0,6,587295,2449665403360,150313945957168045500,
%T A172823 89971474429693406536937414640,388914772864932993317961153232659601617,
%U A172823 9645840692950033151436461750208136925819574909120
%N A172823 Number of 2*n X n 0..4 arrays with row sums 7 and column sums 14
%H A172823 R. H. Hardin, <a href="b172823.txt">Table of n, a(n) for n=1..12</a>
%K A172823 nonn,new
%O A172823 1,2
%A A172823 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172822
%S A172822 1,3,55,10147,21107931,511508441781,150603715387867113,
%T A172822 555249505892003620997475,26159597034338175244538552103907,
%U A172822 16021054958659349444630042183181806132953
%N A172822 Number of n X n 0..4 arrays with row sums n and column sums n
%H A172822 R. H. Hardin, <a href="b172822.txt">Table of n, a(n) for n=1..12</a>
%K A172822 nonn,new
%O A172822 1,2
%A A172822 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172821
%S A172821 1,51,3391,261331,21107931,1825913781,165824491281,15611242525971,
%T A172821 1511084106275011,149563116897254161,15076390343931567861,
%U A172821 1543079518763428325461,159985421854875339469501
%N A172821 Number of n X 5 0..4 arrays with row sums 5 and column sums n
%H A172821 R. H. Hardin, <a href="b172821.txt">Table of n, a(n) for n=1..95</a>
%K A172821 nonn,new
%O A172821 1,2
%A A172821 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172820
%S A172820 1,73789,14823083089,4855940642591941,1543079518763428325461,
%T A172820 594239211348093910990313341,263097755944932890425699642604449,
%U A172820 128871474049189390070331107241079880869
%N A172820 Number of n X 12 0..4 arrays with row sums 12 and column sums n
%H A172820 R. H. Hardin, <a href="b172820.txt">Table of n, a(n) for n=1..13</a>
%K A172820 nonn,new
%O A172820 1,2
%A A172820 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172819
%S A172819 1,3139,19546975,171475190227,1511084106275011,15293664515956438501,
%T A172819 171057032412324570540169,2057906511849020370326921875,
%U A172819 26159597034338175244538552103907,347633505891623648387005776489399049
%N A172819 Number of n X 9 0..4 arrays with row sums 9 and column sums n
%H A172819 R. H. Hardin, <a href="b172819.txt">Table of n, a(n) for n=1..25</a>
%K A172819 nonn,new
%O A172819 1,2
%A A172819 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172818
%S A172818 1,25653,1612210777,157366288875709,15076390343931567861,
%T A172818 1719264773436264439137501,222448511010065224292209227273,
%U A172818 31544072766055727003036345569283997
%N A172818 Number of n X 11 0..4 arrays with row sums 11 and column sums n
%H A172818 R. H. Hardin, <a href="b172818.txt">Table of n, a(n) for n=1..16</a>
%K A172818 nonn,new
%O A172818 1,2
%A A172818 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172817
%S A172817 1,8953,176703337,5158851352489,149563116897254161,
%T A172817 5069664274447290501601,192419094165660969991083673,
%U A172817 7929430337861717806105965593257,347633505891623648387005776489399049
%N A172817 Number of n X 10 0..4 arrays with row sums 10 and column sums n
%H A172817 R. H. Hardin, <a href="b172817.txt">Table of n, a(n) for n=1..19</a>
%K A172817 nonn,new
%O A172817 1,2
%A A172817 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172816
%S A172816 1,19,415,10147,261331,7100821,200148649,5796870115,171475190227,
%T A172816 5158851352489,157366288875709,4855940642591941,151309409779981285,
%U A172816 4754260599949620163,150466404665460290335,4792275283921293214435
%N A172816 Number of n X 4 0..4 arrays with row sums 4 and column sums n
%H A172816 R. H. Hardin, <a href="b172816.txt">Table of n, a(n) for n=1..99</a>
%K A172816 nonn,new
%O A172816 1,2
%A A172816 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172815
%S A172815 1,1107,2186815,5796870115,15611242525971,47436084817844661,
%T A172815 157217801593519529577,555249505892003620997475,
%U A172815 2057906511849020370326921875,7929430337861717806105965593257
%N A172815 Number of n X 8 0..4 arrays with row sums 8 and column sums n
%H A172815 R. H. Hardin, <a href="b172815.txt">Table of n, a(n) for n=1..35</a>
%K A172815 nonn,new
%O A172815 1,2
%A A172815 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172814
%S A172814 1,141,28681,7100821,1825913781,511508441781,152303757391521,
%T A172814 47436084817844661,15293664515956438501,5069664274447290501601,
%U A172814 1719264773436264439137501,594239211348093910990313341
%N A172814 Number of n X 6 0..4 arrays with row sums 6 and column sums n
%H A172814 R. H. Hardin, <a href="b172814.txt">Table of n, a(n) for n=1..61</a>
%K A172814 nonn,new
%O A172814 1,2
%A A172814 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172813
%S A172813 1,393,248137,200148649,165824491281,152303757391521,150603715387867113,
%T A172813 157217801593519529577,171057032412324570540169,
%U A172813 192419094165660969991083673,222448511010065224292209227273
%N A172813 Number of n X 7 0..4 arrays with row sums 7 and column sums n
%H A172813 R. H. Hardin, <a href="b172813.txt">Table of n, a(n) for n=1..45</a>
%K A172813 nonn,new
%O A172813 1,2
%A A172813 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172812
%S A172812 0,0,6,36840,2461011960,753889607031060,897212452713773709765,
%T A172812 3625229476732355845849016640,44424128771680100390106879719766312,
%U A172812 1501841442695075526144022337210256443707200
%N A172812 Number of n X n 0..4 arrays with row sums 11 and column sums 11
%H A172812 R. H. Hardin, <a href="b172812.txt">Table of n, a(n) for n=1..13</a>
%K A172812 nonn,new
%O A172812 1,3
%A A172812 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172811
%S A172811 0,3,217,91140,128843140,511508441781,4960069104562005,
%T A172811 105468145309870490840,4512934309457492170923216,
%U A172811 362415672451851793780617637200,51563823623409518633102950412670240
%N A172811 Number of n X n 0..4 arrays with row sums 6 and column sums 6
%H A172811 R. H. Hardin, <a href="b172811.txt">Table of n, a(n) for n=1..25</a>
%K A172811 nonn,new
%O A172811 1,2
%A A172811 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172810
%S A172810 0,0,1,10147,1380339980,923707339966121,2559427003605480026767,
%T A172810 25779625922082640845623786596,842495245380303684515723877888629640,
%U A172810 81060631348910042232392222017393387380144900
%N A172810 Number of n X n 0..4 arrays with row sums 12 and column sums 12
%H A172810 R. H. Hardin, <a href="b172810.txt">Table of n, a(n) for n=1..12</a>
%K A172810 nonn,new
%O A172810 1,4
%A A172810 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172809
%S A172809 0,4,189,36840,21107931,29990997180,92673325442124,563326170709889664,
%T A172809 6228365853689321408640,117660704031419990309224272,
%U A172809 3609165493134932351223036800520,172329705795296334349960101885305088
%N A172809 Number of n X n 0..4 arrays with row sums 5 and column sums 5
%H A172809 R. H. Hardin, <a href="b172809.txt">Table of n, a(n) for n=1..35</a>
%K A172809 nonn,new
%O A172809 1,2
%A A172809 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172808
%S A172808 0,0,55,157984,2461011960,148699584483880,29715346096043717550,
%T A172808 17246529136870025302912000,26159597034338175244538552103907,
%U A172808 95066344634881957233042525618261985200
%N A172808 Number of n X n 0..4 arrays with row sums 9 and column sums 9
%H A172808 R. H. Hardin, <a href="b172808.txt">Table of n, a(n) for n=1..15</a>
%K A172808 nonn,new
%O A172808 1,3
%A A172808 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172807
%S A172807 0,0,0,2008,514790020,753889607031060,4778708254823045056395,
%T A172807 116674060299842227684197102400,9827935578168069077212412672403655680,
%U A172807 2590181620730810991347647340464728300573016800
%N A172807 Number of n X n 0..4 arrays with row sums 13 and column sums 13
%H A172807 R. H. Hardin, <a href="b172807.txt">Table of n, a(n) for n=1..11</a>
%K A172807 nonn,new
%O A172807 1,4
%A A172807 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172806
%S A172806 1,5,120,10147,2224955,1047649905,936670590450,1455918295922650,
%T A172806 3680232136895819610,14356628851597700179050,82857993930808028192521800,
%U A172806 683327637694741065563262206250,7821620120684573354895941635688250
%N A172806 Number of n X n 0..4 arrays with row sums 4 and column sums 4
%H A172806 R. H. Hardin, <a href="b172806.txt">Table of n, a(n) for n=1..56</a>
%K A172806 nonn,new
%O A172806 1,2
%A A172806 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172805
%S A172805 0,1,120,193573,1380339980,35157552997320,2707357478410889115,
%T A172805 555249505892003620997475,274283912325835718446290655176,
%U A172805 300690927870900038104809199886390475
%N A172805 Number of n X n 0..4 arrays with row sums 8 and column sums 8
%H A172805 R. H. Hardin, <a href="b172805.txt">Table of n, a(n) for n=1..17</a>
%K A172805 nonn,new
%O A172805 1,3
%A A172805 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172804
%S A172804 0,0,0,282,128843140,410969810835855,5879061635576130720965,
%T A172804 340165207656981415224173431200,71757898497428589696054333057077505600,
%U A172804 50113836197854536934306053419046449681977771500
%N A172804 Number of n X n 0..4 arrays with row sums 14 and column sums 14
%K A172804 nonn,new
%O A172804 1,4
%A A172804 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172803
%S A172803 0,0,0,24,21107931,148699584483880,4778708254823045056395,
%T A172803 644431545036931434768794630976,332434916001875361804441620924628744480,
%U A172803 597860889495439071046769406191836215339420282624
%N A172803 Number of n X n 0..4 arrays with row sums 15 and column sums 15
%K A172803 nonn,new
%O A172803 1,4
%A A172803 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172802
%S A172802 0,0,0,1,2224955,35157552997320,2559427003605480026767,
%T A172802 797070772051002306880745221605,987056953959331385275218816790570496400,
%U A172802 4462284367678683525426963348639073144370183155500
%N A172802 Number of n X n 0..4 arrays with row sums 16 and column sums 16
%K A172802 nonn,new
%O A172802 1,5
%A A172802 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172801
%S A172801 0,2,189,157984,514790020,5334492792240,150603715387867113,
%T A172801 10301508972326640532800,1555479697698632517004275840,
%U A172801 480593112323971829410256539190400
%N A172801 Number of n X n 0..4 arrays with row sums 7 and column sums 7
%H A172801 R. H. Hardin, <a href="b172801.txt">Table of n, a(n) for n=1..20</a>
%K A172801 nonn,new
%O A172801 1,2
%A A172801 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172800
%S A172800 0,0,21,91140,2974547521,410969810835855,204186348024605926840,
%T A172800 319377442011213173229914216,1409441402075134083081734769337800,
%U A172800 16021054958659349444630042183181806132953
%N A172800 Number of n X n 0..4 arrays with row sums 10 and column sums 10
%H A172800 R. H. Hardin, <a href="b172800.txt">Table of n, a(n) for n=1..15</a>
%K A172800 nonn,new
%O A172800 1,3
%A A172800 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172799
%S A172799 1579060246400,6882009860546294146142870400,
%T A172799 74896615128945574656737417426177604440960000,
%U A172799 1203363804900576283501636346142713047339604154996148224048000
%N A172799 Number of 8*n X 8 0..3 arrays with row sums 3 and column sums 3*n
%H A172799 R. H. Hardin, <a href="b172799.txt">Table of n, a(n) for n=1..10</a>
%K A172799 nonn,new
%O A172799 1,1
%A A172799 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172798
%S A172798 887226868605,1771064195990555279483745075,
%T A172798 7745310191767932461627415566004172302489000,
%U A172798 47307220244039925184729002585595148532177882871899513973875
%N A172798 Number of 7*n X 7 0..3 arrays with row sums 4 and column sums 4*n
%H A172798 R. H. Hardin, <a href="b172798.txt">Table of n, a(n) for n=1..10</a>
%K A172798 nonn,new
%O A172798 1,1
%A A172798 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172797
%S A172797 4662857360,19483897303054057473600,177485739636656023698217362159890000,
%T A172797 2255071025881975002973276574847623754650721480000,
%U A172797 34489400617513696317244454709620842599380338820310459110031360
%N A172797 Number of 7*n X 7 0..3 arrays with row sums 3 and column sums 3*n
%H A172797 R. H. Hardin, <a href="b172797.txt">Table of n, a(n) for n=1..12</a>
%K A172797 nonn,new
%O A172797 1,1
%A A172797 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172796
%S A172796 248137,276686670102097569600,12577849178653907149465594061673565824000,
%T A172796 633070426668791623743477133016092146118990897169495988019200000,
%U A172796 6041723108140506139185920296627256100375775025140875595591377857737131840561930240000000
%N A172796 Number of 7*n X 3*n 0..3 arrays with row sums 3 and column sums 7
%H A172796 R. H. Hardin, <a href="b172796.txt">Table of n, a(n) for n=1..11</a>
%K A172796 nonn,new
%O A172796 1,1
%A A172796 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172795
%S A172795 1,25288120,8590448906009178865,2089345627982707177699421140008960,
%T A172795 77513216887442246128882408582036570535556173496000,
%U A172795 168135704190434138519823546360671865844450109820045620452225487496000
%N A172795 Number of 7*n X n 0..3 arrays with row sums 3 and column sums 21
%H A172795 R. H. Hardin, <a href="b172795.txt">Table of n, a(n) for n=1..10</a>
%K A172795 nonn,new
%O A172795 1,2
%A A172795 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172794
%S A172794 20592419760,407505677116244252111280,
%T A172794 15496001580602804970376043343562432320,
%U A172794 778626541064162359278560268669084299553822689853360
%N A172794 Number of 6*n X 6 0..3 arrays with row sums 5 and column sums 5*n
%H A172794 R. H. Hardin, <a href="b172794.txt">Table of n, a(n) for n=1..11</a>
%K A172794 nonn,new
%O A172794 1,1
%A A172794 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172793
%S A172793 137225088000,1796185853884657144990080000,
%T A172793 163283938767642016139942363441093164154880000,
%U A172793 35452241260001460008130612784749576025029062217424089840000000
%N A172793 Number of 6*n X 18 0..3 arrays with row sums 3 and column sums n
%H A172793 R. H. Hardin, <a href="b172793.txt">Table of n, a(n) for n=1..11</a>
%K A172793 nonn,new
%O A172793 1,1
%A A172793 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172792
%S A172792 1,1703636,9991003788000025,16391575725348461384454895264,
%T A172792 1999627385492230281454043722065265134189600,
%U A172792 7973459202457374191609067356023716183497438508372391165200
%N A172792 Number of 6*n X n 0..3 arrays with row sums 3 and column sums 18
%H A172792 R. H. Hardin, <a href="b172792.txt">Table of n, a(n) for n=1..11</a>
%K A172792 nonn,new
%O A172792 1,2
%A A172792 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172791
%S A172791 1992760,635481090640800,338059904123009135280850,
%T A172791 224061210725807969378046253140000,
%U A172791 167919620604132417334149114648212316386760
%N A172791 Number of 5*n X 5 0..3 arrays with row sums 4 and column sums 4*n
%H A172791 R. H. Hardin, <a href="b172791.txt">Table of n, a(n) for n=1..19</a>
%K A172791 nonn,new
%O A172791 1,1
%A A172791 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172790
%S A172790 194851,69830279731379332620,520089040184399473996377693211802013900,
%T A172790 2970408787718945344873427655667537554031914545552039003208000,
%U A172790 2477611987308842570549778100041447740458779996145727507089395743767957848477717360000
%N A172790 Number of 5*n X 4*n 0..3 arrays with row sums 4 and column sums 5
%H A172790 R. H. Hardin, <a href="b172790.txt">Table of n, a(n) for n=1..10</a>
%K A172790 nonn,new
%O A172790 1,1
%A A172790 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172789
%S A172789 153040,2217051589200,53461314050652508000,1605078294448526381158693200,
%T A172789 54474595447150914468834015421847040,
%U A172789 1999627385492230281454043722065265134189600
%N A172789 Number of 5*n X 5 0..3 arrays with row sums 3 and column sums 3*n
%H A172789 R. H. Hardin, <a href="b172789.txt">Table of n, a(n) for n=1..19</a>
%K A172789 nonn,new
%O A172789 1,1
%A A172789 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172788
%S A172788 3391,62810128195920,2275745987915517509338826880,
%T A172788 12263459619622095086766632900215307533209600,
%U A172788 2789130810728711760475063805085713493445792640582613331968000
%N A172788 Number of 5*n X 3*n 0..3 arrays with row sums 3 and column sums 5
%H A172788 R. H. Hardin, <a href="b172788.txt">Table of n, a(n) for n=1..17</a>
%K A172788 nonn,new
%O A172788 1,1
%A A172788 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172787
%S A172787 1328792850,13994422270858878448950,432897335001181060850579402825662500,
%T A172787 21667982811499395841985904262389876977509416273750,
%U A172787 1422407573139278002798418134584320442947352453284455417854786600
%N A172787 Number of 5*n X 10 0..3 arrays with row sums 4 and column sums 2*n
%H A172787 R. H. Hardin, <a href="b172787.txt">Table of n, a(n) for n=1..11</a>
%K A172787 nonn,new
%O A172787 1,1
%A A172787 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172786
%S A172786 51,2108358849529,576429489441112352734523835,
%T A172786 81775344380801671912399265731289816687307700,
%U A172786 1422407573139278002798418134584320442947352453284455417854786600
%N A172786 Number of 5*n X 2*n 0..3 arrays with row sums 4 and column sums 10
%H A172786 R. H. Hardin, <a href="b172786.txt">Table of n, a(n) for n=1..10</a>
%K A172786 nonn,new
%O A172786 1,1
%A A172786 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172785
%S A172785 305540235000,54918587341306311174536985000,
%T A172785 85504723470329847456379126545730424791071000000,
%U A172785 349386504852546133438847462576945313875421034231833558901433865000
%N A172785 Number of 5*n X 20 0..3 arrays with row sums 4 and column sums n
%H A172785 R. H. Hardin, <a href="b172785.txt">Table of n, a(n) for n=1..10</a>
%K A172785 nonn,new
%O A172785 1,1
%A A172785 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172784
%S A172784 168168000,369820640830881240000,3937477620391471128913917360384000,
%T A172784 85504723470329847456379126545730424791071000000,
%U A172784 2789130810728711760475063805085713493445792640582613331968000
%N A172784 Number of 5*n X 15 0..3 arrays with row sums 3 and column sums n
%H A172784 R. H. Hardin, <a href="b172784.txt">Table of n, a(n) for n=1..13</a>
%K A172784 nonn,new
%O A172784 1,1
%A A172784 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172783
%S A172783 1,116304,11936862061495,133937216531070195309888,
%T A172783 54474595447150914468834015421847040,
%U A172783 404868757253371611415640265892979872110459075840
%N A172783 Number of 5*n X n 0..3 arrays with row sums 3 and column sums 15
%H A172783 R. H. Hardin, <a href="b172783.txt">Table of n, a(n) for n=1..13</a>
%K A172783 nonn,new
%O A172783 1,2
%A A172783 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172782
%S A172782 0,8953,174322342281087,630707951172870178240070665,
%T A172782 167919620604132417334149114648212316386760,
%U A172782 1636420745348843918820746398768090753188232958726302893825
%N A172782 Number of 5*n X n 0..3 arrays with row sums 4 and column sums 20
%H A172782 R. H. Hardin, <a href="b172782.txt">Table of n, a(n) for n=1..10</a>
%K A172782 nonn,new
%O A172782 1,2
%A A172782 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172781
%S A172781 122975608,54450121092072227480,58723201215471060507377986244320,
%T A172781 92324899878614980043437168195067658467141080,
%U A172781 179432354155518286903008233851199224951768795442512239168
%N A172781 Number of 4*n X 8 0..3 arrays with row sums 6 and column sums 3*n
%H A172781 R. H. Hardin, <a href="b172781.txt">Table of n, a(n) for n=1..11</a>
%K A172781 nonn,new
%O A172781 1,1
%A A172781 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172780
%S A172780 2008,122975608,11012437532800,1160317947294167480,
%T A172780 133937216531070195309888,16391575725348461384454895264,
%U A172780 2089345627982707177699421140008960
%N A172780 Number of 4*n X 4 0..3 arrays with row sums 3 and column sums 3*n
%H A172780 R. H. Hardin, <a href="b172780.txt">Table of n, a(n) for n=1..24</a>
%K A172780 nonn,new
%O A172780 1,1
%A A172780 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172779
%S A172779 415,33833709700,1182802402030354182000,
%T A172779 2253836332035499736062301563800000,
%U A172779 85504723470329847456379126545730424791071000000
%N A172779 Number of 4*n X 3*n 0..3 arrays with row sums 3 and column sums 4
%H A172779 R. H. Hardin, <a href="b172779.txt">Table of n, a(n) for n=1..24</a>
%K A172779 nonn,new
%O A172779 1,1
%A A172779 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172778
%S A172778 369600,250071339672000,569060910292172349004800,
%T A172778 2253836332035499736062301563800000,
%U A172778 12263459619622095086766632900215307533209600
%N A172778 Number of 4*n X 12 0..3 arrays with row sums 3 and column sums n
%H A172778 R. H. Hardin, <a href="b172778.txt">Table of n, a(n) for n=1..16</a>
%K A172778 nonn,new
%O A172778 1,1
%A A172778 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172777
%S A172777 63063000,163081394186253543000,2253836332035499736062301563800000,
%T A172777 65949164854460383682147934685440527008344135000,
%U A172777 2970408787718945344873427655667537554031914545552039003208000
%N A172777 Number of 4*n X 16 0..3 arrays with row sums 4 and column sums n
%H A172777 R. H. Hardin, <a href="b172777.txt">Table of n, a(n) for n=1..12</a>
%K A172777 nonn,new
%O A172777 1,1
%A A172777 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172776
%S A172776 0,70,125290570185,47141783957741721328080,
%T A172776 1083265025247084674355419003777905401,
%U A172776 778626541064162359278560268669084299553822689853360
%N A172776 Number of 4*n X n 0..3 arrays with row sums 5 and column sums 20
%H A172776 R. H. Hardin, <a href="b172776.txt">Table of n, a(n) for n=1..11</a>
%K A172776 nonn,new
%O A172776 1,2
%A A172776 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172775
%S A172775 1,8092,14823083089,1160317947294167480,1605078294448526381158693200,
%T A172775 22683738449062602277870627264790194000,
%U A172775 2255071025881975002973276574847623754650721480000
%N A172775 Number of 4*n X n 0..3 arrays with row sums 3 and column sums 12
%H A172775 R. H. Hardin, <a href="b172775.txt">Table of n, a(n) for n=1..16</a>
%K A172775 nonn,new
%O A172775 1,2
%A A172775 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172774
%S A172774 0,1107,125290570185,946925090316795061315,
%T A172774 224061210725807969378046253140000,
%U A172774 947560483548938454931523739060201682514650275
%N A172774 Number of 4*n X n 0..3 arrays with row sums 4 and column sums 16
%H A172774 R. H. Hardin, <a href="b172774.txt">Table of n, a(n) for n=1..12</a>
%K A172774 nonn,new
%O A172774 1,2
%A A172774 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172773
%S A172773 352128,654393258290520,2538708180710780880644304,
%T A172773 14906195150068924678040454978560280,
%U A172773 110716878274689761475592889715532248440554848
%N A172773 Number of 3*n X 9 0..3 arrays with row sums 6 and column sums 2*n
%H A172773 R. H. Hardin, <a href="b172773.txt">Table of n, a(n) for n=1..13</a>
%K A172773 nonn,new
%O A172773 1,1
%A A172773 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172772
%S A172772 2385,971853075,654393258290520,570651583984812418275,
%T A172772 576429489441112352734523835,640278723754096114457315005223400,
%U A172772 760038862204473915631559446775632482000
%N A172772 Number of 3*n X 6 0..3 arrays with row sums 4 and column sums 2*n
%H A172772 R. H. Hardin, <a href="b172772.txt">Table of n, a(n) for n=1..28</a>
%K A172772 nonn,new
%O A172772 1,1
%A A172772 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172771
%S A172771 7,4705381,654393258290520,3956256528501607375052700,
%T A172771 432897335001181060850579402825662500,
%U A172771 493319321787145287396560260323437116718848470000
%N A172771 Number of 3*n X 2*n 0..3 arrays with row sums 4 and column sums 6
%H A172771 R. H. Hardin, <a href="b172771.txt">Table of n, a(n) for n=1..16</a>
%K A172771 nonn,new
%O A172771 1,1
%A A172771 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172770
%S A172770 34650,3536978063850,1182802402030354182000,
%T A172770 671520101271652710399345719850,520089040184399473996377693211802013900,
%U A172770 493319321787145287396560260323437116718848470000
%N A172770 Number of 3*n X 12 0..3 arrays with row sums 4 and column sums n
%H A172770 R. H. Hardin, <a href="b172770.txt">Table of n, a(n) for n=1..17</a>
%K A172770 nonn,new
%O A172770 1,1
%A A172770 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172769
%S A172769 756756,19292117692187340,2275745987915517509338826880,
%T A172769 520089040184399473996377693211802013900,
%U A172769 176031335522662538972343236688818802372580893815856
%N A172769 Number of 3*n X 15 0..3 arrays with row sums 5 and column sums n
%H A172769 R. H. Hardin, <a href="b172769.txt">Table of n, a(n) for n=1..13</a>
%K A172769 nonn,new
%O A172769 1,1
%A A172769 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172768
%S A172768 1680,747558000,772200774683520,1182802402030354182000,
%T A172768 2275745987915517509338826880,5078640963785630003833500263433600,
%U A172768 12577849178653907149465594061673565824000
%N A172768 Number of 3*n X 9 0..3 arrays with row sums 3 and column sums n
%H A172768 R. H. Hardin, <a href="b172768.txt">Table of n, a(n) for n=1..24</a>
%K A172768 nonn,new
%O A172768 1,1
%A A172768 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172767
%S A172767 0,1,19546975,80349760089959316,11279631181377626877359650100,
%T A172767 31270791853997716490787029515342398010161,
%U A172767 1138966845227847811612995810985909503258495428544062880
%N A172767 Number of 3*n X n 0..3 arrays with row sums 6 and column sums 18
%H A172767 R. H. Hardin, <a href="b172767.txt">Table of n, a(n) for n=1..11</a>
%K A172767 nonn,new
%O A172767 1,3
%A A172767 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172766
%S A172766 0,20,95635680,28021865662196736,180748183717038324996985791,
%T A172766 15496001580602804970376043343562432320,
%U A172766 12274752385434683822697731132401886699781192750720
%N A172766 Number of 3*n X n 0..3 arrays with row sums 5 and column sums 15
%H A172766 R. H. Hardin, <a href="b172766.txt">Table of n, a(n) for n=1..13</a>
%K A172766 nonn,new
%O A172766 1,2
%A A172766 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172765
%S A172765 0,0,352128,28021865662196736,82386976561127733213714195200,
%T A172765 6159222400147050126331043386857585651398400,
%U A172765 8076571943273575432916626559495058426486672272842882658769
%N A172765 Number of 3*n X n 0..3 arrays with row sums 7 and column sums 21
%H A172765 R. H. Hardin, <a href="b172765.txt">Table of n, a(n) for n=1..10</a>
%K A172765 nonn,new
%O A172765 1,3
%A A172765 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172764
%S A172764 0,141,95635680,1557850709802421,338059904123009135280850,
%T A172764 640278723754096114457315005223400,
%U A172764 7745310191767932461627415566004172302489000
%N A172764 Number of 3*n X n 0..3 arrays with row sums 4 and column sums 12
%H A172764 R. H. Hardin, <a href="b172764.txt">Table of n, a(n) for n=1..17</a>
%K A172764 nonn,new
%O A172764 1,2
%A A172764 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172763
%S A172763 1,580,19546975,11012437532800,53461314050652508000,
%T A172763 1482500191777802616831698000,177485739636656023698217362159890000,
%U A172763 74896615128945574656737417426177604440960000
%N A172763 Number of 3*n X n 0..3 arrays with row sums 3 and column sums 9
%H A172763 R. H. Hardin, <a href="b172763.txt">Table of n, a(n) for n=1..23</a>
%K A172763 nonn,new
%O A172763 1,2
%A A172763 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172762
%S A172762 1,19,28681,3017084995,37642786694586601,58256290618618126540488181,
%T A172762 11402923087449460426136950157514212457,
%U A172762 300748551220458753215613602117054376767307058363395
%N A172762 Number of 2*n X n 0..3 arrays with row sums n and column sums 2*n
%H A172762 R. H. Hardin, <a href="b172762.txt">Table of n, a(n) for n=1..10</a>
%K A172762 nonn,new
%O A172762 1,2
%A A172762 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172761
%S A172761 924,6152037276,127676780106968400,3956256528501607375052700,
%T A172761 164782963951568727647594366494224,
%U A172761 8357144767918620569086886647914789979536
%N A172761 Number of 2*n X 12 0..3 arrays with row sums 6 and column sums n
%H A172761 R. H. Hardin, <a href="b172761.txt">Table of n, a(n) for n=1..17</a>
%K A172761 nonn,new
%O A172761 1,1
%A A172761 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172760
%S A172760 252,79796556,62810128195920,69830279731379332620,
%T A172760 98967158653968381405705552,164782963951568727647594366494224,
%U A172760 307432123426436722747172649595952298432
%N A172760 Number of 2*n X 10 0..3 arrays with row sums 5 and column sums n
%H A172760 R. H. Hardin, <a href="b172760.txt">Table of n, a(n) for n=1..22</a>
%K A172760 nonn,new
%O A172760 1,1
%A A172760 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172759
%S A172759 70,1093050,33833709700,1397573379100570,69830279731379332620,
%T A172759 3956256528501607375052700,244951813940405396678166713400,
%U A172759 16202379114136976323568007348596250
%N A172759 Number of 2*n X 8 0..3 arrays with row sums 4 and column sums n
%H A172759 R. H. Hardin, <a href="b172759.txt">Table of n, a(n) for n=1..31</a>
%K A172759 nonn,new
%O A172759 1,1
%A A172759 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172758
%S A172758 20,16260,20933840,33833709700,62810128195920,127676780106968400,
%T A172758 276686670102097569600,628807365128705353776900,
%U A172758 1482500191777802616831698000,3598791496054686645936395190160
%N A172758 Number of 2*n X 6 0..3 arrays with row sums 3 and column sums n
%H A172758 R. H. Hardin, <a href="b172758.txt">Table of n, a(n) for n=1..49</a>
%K A172758 nonn,new
%O A172758 1,1
%A A172758 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172757
%S A172757 3432,492872166072,276686670102097569600,244951813940405396678166713400,
%T A172757 307432123426436722747172649595952298432,
%U A172757 487003260833491255817344373994693922008332858304
%N A172757 Number of 2*n X 14 0..3 arrays with row sums 7 and column sums n
%H A172757 R. H. Hardin, <a href="b172757.txt">Table of n, a(n) for n=1..12</a>
%K A172757 nonn,new
%O A172757 1,1
%A A172757 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172756
%S A172756 0,0,90,3017084995,2023900513877358900,14162844503221350778432935135,
%T A172756 806909026546240414032388715894620684395,
%U A172756 300748551220458753215613602117054376767307058363395
%N A172756 Number of 2*n X n 0..3 arrays with row sums 8 and column sums 16
%H A172756 R. H. Hardin, <a href="b172756.txt">Table of n, a(n) for n=1..11</a>
%K A172756 nonn,new
%O A172756 1,3
%A A172756 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172755
%S A172755 0,1,28681,38578100932,552523141020099900,58256290618618126540488181,
%T A172755 34391373190693800172956484206435649,
%U A172755 92324899878614980043437168195067658467141080
%N A172755 Number of 2*n X n 0..3 arrays with row sums 6 and column sums 12
%H A172755 R. H. Hardin, <a href="b172755.txt">Table of n, a(n) for n=1..15</a>
%K A172755 nonn,new
%O A172755 1,3
%A A172755 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172754
%S A172754 0,0,0,1093050,37642786694586601,14162844503221350778432935135,
%T A172754 53130204996192153080609362878028936125345,
%U A172754 1711672166340730383529033759740996945302163471959773200
%N A172754 Number of 2*n X n 0..3 arrays with row sums 10 and column sums 20
%H A172754 R. H. Hardin, <a href="b172754.txt">Table of n, a(n) for n=1..10</a>
%K A172754 nonn,new
%O A172754 1,4
%A A172754 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172753
%S A172753 1,44,28681,122975608,2217051589200,127676780106968400,
%T A172753 19483897303054057473600,6882009860546294146142870400,
%U A172753 5078640963785630003833500263433600
%N A172753 Number of 2*n X n 0..3 arrays with row sums 3 and column sums 6
%H A172753 R. H. Hardin, <a href="b172753.txt">Table of n, a(n) for n=1..42</a>
%K A172753 nonn,new
%O A172753 1,2
%A A172753 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172752
%S A172752 0,0,2385,19641294288,2023900513877358900,1839907815191199047874666240,
%T A172752 11402923087449460426136950157514212457,
%U A172752 388484402102927887501779717031406502426930399360
%N A172752 Number of 2*n X n 0..3 arrays with row sums 7 and column sums 14
%H A172752 R. H. Hardin, <a href="b172752.txt">Table of n, a(n) for n=1..13</a>
%K A172752 nonn,new
%O A172752 1,3
%A A172752 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172751
%S A172751 0,19,81225,3017084995,635481090640800,570651583984812418275,
%T A172751 1771064195990555279483745075,16202379114136976323568007348596250,
%U A172751 385904790671942888212671079168713127308000
%N A172751 Number of 2*n X n 0..3 arrays with row sums 4 and column sums 8
%H A172751 R. H. Hardin, <a href="b172751.txt">Table of n, a(n) for n=1..24</a>
%K A172751 nonn,new
%O A172751 1,2
%A A172751 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172750
%S A172750 0,6,81225,19641294288,37642786694586601,407505677116244252111280,
%T A172750 19524684484378921494922081853046,
%U A172750 3444203485343904342478939367221160666880
%N A172750 Number of 2*n X n 0..3 arrays with row sums 5 and column sums 10
%H A172750 R. H. Hardin, <a href="b172750.txt">Table of n, a(n) for n=1..18</a>
%K A172750 nonn,new
%O A172750 1,2
%A A172750 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172749
%S A172749 0,0,1,122975608,552523141020099900,28004870901329112246159413600,
%T A172749 13286253592938310343269948335403784033355,
%U A172749 48077185419930811359972085602422918530347454406137600
%N A172749 Number of 2*n X n 0..3 arrays with row sums 9 and column sums 18
%H A172749 R. H. Hardin, <a href="b172749.txt">Table of n, a(n) for n=1..10</a>
%K A172749 nonn,new
%O A172749 1,4
%A A172749 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172748
%S A172748 1,3,55,8515,13130451,210668680701,36012988082548233,
%T A172748 68307608848277646753795,1469748355306450464225368161315,
%U A172748 364459290339560650671344361875961740353
%N A172748 Number of n X n 0..3 arrays with row sums 2*n and column sums 2*n
%K A172748 nonn,new
%O A172748 1,2
%A A172748 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172747
%S A172747 1,25653,1612210777,56983034209069,3165885061421056701,
%T A172747 213886825849469969067741,16000785789980520484761813393,
%U A172747 1313785405202869686066790465746477
%N A172747 Number of n X 11 0..3 arrays with row sums 11 and column sums n
%H A172747 R. H. Hardin, <a href="b172747.txt">Table of n, a(n) for n=1..16</a>
%K A172747 nonn,new
%O A172747 1,2
%A A172747 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172746
%S A172746 1,212941,137176167025,43004935312603525,23391570666645297045541,
%T A172746 16127021731634644897218234421,12543075374928411639565288275440761,
%U A172746 10911763343002007651294794790855296669285
%N A172746 Number of n X 13 0..3 arrays with row sums 13 and column sums n
%H A172746 R. H. Hardin, <a href="b172746.txt">Table of n, a(n) for n=1..13</a>
%K A172746 nonn,new
%O A172746 1,2
%A A172746 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172745
%S A172745 1,8953,176703337,2108358849529,37642786694586601,800511109831448931841,
%T A172745 18674647115259946003114993,473652951543995073451549187737,
%U A172745 12810903082325592313645532408260249
%N A172745 Number of n X 10 0..3 arrays with row sums 10 and column sums n
%H A172745 R. H. Hardin, <a href="b172745.txt">Table of n, a(n) for n=1..20</a>
%K A172745 nonn,new
%O A172745 1,2
%A A172745 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172744
%S A172744 1,3139,19546975,79089365395,455886376926451,3065961180872817781,
%T A172744 22406966527952120859529,176369571737525105395211155,
%U A172744 1469748355306450464225368161315,12810903082325592313645532408260249
%N A172744 Number of n X 9 0..3 arrays with row sums 9 and column sums n
%H A172744 R. H. Hardin, <a href="b172744.txt">Table of n, a(n) for n=1..26</a>
%K A172744 nonn,new
%O A172744 1,2
%A A172744 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172743
%S A172743 1,7,55,415,3391,28681,248137,2186815,19546975,176703337,1612210777,
%T A172743 14823083089,137176167025,1276548044695,11936862061495,112092031526335,
%U A172743 1056512066133055,9991003788000025,94760438267696425,901154863603211665
%N A172743 Number of n X 3 0..3 arrays with row sums 3 and column sums n
%H A172743 R. H. Hardin, <a href="b172743.txt">Table of n, a(n) for n=1..99</a>
%K A172743 nonn,new
%O A172743 1,2
%A A172743 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172742
%S A172742 1,51,3391,194851,13130451,950563701,71936181801,5646492993891,
%T A172742 455886376926451,37642786694586601,3165885061421056701,
%U A172742 270372933455657475301,23391570666645297045541,2046340229879039857971651
%N A172742 Number of n X 5 0..3 arrays with row sums 5 and column sums n
%H A172742 R. H. Hardin, <a href="b172742.txt">Table of n, a(n) for n=1..95</a>
%K A172742 nonn,new
%O A172742 1,2
%A A172742 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172741
%S A172741 1,73789,14823083089,1557850709802421,270372933455657475301,
%T A172741 58256290618618126540488181,14029474438668037231099897238329,
%U A172741 3743451332951292699814807453286627989
%N A172741 Number of n X 12 0..3 arrays with row sums 12 and column sums n
%H A172741 R. H. Hardin, <a href="b172741.txt">Table of n, a(n) for n=1..15</a>
%K A172741 nonn,new
%O A172741 1,2
%A A172741 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172740
%S A172740 1,19,415,8515,194851,4705381,117546409,3017084995,79089365395,
%T A172740 2108358849529,56983034209069,1557850709802421,43004935312603525,
%U A172740 1197072354214287955,33562172767661519455,946925090316795061315
%N A172740 Number of n X 4 0..3 arrays with row sums 4 and column sums n
%H A172740 R. H. Hardin, <a href="b172740.txt">Table of n, a(n) for n=1..99</a>
%K A172740 nonn,new
%O A172740 1,2
%A A172740 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172739
%S A172739 1,1107,2186815,3017084995,5646492993891,12077564886803781,
%T A172739 27807404215104008937,68307608848277646753795,
%U A172739 176369571737525105395211155,473652951543995073451549187737
%N A172739 Number of n X 8 0..3 arrays with row sums 8 and column sums n
%H A172739 R. H. Hardin, <a href="b172739.txt">Table of n, a(n) for n=1..35</a>
%K A172739 nonn,new
%O A172739 1,2
%A A172739 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172738
%S A172738 1,141,28681,4705381,950563701,210668680701,49294458374841,
%T A172738 12077564886803781,3065961180872817781,800511109831448931841,
%U A172738 213886825849469969067741,58256290618618126540488181
%N A172738 Number of n X 6 0..3 arrays with row sums 6 and column sums n
%H A172738 R. H. Hardin, <a href="b172738.txt">Table of n, a(n) for n=1..61</a>
%K A172738 nonn,new
%O A172738 1,2
%A A172738 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172737
%S A172737 1,393,248137,117546409,71936181801,49294458374841,36012988082548233,
%T A172737 27807404215104008937,22406966527952120859529,18674647115259946003114993,
%U A172737 16000785789980520484761813393,14029474438668037231099897238329
%N A172737 Number of n X 7 0..3 arrays with row sums 7 and column sums n
%H A172737 R. H. Hardin, <a href="b172737.txt">Table of n, a(n) for n=1..45</a>
%K A172737 nonn,new
%O A172737 1,2
%A A172737 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172736
%S A172736 0,2,87,20112,13130451,20592419760,68480747646822,439844417498287872,
%T A172736 5071434436872671368032,98967158653968381405705552,
%U A172736 3114311420263883442781958758440,151767050607668644623432327234864384
%N A172736 Number of n X n 0..3 arrays with row sums 5 and column sums 5
%H A172736 R. H. Hardin, <a href="b172736.txt">Table of n, a(n) for n=1..35</a>
%K A172736 nonn,new
%O A172736 1,2
%A A172736 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172735
%S A172735 0,0,1,2008,45934340,3875209794400,1031515670404216935,
%T A172735 771548465309275841582080,1469748355306450464225368161315,
%U A172735 6564303619112746449775184462681016600
%N A172735 Number of n X n 0..3 arrays with row sums 9 and column sums 9
%H A172735 R. H. Hardin, <a href="b172735.txt">Table of n, a(n) for n=1..15</a>
%K A172735 nonn,new
%O A172735 1,4
%A A172735 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172734
%S A172734 0,0,0,24,1992760,1070625363840,1995721130136414975,
%T A172734 11790866860533363630752256,202230411632955453212580408141744,
%U A172734 9272631275636047653190234511763264499200
%N A172734 Number of n X n 0..3 arrays with row sums 11 and column sums 11
%H A172734 R. H. Hardin, <a href="b172734.txt">Table of n, a(n) for n=1..13</a>
%K A172734 nonn,new
%O A172734 1,4
%A A172734 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172733
%S A172733 0,0,6,8515,84202220,2805859246095,272039989206828885,
%T A172733 68307608848277646753795,40385466638839151897733367128,
%U A172733 51971656438500975228442173361615800
%N A172733 Number of n X n 0..3 arrays with row sums 8 and column sums 8
%H A172733 R. H. Hardin, <a href="b172733.txt">Table of n, a(n) for n=1..17</a>
%K A172733 nonn,new
%O A172733 1,3
%A A172733 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172732
%S A172732 0,0,0,1,153040,210668680701,1031515670404216935,
%T A172732 16533561074241057523755540,808064031263375768787771957383712,
%U A172732 111562091000778108490941203120508002739900
%N A172732 Number of n X n 0..3 arrays with row sums 12 and column sums 12
%H A172732 R. H. Hardin, <a href="b172732.txt">Table of n, a(n) for n=1..12</a>
%K A172732 nonn,new
%O A172732 1,5
%A A172732 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172731
%S A172731 0,0,0,0,6210,20592419760,272039989206828885,11790866860533363630752256,
%T A172731 1609379339428507638996616962234528,
%U A172731 649308484381906359124692131947427492018400
%N A172731 Number of n X n 0..3 arrays with row sums 13 and column sums 13
%H A172731 R. H. Hardin, <a href="b172731.txt">Table of n, a(n) for n=1..11</a>
%K A172731 nonn,new
%O A172731 1,5
%A A172731 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172730
%S A172730 0,0,0,0,120,971853075,36012988082548233,4263492449580925474047120,
%T A172730 1609379339428507638996616962234528,
%U A172730 1856926911618773209041555011161165016996700
%N A172730 Number of n X n 0..3 arrays with row sums 14 and column sums 14
%H A172730 R. H. Hardin, <a href="b172730.txt">Table of n, a(n) for n=1..11</a>
%K A172730 nonn,new
%O A172730 1,5
%A A172730 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172729
%S A172729 0,0,0,0,1,20933840,2310106618930215,771548465309275841582080,
%T A172729 808064031263375768787771957383712,
%U A172729 2633262235838582855557285384905264898523904
%N A172729 Number of n X n 0..3 arrays with row sums 15 and column sums 15
%H A172729 R. H. Hardin, <a href="b172729.txt">Table of n, a(n) for n=1..11</a>
%K A172729 nonn,new
%O A172729 1,6
%A A172729 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172728
%S A172728 0,0,0,0,0,202410,68480747646822,68307608848277646753795,
%T A172728 202230411632955453212580408141744,
%U A172728 1856926911618773209041555011161165016996700
%N A172728 Number of n X n 0..3 arrays with row sums 16 and column sums 16
%K A172728 nonn,new
%O A172728 1,6
%A A172728 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172727
%S A172727 0,0,0,0,0,720,887226868605,2867622077261705078784,
%T A172727 24875052693330062712109843270944,
%U A172727 649308484381906359124692131947427492018400
%N A172727 Number of n X n 0..3 arrays with row sums 17 and column sums 17
%K A172727 nonn,new
%O A172727 1,6
%A A172727 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172726
%S A172726 0,0,0,0,0,1,4662857360,54450121092072227480,
%T A172726 1469748355306450464225368161315,
%U A172726 111562091000778108490941203120508002739900
%N A172726 Number of n X n 0..3 arrays with row sums 18 and column sums 18
%K A172726 nonn,new
%O A172726 1,7
%A A172726 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172725
%S A172725 0,0,0,282,13130451,2805859246095,1995721130136414975,
%T A172725 4263492449580925474047120,24875052693330062712109843270944,
%U A172725 364459290339560650671344361875961740353
%N A172725 Number of n X n 0..3 arrays with row sums 10 and column sums 10
%H A172725 R. H. Hardin, <a href="b172725.txt">Table of n, a(n) for n=1..15</a>
%K A172725 nonn,new
%O A172725 1,4
%A A172725 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172724
%S A172724 0,0,21,20112,84202220,1070625363840,36012988082548233,
%T A172724 2867622077261705078784,494207026605606690167323392,
%U A172724 171330275896315127362166522313600
%N A172724 Number of n X n 0..3 arrays with row sums 7 and column sums 7
%H A172724 R. H. Hardin, <a href="b172724.txt">Table of n, a(n) for n=1..21</a>
%K A172724 nonn,new
%O A172724 1,3
%A A172724 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172723
%S A172723 0,3,87,8515,1992760,971853075,887226868605,1397573379100570,
%T A172723 3564687080800036800,13994422270858878448950,81142871841524279822631750,
%U A172723 671520101271652710399345719850,7707138964962257026620598104320400
%N A172723 Number of n X n 0..3 arrays with row sums 4 and column sums 4
%H A172723 R. H. Hardin, <a href="b172723.txt">Table of n, a(n) for n=1..55</a>
%K A172723 nonn,new
%O A172723 1,2
%A A172723 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172722
%S A172722 0,1,55,27636,45934340,210668680701,2310106618930215,
%T A172722 54450121092072227480,2538708180710780880644304,
%U A172722 218999995866993338639652628800,33082630485031588732233929380223040
%N A172722 Number of n X n 0..3 arrays with row sums 6 and column sums 6
%H A172722 R. H. Hardin, <a href="b172722.txt">Table of n, a(n) for n=1..25</a>
%K A172722 nonn,new
%O A172722 1,3
%A A172722 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172721
%S A172721 41514583320,2402820978940192425615000,
%T A172721 384184227197088213207839624049360408000,
%U A172721 95131219937961621952706904277268129938808819520975000
%N A172721 Number of 9*n X 9 0..2 arrays with row sums 2 and column sums 2*n
%H A172721 R. H. Hardin, <a href="b172721.txt">Table of n, a(n) for n=1..11</a>
%K A172721 nonn,new
%O A172721 1,1
%A A172721 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172720
%S A172720 3139,3423871305383100,34335031273255183438800013252500,
%T A172720 141178303808277520681079289672968415090175698600000,
%U A172720 52186319789297652611103763179819584756687013016766057690192910549000000
%N A172720 Number of 9*n X 2*n 0..2 arrays with row sums 2 and column sums 9
%H A172720 R. H. Hardin, <a href="b172720.txt">Table of n, a(n) for n=1..11</a>
%K A172720 nonn,new
%O A172720 1,1
%A A172720 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172719
%S A172719 12504636144000,9055359650665478876752602576000,
%T A172719 46455761324619133018320834819622638940550400000000,
%U A172719 571951409844641024133465142141641285731325047530743069428902050000000
%N A172719 Number of 9*n X 18 0..2 arrays with row sums 2 and column sums n
%H A172719 R. H. Hardin, <a href="b172719.txt">Table of n, a(n) for n=1..10</a>
%K A172719 nonn,new
%O A172719 1,1
%A A172719 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172718
%S A172718 1,44152809,12331175198408791725,1237527402223677679598899948925700,
%T A172718 10500268455366221173591541192738752806765261187500,
%U A172718 3138480844349933121860864061245246387668619696538799391771830312500
%N A172718 Number of 9*n X n 0..2 arrays with row sums 2 and column sums 18
%H A172718 R. H. Hardin, <a href="b172718.txt">Table of n, a(n) for n=1..10</a>
%K A172718 nonn,new
%O A172718 1,2
%A A172718 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172717
%S A172717 1310799454720,3321735626954282018581900800,
%T A172717 20877662827076969301744211245461828236480000,
%U A172717 193433401300524863973681384616930307135746244016427424448000
%N A172717 Number of 8*n X 8 0..2 arrays with row sums 3 and column sums 3*n
%H A172717 R. H. Hardin, <a href="b172717.txt">Table of n, a(n) for n=1..10</a>
%K A172717 nonn,new
%O A172717 1,1
%A A172717 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172716
%S A172716 81729648000,102681435747106627787376000,
%T A172716 720289186703359375552628986978410240000000,
%U A172716 10921172797213788502287477185814678963894090694591590000000
%N A172716 Number of 8*n X 16 0..2 arrays with row sums 2 and column sums n
%H A172716 R. H. Hardin, <a href="b172716.txt">Table of n, a(n) for n=1..11</a>
%K A172716 nonn,new
%O A172716 1,1
%A A172716 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172715
%S A172715 1,5196627,64089909936535329,147261383858486597256254538810,
%T A172715 17441993280095359392117664251165245279447250,
%U A172715 48930886220271330542271419741692768122929164062703692950250
%N A172715 Number of 8*n X n 0..2 arrays with row sums 2 and column sums 16
%H A172715 R. H. Hardin, <a href="b172715.txt">Table of n, a(n) for n=1..11</a>
%K A172715 nonn,new
%O A172715 1,2
%A A172715 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172714
%S A172714 344827336455,134169254007831527780970675,
%T A172714 115401909362364329779895671658357933292000,
%U A172714 138888194169408560414765498059420203413991485926907191875
%N A172714 Number of 7*n X 7 0..2 arrays with row sums 4 and column sums 4*n
%H A172714 R. H. Hardin, <a href="b172714.txt">Table of n, a(n) for n=1..10</a>
%K A172714 nonn,new
%O A172714 1,1
%A A172714 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172713
%S A172713 3758757240,8610740229284196003000,42758542523186088601256526892620000,
%T A172713 295799727163686817428226118102222383709995755000,
%U A172713 2462059619716250949213618993808700317296159580533060054282240
%N A172713 Number of 7*n X 7 0..2 arrays with row sums 3 and column sums 3*n
%H A172713 R. H. Hardin, <a href="b172713.txt">Table of n, a(n) for n=1..12</a>
%K A172713 nonn,new
%O A172713 1,1
%A A172713 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172712
%S A172712 35617,87186713382942888000,5516710929099784554483904755922151424000,
%T A172712 333637887279822868851213302567250125738747164586627567616000000,
%U A172712 3578764279810021543463667726214338048478657324628002393015738511636185172262912000000000
%N A172712 Number of 7*n X 3*n 0..2 arrays with row sums 3 and column sums 7
%H A172712 R. H. Hardin, <a href="b172712.txt">Table of n, a(n) for n=1..11</a>
%K A172712 nonn,new
%O A172712 1,1
%A A172712 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172711
%S A172711 9135630,18126466426218150,76497104228450459248094400,
%T A172711 447362835296127429187676764430583750,
%U A172711 3140955188547844588635579044406639849274718880
%N A172711 Number of 7*n X 7 0..2 arrays with row sums 2 and column sums 2*n
%H A172711 R. H. Hardin, <a href="b172711.txt">Table of n, a(n) for n=1..14</a>
%K A172711 nonn,new
%O A172711 1,1
%A A172711 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172710
%S A172710 393,492872166072,737005538936597762145600,
%T A172710 117847210656873992022720603629873856000,
%U A172710 620630259068078856134953474839665823595181960916480000
%N A172710 Number of 7*n X 2*n 0..2 arrays with row sums 2 and column sums 7
%H A172710 R. H. Hardin, <a href="b172710.txt">Table of n, a(n) for n=1..14</a>
%K A172710 nonn,new
%O A172710 1,1
%A A172710 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172709
%S A172709 681080400,1908371363842760216400,23511842995969107700302647865600000,
%T A172709 563878027583411427270722913947560641974771250000,
%U A172709 19762924171488525106388046998057857689799517215013510114150400
%N A172709 Number of 7*n X 14 0..2 arrays with row sums 2 and column sums n
%H A172709 R. H. Hardin, <a href="b172709.txt">Table of n, a(n) for n=1..13</a>
%K A172709 nonn,new
%O A172709 1,1
%A A172709 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172708
%S A172708 0,3432,8385127334900305,6691944921318182113271713149120,
%T A172708 560802479516289977082608657662771140464350896000,
%U A172708 2224408910230139891062284534118773609167351452658037731567555400000
%N A172708 Number of 7*n X n 0..2 arrays with row sums 3 and column sums 21
%H A172708 R. H. Hardin, <a href="b172708.txt">Table of n, a(n) for n=1..10</a>
%K A172708 nonn,new
%O A172708 1,2
%A A172708 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172707
%S A172707 1,616227,338183208699840,17928312105849154413381000,
%T A172707 29871310888030132154949642916374456000,
%U A172707 792606936905424716827805609592848631050897983368000
%N A172707 Number of 7*n X n 0..2 arrays with row sums 2 and column sums 14
%H A172707 R. H. Hardin, <a href="b172707.txt">Table of n, a(n) for n=1..13</a>
%K A172707 nonn,new
%O A172707 1,2
%A A172707 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172706
%S A172706 2030271480,1563133410663239958840,2333327772435239191435579651896000,
%T A172706 4611389774049135678307206605901825339706084920,
%U A172706 10650335487681202470770878014052471383678883855546130772480
%N A172706 Number of 6*n X 6 0..2 arrays with row sums 5 and column sums 5*n
%H A172706 R. H. Hardin, <a href="b172706.txt">Table of n, a(n) for n=1..12</a>
%K A172706 nonn,new
%O A172706 1,1
%A A172706 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172705
%S A172705 7484400,62347376347779600,1796185853884657144990080000,
%T A172705 90708013716277730071013162104546890000,
%U A172705 6309193293073551526242474424812731974830376454400
%N A172705 Number of 6*n X 12 0..2 arrays with row sums 2 and column sums n
%H A172705 R. H. Hardin, <a href="b172705.txt">Table of n, a(n) for n=1..15</a>
%K A172705 nonn,new
%O A172705 1,1
%A A172705 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172704
%S A172704 137225088000,1796185853884657144990080000,
%T A172704 151012959794833573952000702864304916267008000,
%U A172704 30037987606650431798215986206016098569029061818175150320000000
%N A172704 Number of 6*n X 18 0..2 arrays with row sums 3 and column sums n
%H A172704 R. H. Hardin, <a href="b172704.txt">Table of n, a(n) for n=1..11</a>
%K A172704 nonn,new
%O A172704 1,1
%A A172704 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172703
%S A172703 1,73789,1820016119376,2248575441654260591964,
%T A172703 53234949493963470885785898137100,
%U A172703 13495472374334172242190334756526625738793200
%N A172703 Number of 6*n X n 0..2 arrays with row sums 2 and column sums 12
%H A172703 R. H. Hardin, <a href="b172703.txt">Table of n, a(n) for n=1..15</a>
%K A172703 nonn,new
%O A172703 1,2
%A A172703 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172702
%S A172702 0,924,28425752310361,128167620006030442731491136,
%T A172702 31265195228897742672250371317378119778400,
%U A172702 208166284750977846936978491445694459933214981665738410000
%N A172702 Number of 6*n X n 0..2 arrays with row sums 3 and column sums 18
%H A172702 R. H. Hardin, <a href="b172702.txt">Table of n, a(n) for n=1..11</a>
%K A172702 nonn,new
%O A172702 1,2
%A A172702 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172701
%S A172701 628080,31237467004800,2630024635942177123650,
%T A172701 276584170067890425950927140800,32920939338005930859094601223181654080,
%U A172701 4240699386876069215051650356835022129252074650
%N A172701 Number of 5*n X 5 0..2 arrays with row sums 4 and column sums 4*n
%H A172701 R. H. Hardin, <a href="b172701.txt">Table of n, a(n) for n=1..19</a>
%K A172701 nonn,new
%O A172701 1,1
%A A172701 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172700
%S A172700 32611,20033574566576932620,203165913750358400342600887093908501900,
%T A172700 1403289426719396371385462622642366104137840568753800490568000,
%U A172700 1329376191210725010990283299434694464336896476138824800813517191378992025207893360000
%N A172700 Number of 5*n X 4*n 0..2 arrays with row sums 4 and column sums 5
%H A172700 R. H. Hardin, <a href="b172700.txt">Table of n, a(n) for n=1..10</a>
%K A172700 nonn,new
%O A172700 1,1
%A A172700 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172699
%S A172699 111920,755265546000,8432855707512236000,117188101836293753504038800,
%T A172699 1840612120873072393197860408401920,
%U A172699 31265195228897742672250371317378119778400
%N A172699 Number of 5*n X 5 0..2 arrays with row sums 3 and column sums 3*n
%H A172699 R. H. Hardin, <a href="b172699.txt">Table of n, a(n) for n=1..19</a>
%K A172699 nonn,new
%O A172699 1,1
%A A172699 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172698
%S A172698 991,31091880269520,1392491976667637592480814080,
%T A172698 8412911327063605695441946427825386408857600,
%U A172698 2054966035616319299085926097126189858441720582082825402368000
%N A172698 Number of 5*n X 3*n 0..2 arrays with row sums 3 and column sums 5
%H A172698 R. H. Hardin, <a href="b172698.txt">Table of n, a(n) for n=1..17</a>
%K A172698 nonn,new
%O A172698 1,1
%A A172698 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172697
%S A172697 6210,1328792850,467046072593100,203571289613781911250,
%T A172697 100139724386322263262579960,53234949493963470885785898137100,
%U A172697 29871310888030132154949642916374456000
%N A172697 Number of 5*n X 5 0..2 arrays with row sums 2 and column sums 2*n
%H A172697 R. H. Hardin, <a href="b172697.txt">Table of n, a(n) for n=1..19</a>
%K A172697 nonn,new
%O A172697 1,1
%A A172697 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172696
%S A172696 1328792850,6736902064011471514050,105023948177499505831398834880267500,
%T A172696 2652858790156350652048342955686055807529567941250,
%U A172696 87922422724198361142300055273769637829603366818705131011416600
%N A172696 Number of 5*n X 10 0..2 arrays with row sums 4 and column sums 2*n
%H A172696 R. H. Hardin, <a href="b172696.txt">Table of n, a(n) for n=1..11</a>
%K A172696 nonn,new
%O A172696 1,1
%A A172696 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172695
%S A172695 51,79796556,19292117692187340,130096688857163908226867520,
%T A172695 10585995418849686886860491452609238400,
%U A172695 6309193293073551526242474424812731974830376454400
%N A172695 Number of 5*n X 2*n 0..2 arrays with row sums 2 and column sums 5
%H A172695 R. H. Hardin, <a href="b172695.txt">Table of n, a(n) for n=1..19</a>
%K A172695 nonn,new
%O A172695 1,1
%A A172695 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172694
%S A172694 1,30189264889,14988373385805104784584085,
%T A172694 3448359090155531650840981107763278132248500,
%U A172694 87922422724198361142300055273769637829603366818705131011416600
%N A172694 Number of 5*n X 2*n 0..2 arrays with row sums 4 and column sums 10
%H A172694 R. H. Hardin, <a href="b172694.txt">Table of n, a(n) for n=1..10</a>
%K A172694 nonn,new
%O A172694 1,2
%A A172694 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172693
%S A172693 168168000,369820640830881240000,3581747697361314888194118659328000,
%T A172693 70118545603035216189674302236706595177583000000,
%U A172693 2054966035616319299085926097126189858441720582082825402368000
%N A172693 Number of 5*n X 15 0..2 arrays with row sums 3 and column sums n
%H A172693 R. H. Hardin, <a href="b172693.txt">Table of n, a(n) for n=1..13</a>
%K A172693 nonn,new
%O A172693 1,1
%A A172693 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172692
%S A172692 113400,3930730108200,369820640830881240000,
%T A172692 54918587341306311174536985000,10585995418849686886860491452609238400,
%U A172692 2415977451999318332950627138384873223959560000
%N A172692 Number of 5*n X 10 0..2 arrays with row sums 2 and column sums n
%H A172692 R. H. Hardin, <a href="b172692.txt">Table of n, a(n) for n=1..19</a>
%K A172692 nonn,new
%O A172692 1,1
%A A172692 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172691
%S A172691 305540235000,54918587341306311174536985000,
%T A172691 70118545603035216189674302236706595177583000000,
%U A172691 232273808505581071157606795985678847488890659685774558806222665000
%N A172691 Number of 5*n X 20 0..2 arrays with row sums 4 and column sums n
%H A172691 R. H. Hardin, <a href="b172691.txt">Table of n, a(n) for n=1..10</a>
%K A172691 nonn,new
%O A172691 1,1
%A A172691 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172690
%S A172690 0,252,98984058271,2556622439429326953984,
%T A172690 1840612120873072393197860408401920,
%U A172690 20856759061804812713740241716791439872159072000
%N A172690 Number of 5*n X n 0..2 arrays with row sums 3 and column sums 15
%H A172690 R. H. Hardin, <a href="b172690.txt">Table of n, a(n) for n=1..13</a>
%K A172690 nonn,new
%O A172690 1,2
%A A172690 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172689
%S A172689 1,8953,10060071021,293631119403639732,100139724386322263262579960,
%T A172689 245885257930209910994050195049583660,
%U A172689 3140955188547844588635579044406639849274718880
%N A172689 Number of 5*n X n 0..2 arrays with row sums 2 and column sums 10
%H A172689 R. H. Hardin, <a href="b172689.txt">Table of n, a(n) for n=1..19</a>
%K A172689 nonn,new
%O A172689 1,2
%A A172689 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172688
%S A172688 0,1,10060071021,67224767490120483872521,
%T A172688 32920939338005930859094601223181654080,
%U A172688 565765822563970936585370178796808848810196343004022625
%N A172688 Number of 5*n X n 0..2 arrays with row sums 4 and column sums 20
%H A172688 R. H. Hardin, <a href="b172688.txt">Table of n, a(n) for n=1..10</a>
%K A172688 nonn,new
%O A172688 1,3
%A A172688 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172687
%S A172687 34145664,1418365816378741440,137649612557218268634666665280,
%T A172687 19578536057961112914067888563728861113920,
%U A172687 3448690205704401268993875180062056328713917132382464
%N A172687 Number of 4*n X 8 0..2 arrays with row sums 6 and column sums 3*n
%H A172687 R. H. Hardin, <a href="b172687.txt">Table of n, a(n) for n=1..11</a>
%K A172687 nonn,new
%O A172687 1,1
%A A172687 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172686
%S A172686 1344,34145664,1252634243520,54067759732546560,2556622439429326953984,
%T A172686 128167620006030442731491136,6691944921318182113271713149120,
%U A172686 360021667477153691393072311177900800
%N A172686 Number of 4*n X 4 0..2 arrays with row sums 3 and column sums 3*n
%H A172686 R. H. Hardin, <a href="b172686.txt">Table of n, a(n) for n=1..24</a>
%K A172686 nonn,new
%O A172686 1,1
%A A172686 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172685
%S A172685 175,20982583300,853588467101915622000,
%T A172685 1760753561430175391642594031000000,
%U A172685 70118545603035216189674302236706595177583000000
%N A172685 Number of 4*n X 3*n 0..2 arrays with row sums 3 and column sums 4
%H A172685 R. H. Hardin, <a href="b172685.txt">Table of n, a(n) for n=1..24</a>
%K A172685 nonn,new
%O A172685 1,1
%A A172685 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172684
%S A172684 2520,545007960,250071339672000,163081394186253543000,
%T A172684 130096688857163908226867520,118274738663434470504494036529600,
%U A172684 117847210656873992022720603629873856000
%N A172684 Number of 4*n X 8 0..2 arrays with row sums 2 and column sums n
%H A172684 R. H. Hardin, <a href="b172684.txt">Table of n, a(n) for n=1..24</a>
%K A172684 nonn,new
%O A172684 1,1
%A A172684 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172683
%S A172683 63063000,163081394186253543000,1760753561430175391642594031000000,
%T A172683 40125878309014654627871871114724921424759175000,
%U A172683 1403289426719396371385462622642366104137840568753800490568000
%N A172683 Number of 4*n X 16 0..2 arrays with row sums 4 and column sums n
%H A172683 R. H. Hardin, <a href="b172683.txt">Table of n, a(n) for n=1..12</a>
%K A172683 nonn,new
%O A172683 1,1
%A A172683 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172682
%S A172682 369600,250071339672000,504669505260741099417600,
%T A172682 1760753561430175391642594031000000,
%U A172682 8412911327063605695441946427825386408857600
%N A172682 Number of 4*n X 12 0..2 arrays with row sums 3 and column sums n
%H A172682 R. H. Hardin, <a href="b172682.txt">Table of n, a(n) for n=1..16</a>
%K A172682 nonn,new
%O A172682 1,1
%A A172682 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172681
%S A172681 0,0,34650,54067759732546560,3138774474975381117998859525561,
%T A172681 4611389774049135678307206605901825339706084920,
%U A172681 118146008179337031214747923950181684918919939516291368598887900
%N A172681 Number of 4*n X n 0..2 arrays with row sums 5 and column sums 20
%H A172681 R. H. Hardin, <a href="b172681.txt">Table of n, a(n) for n=1..11</a>
%K A172681 nonn,new
%O A172681 1,3
%A A172681 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172680
%S A172680 0,70,358198369,54067759732546560,117188101836293753504038800,
%T A172680 2305398394297691701702591298576921200,
%U A172680 295799727163686817428226118102222383709995755000
%N A172680 Number of 4*n X n 0..2 arrays with row sums 3 and column sums 12
%H A172680 R. H. Hardin, <a href="b172680.txt">Table of n, a(n) for n=1..16</a>
%K A172680 nonn,new
%O A172680 1,2
%A A172680 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172679
%S A172679 0,1,57775905,716068213864217155,276584170067890425950927140800,
%T A172679 1842950731180774733190378188820331816058325,
%U A172679 138888194169408560414765498059420203413991485926907191875
%N A172679 Number of 4*n X n 0..2 arrays with row sums 4 and column sums 16
%H A172679 R. H. Hardin, <a href="b172679.txt">Table of n, a(n) for n=1..12</a>
%K A172679 nonn,new
%O A172679 1,3
%A A172679 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172678
%S A172678 1,1107,57775905,40629560387130,203571289613781911250,
%T A172678 4937928427617947420104982250,447362835296127429187676764430583750,
%U A172678 125661678519106774927206307245894357500775000
%N A172678 Number of 4*n X n 0..2 arrays with row sums 2 and column sums 8
%H A172678 R. H. Hardin, <a href="b172678.txt">Table of n, a(n) for n=1..24</a>
%K A172678 nonn,new
%O A172678 1,2
%A A172678 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172677
%S A172677 2385,342829125,96307175477520,35265186498828525525,
%T A172677 14988373385805104784584085,7011583640699581889649892843200,
%U A172677 3507072475491201687990422459298120000
%N A172677 Number of 3*n X 6 0..2 arrays with row sums 4 and column sums 2*n
%H A172677 R. H. Hardin, <a href="b172677.txt">Table of n, a(n) for n=1..28</a>
%K A172677 nonn,new
%O A172677 1,1
%A A172677 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172676
%S A172676 352128,96307175477520,78459796857685317493440,
%T A172676 97830370630931874758353953631440,
%U A172676 154847134246966332451923962693083298516928
%N A172676 Number of 3*n X 9 0..2 arrays with row sums 6 and column sums 2*n
%H A172676 R. H. Hardin, <a href="b172676.txt">Table of n, a(n) for n=1..13</a>
%K A172676 nonn,new
%O A172676 1,1
%A A172676 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172675
%S A172675 21,2385,352128,57775905,10060071021,1820016119376,338183208699840,
%T A172675 64089909936535329,12331175198408791725,2401214665364782652385,
%U A172675 472159936393091112404160,93594776429965445731933200
%N A172675 Number of 3*n X 3 0..2 arrays with row sums 2 and column sums 2*n
%H A172675 R. H. Hardin, <a href="b172675.txt">Table of n, a(n) for n=1..33</a>
%K A172675 nonn,new
%O A172675 1,1
%A A172675 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172674
%S A172674 1,481381,96307175477520,772712857320523525028700,
%T A172674 105023948177499505831398834880267500,
%U A172674 141697641014910453193894333591821632439066870000
%N A172674 Number of 3*n X 2*n 0..2 arrays with row sums 4 and column sums 6
%H A172674 R. H. Hardin, <a href="b172674.txt">Table of n, a(n) for n=1..16</a>
%K A172674 nonn,new
%O A172674 1,2
%A A172674 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172673
%S A172673 1680,747558000,655551508577280,853588467101915622000,
%T A172673 1392491976667637592480814080,2631822682451717147804985743769600,
%U A172673 5516710929099784554483904755922151424000
%N A172673 Number of 3*n X 9 0..2 arrays with row sums 3 and column sums n
%H A172673 R. H. Hardin, <a href="b172673.txt">Table of n, a(n) for n=1..24</a>
%K A172673 nonn,new
%O A172673 1,1
%A A172673 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172672
%S A172672 34650,3536978063850,853588467101915622000,
%T A172672 356683733303639928299266819050,203165913750358400342600887093908501900,
%U A172672 141697641014910453193894333591821632439066870000
%N A172672 Number of 3*n X 12 0..2 arrays with row sums 4 and column sums n
%H A172672 R. H. Hardin, <a href="b172672.txt">Table of n, a(n) for n=1..17</a>
%K A172672 nonn,new
%O A172672 1,1
%A A172672 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172671
%S A172671 90,202410,747558000,3536978063850,19292117692187340,
%T A172671 115428185943399529200,737005538936597762145600,
%U A172671 4937928427617947420104982250,34335031273255183438800013252500
%N A172671 Number of 3*n X 6 0..2 arrays with row sums 2 and column sums n
%H A172671 R. H. Hardin, <a href="b172671.txt">Table of n, a(n) for n=1..33</a>
%K A172671 nonn,new
%O A172671 1,1
%A A172671 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172670
%S A172670 756756,19292117692187340,1392491976667637592480814080,
%T A172670 203165913750358400342600887093908501900,
%U A172670 43953912340166344641202095832376782984770776827056
%N A172670 Number of 3*n X 15 0..2 arrays with row sums 5 and column sums n
%H A172670 R. H. Hardin, <a href="b172670.txt">Table of n, a(n) for n=1..13</a>
%K A172670 nonn,new
%O A172670 1,1
%A A172670 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172669
%S A172669 0,20,1376095,1252634243520,8432855707512236000,
%T A172669 297155971199028097720850000,42758542523186088601256526892620000,
%U A172669 20877662827076969301744211245461828236480000
%N A172669 Number of 3*n X n 0..2 arrays with row sums 3 and column sums 9
%H A172669 R. H. Hardin, <a href="b172669.txt">Table of n, a(n) for n=1..23</a>
%K A172669 nonn,new
%O A172669 1,2
%A A172669 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172668
%S A172668 1,141,352128,6152037276,467046072593100,115428185943399529200,
%T A172668 76497104228450459248094400,118274738663434470504494036529600,
%U A172668 384184227197088213207839624049360408000
%N A172668 Number of 3*n X n 0..2 arrays with row sums 2 and column sums 6
%H A172668 R. H. Hardin, <a href="b172668.txt">Table of n, a(n) for n=1..33</a>
%K A172668 nonn,new
%O A172668 1,2
%A A172668 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172667
%S A172667 0,0,1680,1252634243520,15966070888767791932191,
%T A172667 2333327772435239191435579651896000,
%U A172667 2927463235524353975393755326111103021707295200
%N A172667 Number of 3*n X n 0..2 arrays with row sums 5 and column sums 15
%H A172667 R. H. Hardin, <a href="b172667.txt">Table of n, a(n) for n=1..14</a>
%K A172667 nonn,new
%O A172667 1,3
%A A172667 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172666
%S A172666 0,0,0,369600,8432855707512236000,2333327772435239191435579651896000,
%T A172666 7912042661932746320895028318655823357822209601009,
%U A172666 288661272140059402021783839980598829830361488723258925444306851840
%N A172666 Number of 3*n X n 0..2 arrays with row sums 7 and column sums 21
%H A172666 R. H. Hardin, <a href="b172666.txt">Table of n, a(n) for n=1..10</a>
%K A172666 nonn,new
%O A172666 1,4
%A A172666 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172665
%S A172665 0,0,0,1,467046072593100,7011583640699581889649892843200,
%T A172665 1126052188643363954802820836947166068046894450600,
%U A172665 2120920516514529364095866499134381437091878416644933740597080005445
%N A172665 Number of 3*n X n 0..2 arrays with row sums 8 and column sums 24
%H A172665 R. H. Hardin, <a href="b172665.txt">Table of n, a(n) for n=1..10</a>
%K A172665 nonn,new
%O A172665 1,5
%A A172665 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172664
%S A172664 0,1,352128,8365213132981,2630024635942177123650,
%T A172664 7011583640699581889649892843200,
%U A172664 115401909362364329779895671658357933292000
%N A172664 Number of 3*n X n 0..2 arrays with row sums 4 and column sums 12
%H A172664 R. H. Hardin, <a href="b172664.txt">Table of n, a(n) for n=1..17</a>
%K A172664 nonn,new
%O A172664 1,3
%A A172664 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172663
%S A172663 0,0,1,6152037276,2630024635942177123650,
%T A172663 16157198427400511406055803977773761,
%U A172663 1126052188643363954802820836947166068046894450600
%N A172663 Number of 3*n X n 0..2 arrays with row sums 6 and column sums 18
%H A172663 R. H. Hardin, <a href="b172663.txt">Table of n, a(n) for n=1..11</a>
%K A172663 nonn,new
%O A172663 1,4
%A A172663 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172662
%S A172662 1,19,5881,115321795,101471705778601,5520471267708881730181,
%T A172662 17949532724881770551236183389177,
%U A172662 3781994369388530584295658367829503216546371
%N A172662 Number of 2*n X n 0..2 arrays with row sums n and column sums 2*n
%H A172662 R. H. Hardin, <a href="b172662.txt">Table of n, a(n) for n=1..10</a>
%K A172662 nonn,new
%O A172662 1,2
%A A172662 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172661
%S A172661 20,16260,16214000,20982583300,31091880269520,50432680389260400,
%T A172661 87186713382942888000,158040497103890978362500,
%U A172661 297155971199028097720850000,575241665325478123466841078160
%N A172661 Number of 2*n X 6 0..2 arrays with row sums 3 and column sums n
%H A172661 R. H. Hardin, <a href="b172661.txt">Table of n, a(n) for n=1..49</a>
%K A172661 nonn,new
%O A172661 1,1
%A A172661 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172660
%S A172660 6,282,16260,1093050,79796556,6152037276,492872166072,40629560387130,
%T A172660 3423871305383100,293631119403639732,25543657090245824712,
%U A172660 2248575441654260591964,199926155194724792622600
%N A172660 Number of 2*n X 4 0..2 arrays with row sums 2 and column sums n
%H A172660 R. H. Hardin, <a href="b172660.txt">Table of n, a(n) for n=1..49</a>
%K A172660 nonn,new
%O A172660 1,1
%A A172660 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172659
%S A172659 70,1093050,20982583300,589985590963930,20033574566576932620,
%T A172659 772712857320523525028700,32594623818348634068046620600,
%U A172659 1469515974755674798685294756444250
%N A172659 Number of 2*n X 8 0..2 arrays with row sums 4 and column sums n
%H A172659 R. H. Hardin, <a href="b172659.txt">Table of n, a(n) for n=1..31</a>
%K A172659 nonn,new
%O A172659 1,1
%A A172659 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172658
%S A172658 3432,492872166072,87186713382942888000,32594623818348634068046620600,
%T A172658 17207115363089145896738850914404036032,
%U A172658 11524700002163865806460331965810529276252403904
%N A172658 Number of 2*n X 14 0..2 arrays with row sums 7 and column sums n
%H A172658 R. H. Hardin, <a href="b172658.txt">Table of n, a(n) for n=1..12</a>
%K A172658 nonn,new
%O A172658 1,1
%A A172658 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172657
%S A172657 924,6152037276,50432680389260400,772712857320523525028700,
%T A172657 15872049782314898323485642439824,
%U A172657 398638294522125203247923680009132025136
%N A172657 Number of 2*n X 12 0..2 arrays with row sums 6 and column sums n
%H A172657 R. H. Hardin, <a href="b172657.txt">Table of n, a(n) for n=1..17</a>
%K A172657 nonn,new
%O A172657 1,1
%A A172657 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172656
%S A172656 252,79796556,31091880269520,20033574566576932620,
%T A172656 16426290801521822874931152,15872049782314898323485642439824,
%U A172656 17207115363089145896738850914404036032
%N A172656 Number of 2*n X 10 0..2 arrays with row sums 5 and column sums n
%H A172656 R. H. Hardin, <a href="b172656.txt">Table of n, a(n) for n=1..22</a>
%K A172656 nonn,new
%O A172656 1,1
%A A172656 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172655
%S A172655 0,0,0,0,0,1,18126466426218150,1469515974755674798685294756444250,
%T A172655 285750466619523879017857522597582662923413783276224,
%U A172655 224061277556863094343442873695698306048075505046254925957562566026000
%N A172655 Number of 2*n X n 0..2 arrays with row sums 12 and column sums 24
%K A172655 nonn,new
%O A172655 1,7
%A A172655 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172654
%S A172654 0,0,0,1,1328792850,35265186498828525525,5001985846592510105642103476685,
%T A172654 3781994369388530584295658367829503216546371,
%U A172654 13935136089336578003371937448487249553674496163189003360
%N A172654 Number of 2*n X n 0..2 arrays with row sums 8 and column sums 16
%H A172654 R. H. Hardin, <a href="b172654.txt">Table of n, a(n) for n=1..12</a>
%K A172654 nonn,new
%O A172654 1,5
%A A172654 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172653
%S A172653 0,1,2385,115321795,31237467004800,35265186498828525525,
%T A172653 134169254007831527780970675,1469515974755674798685294756444250,
%U A172653 41035736539141994695629689682970685292000
%N A172653 Number of 2*n X n 0..2 arrays with row sums 4 and column sums 8
%H A172653 R. H. Hardin, <a href="b172653.txt">Table of n, a(n) for n=1..24</a>
%K A172653 nonn,new
%O A172653 1,3
%A A172653 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172652
%S A172652 0,0,0,2520,755265546000,1563133410663239958840,
%T A172652 17949532724881770551236183389177,
%U A172652 1021718304724615708073668620385984867252800
%N A172652 Number of 2*n X n 0..2 arrays with row sums 7 and column sums 14
%H A172652 R. H. Hardin, <a href="b172652.txt">Table of n, a(n) for n=1..13</a>
%K A172652 nonn,new
%O A172652 1,4
%A A172652 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172651
%S A172651 1,19,2385,1093050,1328792850,3536978063850,18126466426218150,
%T A172651 163081394186253543000,2402820978940192425615000,
%U A172651 54918587341306311174536985000,1864314763102041137068549803435000
%N A172651 Number of 2*n X n 0..2 arrays with row sums 2 and column sums 4
%H A172651 R. H. Hardin, <a href="b172651.txt">Table of n, a(n) for n=1..49</a>
%K A172651 nonn,new
%O A172651 1,2
%A A172651 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172650
%S A172650 0,0,90,34145664,101471705778601,1563133410663239958840,
%T A172650 101720498887190010630546323580,23601511081513772065907556838530960960,
%U A172650 17017920144305424073117333502327304544668289632
%N A172650 Number of 2*n X n 0..2 arrays with row sums 5 and column sums 10
%H A172650 R. H. Hardin, <a href="b172650.txt">Table of n, a(n) for n=1..18</a>
%K A172650 nonn,new
%O A172650 1,3
%A A172650 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172649
%S A172649 0,6,5881,34145664,755265546000,50432680389260400,8610740229284196003000,
%T A172649 3321735626954282018581900800,2631822682451717147804985743769600,
%U A172649 3970498303510674585925708059712525440000
%N A172649 Number of 2*n X n 0..2 arrays with row sums 3 and column sums 6
%H A172649 R. H. Hardin, <a href="b172649.txt">Table of n, a(n) for n=1..41</a>
%K A172649 nonn,new
%O A172649 1,2
%A A172649 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172648
%S A172648 0,0,0,0,113400,50432680389260400,101720498887190010630546323580,
%T A172648 1021718304724615708073668620385984867252800,
%U A172648 52487498990202837315687104908805779583934003413314100425
%N A172648 Number of 2*n X n 0..2 arrays with row sums 9 and column sums 18
%H A172648 R. H. Hardin, <a href="b172648.txt">Table of n, a(n) for n=1..11</a>
%K A172648 nonn,new
%O A172648 1,5
%A A172648 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172647
%S A172647 0,0,1,1093050,31237467004800,5520471267708881730181,
%T A172647 5001985846592510105642103476685,
%U A172647 19578536057961112914067888563728861113920
%N A172647 Number of 2*n X n 0..2 arrays with row sums 6 and column sums 12
%H A172647 R. H. Hardin, <a href="b172647.txt">Table of n, a(n) for n=1..15</a>
%K A172647 nonn,new
%O A172647 1,4
%A A172647 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172646
%S A172646 0,0,0,0,1,3536978063850,134169254007831527780970675,
%T A172646 19578536057961112914067888563728861113920,
%U A172646 13935136089336578003371937448487249553674496163189003360
%N A172646 Number of 2*n X n 0..2 arrays with row sums 10 and column sums 20
%H A172646 R. H. Hardin, <a href="b172646.txt">Table of n, a(n) for n=1..10</a>
%K A172646 nonn,new
%O A172646 1,6
%A A172646 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172645
%S A172645 1,3,31,2371,1084851,3680774301,91358224634433,17470072106054582211,
%T A172645 26088526624958727703324771,310419652143758898175543447421953,
%U A172645 29785621316391113552729016416250323294253
%N A172645 Number of n X n 0..2 arrays with row sums n and column sums n
%H A172645 R. H. Hardin, <a href="b172645.txt">Table of n, a(n) for n=1..14</a>
%K A172645 nonn,new
%O A172645 1,2
%A A172645 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172644
%S A172644 1,25653,55644337,499616140189,4307041366439901,48321459222257750541,
%T A172644 596268326872362183077193,8132104885895774387439086637,
%U A172644 118790606468949916540947848175709,1838094093862096792566838713252275953
%N A172644 Number of n X 11 0..2 arrays with row sums 11 and column sums n
%H A172644 R. H. Hardin, <a href="b172644.txt">Table of n, a(n) for n=1..20</a>
%K A172644 nonn,new
%O A172644 1,2
%A A172644 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172643
%S A172643 1,212941,2320792657,141444270208645,8107975032968711941,
%T A172643 641062514242765723819621,56688799539093817446748651801,
%U A172643 5657362648396875453954444188314837
%N A172643 Number of n X 13 0..2 arrays with row sums 13 and column sums n
%H A172643 R. H. Hardin, <a href="b172643.txt">Table of n, a(n) for n=1..16</a>
%K A172643 nonn,new
%O A172643 1,2
%A A172643 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172642
%S A172642 1,19,175,2371,32611,481381,7343449,115321795,1849858771,30189264889,
%T A172642 499616140189,8365213132981,141444270208645,2411829493241299,
%U A172642 41425627536496495,716068213864217155,12447291755448613315
%N A172642 Number of n X 4 0..2 arrays with row sums 4 and column sums n
%H A172642 R. H. Hardin, <a href="b172642.txt">Table of n, a(n) for n=1..99</a>
%K A172642 nonn,new
%O A172642 1,2
%A A172642 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172641
%S A172641 1,3139,1376095,1849858771,2435292751411,3938448319935781,
%T A172641 6887820148172502169,13039980402350138484115,26088526624958727703324771,
%U A172641 54649595712626140857519780409,118790606468949916540947848175709
%N A172641 Number of n X 9 0..2 arrays with row sums 9 and column sums n
%H A172641 R. H. Hardin, <a href="b172641.txt">Table of n, a(n) for n=1..30</a>
%K A172641 nonn,new
%O A172641 1,2
%A A172641 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172640
%S A172640 1,73789,358198369,8365213132981,185658731463324901,
%T A172640 5520471267708881730181,182062424087215419645579529,
%U A172640 6705797924085107855686463036869,266324891937340036349722115366403541
%N A172640 Number of n X 12 0..2 arrays with row sums 12 and column sums n
%H A172640 R. H. Hardin, <a href="b172640.txt">Table of n, a(n) for n=1..18</a>
%K A172640 nonn,new
%O A172640 1,2
%A A172640 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172639
%S A172639 1,51,991,32611,1084851,39612501,1509893001,59794281891,2435292751411,
%T A172639 101471705778601,4307041366439901,185658731463324901,8107975032968711941,
%U A172639 358060848206529563811,15966070888767791932191,717973394997481520889891
%N A172639 Number of n X 5 0..2 arrays with row sums 5 and column sums n
%H A172639 R. H. Hardin, <a href="b172639.txt">Table of n, a(n) for n=1..99</a>
%K A172639 nonn,new
%O A172639 1,2
%A A172639 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172638
%S A172638 1,1107,219871,115321795,59794281891,37015866368181,24567498558526617,
%T A172638 17470072106054582211,13039980402350138484115,10132345044562368521279737,
%U A172638 8132104885895774387439086637,6705797924085107855686463036869
%N A172638 Number of n X 8 0..2 arrays with row sums 8 and column sums n
%H A172638 R. H. Hardin, <a href="b172638.txt">Table of n, a(n) for n=1..37</a>
%K A172638 nonn,new
%O A172638 1,2
%A A172638 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172637
%S A172637 1,8953,8710537,30189264889,101471705778601,431222728237019041,
%T A172637 1998635481349606292593,10132345044562368521279737,
%U A172637 54649595712626140857519780409,310419652143758898175543447421953
%N A172637 Number of n X 10 0..2 arrays with row sums 10 and column sums n
%H A172637 R. H. Hardin, <a href="b172637.txt">Table of n, a(n) for n=1..25</a>
%K A172637 nonn,new
%O A172637 1,2
%A A172637 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172636
%S A172636 1,393,35617,7343449,1509893001,360255871641,91358224634433,
%T A172636 24567498558526617,6887820148172502169,1998635481349606292593,
%U A172636 596268326872362183077193,182062424087215419645579529
%N A172636 Number of n X 7 0..2 arrays with row sums 7 and column sums n
%H A172636 R. H. Hardin, <a href="b172636.txt">Table of n, a(n) for n=1..47</a>
%K A172636 nonn,new
%O A172636 1,2
%A A172636 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172635
%S A172635 1,616227,15120204295,2411829493241299,358060848206529563811,
%T A172635 75490918169569448369461821,17949532724881770551236183389177,
%U A172635 4867117722741777809293028167592172435
%N A172635 Number of n X 14 0..2 arrays with row sums 14 and column sums n
%H A172635 R. H. Hardin, <a href="b172635.txt">Table of n, a(n) for n=1..14</a>
%K A172635 nonn,new
%O A172635 1,2
%A A172635 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172634
%S A172634 1,7,31,175,991,5881,35617,219871,1376095,8710537,55644337,358198369,
%T A172634 2320792657,15120204295,98984058271,650725327231,4293779332927,
%U A172634 28425752310361,188739799967425,1256510215733185,8385127334900305
%N A172634 Number of n X 3 0..2 arrays with row sums 3 and column sums n
%H A172634 R. H. Hardin, <a href="b172634.txt">Table of n, a(n) for n=1..99</a>
%K A172634 nonn,new
%O A172634 1,2
%A A172634 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172633
%S A172633 1,141,5881,481381,39612501,3680774301,360255871641,37015866368181,
%T A172633 3938448319935781,431222728237019041,48321459222257750541,
%U A172633 5520471267708881730181,641062514242765723819621
%N A172633 Number of n X 6 0..2 arrays with row sums 6 and column sums n
%H A172633 R. H. Hardin, <a href="b172633.txt">Table of n, a(n) for n=1..65</a>
%K A172633 nonn,new
%O A172633 1,2
%A A172633 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172632
%S A172632 0,0,0,0,0,0,0,40320,655551508577280,16426290801521822874931152,
%T A172632 682948436592331311886782349445512800,
%U A172632 56889017685903899457289215621889880980797696000
%N A172632 Number of n X n 0..2 arrays with row sums 15 and column sums 15
%H A172632 R. H. Hardin, <a href="b172632.txt">Table of n, a(n) for n=1..13</a>
%K A172632 nonn,new
%O A172632 1,8
%A A172632 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172631
%S A172631 0,0,0,0,0,0,0,1,41514583320,6736902064011471514050,
%T A172631 1381222121393845839281152868103600,
%U A172631 487937255272130398045688257841450313043450725
%N A172631 Number of n X n 0..2 arrays with row sums 16 and column sums 16
%H A172631 R. H. Hardin, <a href="b172631.txt">Table of n, a(n) for n=1..13</a>
%K A172631 nonn,new
%O A172631 1,9
%A A172631 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172630
%S A172630 0,0,0,0,0,0,0,0,1,3930730108200,41169485539316907099731250,
%T A172630 398638294522125203247923680009132025136,
%U A172630 5601898113262596673984396810254752776141437331611270
%N A172630 Number of n X n 0..2 arrays with row sums 18 and column sums 18
%K A172630 nonn,new
%O A172630 1,10
%A A172630 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172629
%S A172629 0,0,0,1,6210,342829125,49846153939785,17470072106054582211,
%T A172629 13807842729396124460629536,23086954810641629693375704674000,
%U A172629 77065768681699399471388989062347000400
%N A172629 Number of n X n 0..2 arrays with row sums 8 and column sums 8
%H A172629 R. H. Hardin, <a href="b172629.txt">Table of n, a(n) for n=1..18</a>
%K A172629 nonn,new
%O A172629 1,5
%A A172629 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172628
%S A172628 0,0,0,24,111920,2030271480,91358224634433,9350902053107858880,
%T A172628 2018328183128437950016800,859819730886010540636567920000,
%U A172628 682948436592331311886782349445512800
%N A172628 Number of n X n 0..2 arrays with row sums 7 and column sums 7
%H A172628 R. H. Hardin, <a href="b172628.txt">Table of n, a(n) for n=1..21</a>
%K A172628 nonn,new
%O A172628 1,4
%A A172628 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172627
%S A172627 0,0,1,282,628080,3680774301,49846153939785,1418365816378741440,
%T A172627 78459796857685317493440,7915651642356094020231766800,
%U A172627 1381222121393845839281152868103600
%N A172627 Number of n X n 0..2 arrays with row sums 6 and column sums 6
%H A172627 R. H. Hardin, <a href="b172627.txt">Table of n, a(n) for n=1..25</a>
%K A172627 nonn,new
%O A172627 1,4
%A A172627 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172626
%S A172626 0,0,0,0,120,16214000,7873709050620,9350902053107858880,
%T A172626 26088526624958727703324771,162526525004284727135887319788800,
%U A172626 2143395673887060254676185406326252145600
%N A172626 Number of n X n 0..2 arrays with row sums 9 and column sums 9
%H A172626 R. H. Hardin, <a href="b172626.txt">Table of n, a(n) for n=1..17</a>
%K A172626 nonn,new
%O A172626 1,5
%A A172626 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172625
%S A172625 0,0,6,1344,1084851,2030271480,7873709050620,57952896096182976,
%T A172625 754663198156942150944,16426290801521822874931152,
%U A172625 570338548776486677507756325120,30378597441131419114905188023754880
%N A172625 Number of n X n 0..2 arrays with row sums 5 and column sums 5
%H A172625 R. H. Hardin, <a href="b172625.txt">Table of n, a(n) for n=1..35</a>
%K A172625 nonn,new
%O A172625 1,3
%A A172625 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172623
%S A172623 0,0,0,0,1,202410,344827336455,1418365816378741440,
%T A172623 13807842729396124460629536,310419652143758898175543447421953,
%U A172623 15464071313054035722739424623204673044920
%N A172623 Number of n X n 0..2 arrays with row sums 10 and column sums 10
%H A172623 R. H. Hardin, <a href="b172623.txt">Table of n, a(n) for n=1..15</a>
%K A172623 nonn,new
%O A172623 1,6
%A A172623 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172622
%S A172622 0,1,21,2371,628080,342829125,344827336455,589985590963930,
%T A172622 1614966163545077280,6736902064011471514050,41169485539316907099731250,
%U A172622 356683733303639928299266819050,4261817244676586430350506540348800
%N A172622 Number of n X n 0..2 arrays with row sums 4 and column sums 4
%H A172622 R. H. Hardin, <a href="b172622.txt">Table of n, a(n) for n=1..56</a>
%K A172622 nonn,new
%O A172622 1,3
%A A172622 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172621
%S A172621 0,0,0,0,0,720,3758757240,57952896096182976,2018328183128437950016800,
%T A172621 162526525004284727135887319788800,
%U A172621 29785621316391113552729016416250323294253
%N A172621 Number of n X n 0..2 arrays with row sums 11 and column sums 11
%H A172621 R. H. Hardin, <a href="b172621.txt">Table of n, a(n) for n=1..14</a>
%K A172621 nonn,new
%O A172621 1,6
%A A172621 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172620
%S A172620 0,0,0,0,0,1,9135630,589985590963930,78459796857685317493440,
%T A172620 23086954810641629693375704674000,
%U A172620 15464071313054035722739424623204673044920
%N A172620 Number of n X n 0..2 arrays with row sums 12 and column sums 12
%H A172620 R. H. Hardin, <a href="b172620.txt">Table of n, a(n) for n=1..14</a>
%K A172620 nonn,new
%O A172620 1,7
%A A172620 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172619
%S A172619 0,0,0,0,0,0,1,545007960,1614966163545077280,
%T A172619 7915651642356094020231766800,77065768681699399471388989062347000400,
%U A172619 1630901058155542715672439839825819503867780061400
%N A172619 Number of n X n 0..2 arrays with row sums 14 and column sums 14
%H A172619 R. H. Hardin, <a href="b172619.txt">Table of n, a(n) for n=1..14</a>
%K A172619 nonn,new
%O A172619 1,8
%A A172619 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172618
%S A172618 0,0,0,0,0,0,5040,1310799454720,754663198156942150944,
%T A172618 859819730886010540636567920000,2143395673887060254676185406326252145600,
%U A172618 12011244510939290610945342901003078567310643860160
%N A172618 Number of n X n 0..2 arrays with row sums 13 and column sums 13
%H A172618 R. H. Hardin, <a href="b172618.txt">Table of n, a(n) for n=1..13</a>
%K A172618 nonn,new
%O A172618 1,7
%A A172618 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172617
%S A172617 14398171200,109829050417159537464000,
%T A172617 2343976695927269878444049332219968000,
%U A172617 77695253876546044332382434342006457448342441400000
%N A172617 Number of 9*n X 9 0..1 arrays with row sums 2 and column sums 2*n
%H A172617 R. H. Hardin, <a href="b172617.txt">Table of n, a(n) for n=1..11</a>
%K A172617 nonn,new
%O A172617 1,1
%A A172617 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172616
%S A172616 1,842090474940,9561215418596022668009737500,
%T A172616 41499275921837441873503310658937755719615400000,
%U A172616 15806556884633733578544972910460376919497070823113887278535399000000
%N A172616 Number of 9*n X 2*n 0..1 arrays with row sums 2 and column sums 9
%H A172616 R. H. Hardin, <a href="b172616.txt">Table of n, a(n) for n=1..11</a>
%K A172616 nonn,new
%O A172616 1,2
%A A172616 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172615
%S A172615 12504636144000,3237415247416050491577971184000,
%T A172615 6055976192395031960092036887782708145734400000000,
%U A172615 27298383003921247460630029918586935761007187118066456599356950000000
%N A172615 Number of 9*n X 18 0..1 arrays with row sums 2 and column sums n
%H A172615 R. H. Hardin, <a href="b172615.txt">Table of n, a(n) for n=1..10</a>
%K A172615 nonn,new
%O A172615 1,1
%A A172615 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172614
%S A172614 0,1,227873431500,30862498453931119524941700,
%T A172614 303467065300162961503136931771371140440000,
%U A172614 98744132720453916340813668495424059077053759331765698612500
%N A172614 Number of 9*n X n 0..1 arrays with row sums 2 and column sums 18
%H A172614 R. H. Hardin, <a href="b172614.txt">Table of n, a(n) for n=1..10</a>
%K A172614 nonn,new
%O A172614 1,3
%A A172614 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172613
%S A172613 1,48620,227873431500,21452752266265320000,19010638202652030712978200000,
%T A172613 101097362223624462291180422369532000000,
%U A172613 2392741010223442438553822446842770682716580000000
%N A172613 Number of 9*n X n 0..1 arrays with row sums 1 and column sums 9
%H A172613 R. H. Hardin, <a href="b172613.txt">Table of n, a(n) for n=1..11</a>
%K A172613 nonn,new
%O A172613 1,2
%A A172613 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172612
%S A172612 24046189440,231528006523766358806400,
%T A172612 5638345005774311804302107347272704000,
%U A172612 203269774721184658406475682049947158681837795504000
%N A172612 Number of 8*n X 8 0..1 arrays with row sums 3 and column sums 3*n
%H A172612 R. H. Hardin, <a href="b172612.txt">Table of n, a(n) for n=1..10</a>
%K A172612 nonn,new
%O A172612 1,1
%A A172612 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172611
%S A172611 81729648000,36574751938491748341360000,
%T A172611 93453554057243260025029337978773248000000,
%U A172611 518468101226735093154139029192029181133104846891750000000
%N A172611 Number of 8*n X 16 0..1 arrays with row sums 2 and column sums n
%H A172611 R. H. Hardin, <a href="b172611.txt">Table of n, a(n) for n=1..11</a>
%K A172611 nonn,new
%O A172611 1,1
%A A172611 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172610
%S A172610 0,1,9465511770,28342398385058078078010,
%T A172610 3829361053685468612006211542840385000,
%U A172610 11596575535834069329340945743908684169826573155948750
%N A172610 Number of 8*n X n 0..1 arrays with row sums 2 and column sums 16
%H A172610 R. H. Hardin, <a href="b172610.txt">Table of n, a(n) for n=1..11</a>
%K A172610 nonn,new
%O A172610 1,3
%A A172610 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172609
%S A172609 1,12870,9465511770,99561092450391000,7656714453153197981835000,
%T A172609 2889253496242619386328267523990000,
%U A172609 4104167472585675600759440022842715359250000
%N A172609 Number of 8*n X n 0..1 arrays with row sums 1 and column sums 8
%H A172609 R. H. Hardin, <a href="b172609.txt">Table of n, a(n) for n=1..12</a>
%K A172609 nonn,new
%O A172609 1,2
%A A172609 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172608
%S A172608 68938800,617944896216648000,12218892645550878420052920000,
%T A172608 337799058758556439723480176591336120000,
%U A172608 11251027489082268563385494644966780470231583948800
%N A172608 Number of 7*n X 7 0..1 arrays with row sums 3 and column sums 3*n
%H A172608 R. H. Hardin, <a href="b172608.txt">Table of n, a(n) for n=1..13</a>
%K A172608 nonn,new
%O A172608 1,1
%A A172608 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172607
%S A172607 1,1026608414145600,58142871081888953760431694254592000,
%T A172607 3201055426099561623149772782827411540384767799931904000000,
%U A172607 32039970364449358738657145679393908929276464603866377637139985914971871928320000000
%N A172607 Number of 7*n X 3*n 0..1 arrays with row sums 3 and column sums 7
%H A172607 R. H. Hardin, <a href="b172607.txt">Table of n, a(n) for n=1..11</a>
%K A172607 nonn,new
%O A172607 1,2
%A A172607 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172606
%S A172606 3110940,807998767676100,452489654840076972518400,
%T A172606 352231966693098849132080247442500,
%U A172606 329566582178898535416815000275618647613440
%N A172606 Number of 7*n X 7 0..1 arrays with row sums 2 and column sums 2*n
%H A172606 R. H. Hardin, <a href="b172606.txt">Table of n, a(n) for n=1..14</a>
%K A172606 nonn,new
%O A172606 1,1
%A A172606 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172605
%S A172605 1,936369720,1548539246648239560000,259207529217195001892051045386944000,
%T A172605 1401029485328289844705736395976227319651581140480000,
%U A172605 122551057241825639587910301883432838920696717566795677090154147840000
%N A172605 Number of 7*n X 2*n 0..1 arrays with row sums 2 and column sums 7
%H A172605 R. H. Hardin, <a href="b172605.txt">Table of n, a(n) for n=1..14</a>
%K A172605 nonn,new
%O A172605 1,2
%A A172605 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172604
%S A172604 681080400,676508133623135814000,3031982831164890119435183865600000,
%T A172604 26585119285914456883176777688331471089227750000,
%U A172604 341296897608554687055256179810792324655491010769411003750400
%N A172604 Number of 7*n X 14 0..1 arrays with row sums 2 and column sums n
%H A172604 R. H. Hardin, <a href="b172604.txt">Table of n, a(n) for n=1..13</a>
%K A172604 nonn,new
%O A172604 1,1
%A A172604 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172603
%S A172603 1,3432,399072960,472518347558400,3177459078523411968000,
%T A172603 85722533226982363751829504000,7363615666157189603982585462030336000,
%U A172603 1707750599894443404262670865631874246246400000
%N A172603 Number of 7*n X n 0..1 arrays with row sums 1 and column sums 7
%H A172603 R. H. Hardin, <a href="b172603.txt">Table of n, a(n) for n=1..14</a>
%K A172603 nonn,new
%O A172603 1,2
%A A172603 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172602
%S A172602 0,1,399072960,26630804377937061000,49825573548689359631837113344000,
%T A172602 1415189158639246716651027917944817871202200000,
%U A172602 396138136990560832867276344563606859994639454544654153984000
%N A172602 Number of 7*n X n 0..1 arrays with row sums 2 and column sums 14
%H A172602 R. H. Hardin, <a href="b172602.txt">Table of n, a(n) for n=1..13</a>
%K A172602 nonn,new
%O A172602 1,3
%A A172602 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172601
%S A172601 0,0,1,472518347558400,49825573548689359631837113344000,
%T A172601 187524788254217999458899339512832890227588026696000,
%U A172601 17188521061665681913347495181567174669437929715236078389999795132800000
%N A172601 Number of 7*n X n 0..1 arrays with row sums 3 and column sums 21
%H A172601 R. H. Hardin, <a href="b172601.txt">Table of n, a(n) for n=1..10</a>
%K A172601 nonn,new
%O A172601 1,4
%A A172601 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172600
%S A172600 137225088000,229747284991066934931840000,
%T A172600 2767806480542211571651550187279222472704000,
%U A172600 80261625461932627702056015943137363301590520498062576000000
%N A172600 Number of 6*n X 18 0..1 arrays with row sums 3 and column sums n
%H A172600 R. H. Hardin, <a href="b172600.txt">Table of n, a(n) for n=1..11</a>
%K A172600 nonn,new
%O A172600 1,1
%A A172600 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172599
%S A172599 7484400,21959547410077200,229747284991066934931840000,
%T A172599 4237239732072431006302896746240010000,
%U A172599 107858549105202487690102571993535153527817734400
%N A172599 Number of 6*n X 12 0..1 arrays with row sums 2 and column sums n
%H A172599 R. H. Hardin, <a href="b172599.txt">Table of n, a(n) for n=1..15</a>
%K A172599 nonn,new
%O A172599 1,1
%A A172599 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172598
%S A172598 0,0,1,2308743493056,674741194624071430134879600,
%T A172598 4286480050914725856131217361234536200970000,
%U A172598 422006703164863149555124419847097220761027785050113408280000
%N A172598 Number of 6*n X n 0..1 arrays with row sums 3 and column sums 18
%H A172598 R. H. Hardin, <a href="b172598.txt">Table of n, a(n) for n=1..11</a>
%K A172598 nonn,new
%O A172598 1,4
%A A172598 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172597
%S A172597 0,1,17153136,25780447171287900,674741194624071430134879600,
%T A172597 181567587344159723781226957237357470000,
%U A172597 346851737504423205666773757122219688261768787084800
%N A172597 Number of 6*n X n 0..1 arrays with row sums 2 and column sums 12
%H A172597 R. H. Hardin, <a href="b172597.txt">Table of n, a(n) for n=1..15</a>
%K A172597 nonn,new
%O A172597 1,3
%A A172597 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172596
%S A172596 1,273957842462220,2460737690938850798370391240644900,
%T A172596 15195407946323194013765595656097788013666363361791048000,
%U A172596 13223617955495651593814858107477805060086684838739019514933477474642809905360000
%N A172596 Number of 5*n X 4*n 0..1 arrays with row sums 4 and column sums 5
%H A172596 R. H. Hardin, <a href="b172596.txt">Table of n, a(n) for n=1..10</a>
%K A172596 nonn,new
%O A172596 1,2
%A A172596 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172595
%S A172595 1,14367744720,607807501825156719166080,
%T A172595 3502030404249297872766171625530047385600,
%U A172595 826383627873354098496906047829953892578069546869459968000
%N A172595 Number of 5*n X 3*n 0..1 arrays with row sums 3 and column sums 5
%H A172595 R. H. Hardin, <a href="b172595.txt">Table of n, a(n) for n=1..18</a>
%K A172595 nonn,new
%O A172595 1,2
%A A172595 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172594
%S A172594 2040,56586600,2604964362000,149144498504001000,9647422924194982967040,
%T A172594 674741194624071430134879600,49825573548689359631837113344000,
%U A172594 3829361053685468612006211542840385000
%N A172594 Number of 5*n X 5 0..1 arrays with row sums 2 and column sums 2*n
%H A172594 R. H. Hardin, <a href="b172594.txt">Table of n, a(n) for n=1..19</a>
%K A172594 nonn,new
%O A172594 1,1
%A A172594 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172593
%S A172593 56586600,1289144584143523800,91880966659006902855367890000,
%T A172593 10697695830125272790880922760946584535000,
%U A172593 1639181069042816463750016605763302490015414336761600
%N A172593 Number of 5*n X 10 0..1 arrays with row sums 4 and column sums 2*n
%H A172593 R. H. Hardin, <a href="b172593.txt">Table of n, a(n) for n=1..11</a>
%K A172593 nonn,new
%O A172593 1,1
%A A172593 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172592
%S A172592 0,1,306407299538340,74580431332898028185458459628980,
%T A172592 1639181069042816463750016605763302490015414336761600,
%U A172592 1589885820403830809743559212333580058709307314045492378010426129831107500
%N A172592 Number of 5*n X 2*n 0..1 arrays with row sums 4 and column sums 10
%H A172592 R. H. Hardin, <a href="b172592.txt">Table of n, a(n) for n=1..10</a>
%K A172592 nonn,new
%O A172592 1,3
%A A172592 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172591
%S A172591 1,1172556,306407299538340,2144953893641078315315520,
%T A172591 178394712594906480448637769546038400,
%U A172591 107858549105202487690102571993535153527817734400
%N A172591 Number of 5*n X 2*n 0..1 arrays with row sums 2 and column sums 5
%H A172591 R. H. Hardin, <a href="b172591.txt">Table of n, a(n) for n=1..19</a>
%K A172591 nonn,new
%O A172591 1,2
%A A172591 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172590
%S A172590 168168000,46764764308702440000,65662040698002721810659005184000,
%T A172590 189746842198224628153363826062842999921000000,
%U A172590 826383627873354098496906047829953892578069546869459968000
%N A172590 Number of 5*n X 15 0..1 arrays with row sums 3 and column sums n
%H A172590 R. H. Hardin, <a href="b172590.txt">Table of n, a(n) for n=1..13</a>
%K A172590 nonn,new
%O A172590 1,1
%A A172590 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172589
%S A172589 113400,1371785398200,46764764308702440000,2532230252503738514963235000,
%T A172589 178394712594906480448637769546038400,
%U A172589 14896749360215409445279324463510987496360000
%N A172589 Number of 5*n X 10 0..1 arrays with row sums 2 and column sums n
%H A172589 R. H. Hardin, <a href="b172589.txt">Table of n, a(n) for n=1..19</a>
%K A172589 nonn,new
%O A172589 1,1
%A A172589 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172588
%S A172588 305540235000,2532230252503738514963235000,
%T A172588 189746842198224628153363826062842999921000000,
%U A172588 37911589613425952733393718264069147678877877626169022024515000
%N A172588 Number of 5*n X 20 0..1 arrays with row sums 4 and column sums n
%H A172588 R. H. Hardin, <a href="b172588.txt">Table of n, a(n) for n=1..10</a>
%K A172588 nonn,new
%O A172588 1,1
%A A172588 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172587
%S A172587 0,0,1,11732745024,9647422924194982967040,
%T A172587 104923772318750486122754981962387200,
%U A172587 11251027489082268563385494644966780470231583948800
%N A172587 Number of 5*n X n 0..1 arrays with row sums 3 and column sums 15
%H A172587 R. H. Hardin, <a href="b172587.txt">Table of n, a(n) for n=1..13</a>
%K A172587 nonn,new
%O A172587 1,4
%A A172587 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172586
%S A172586 0,1,756756,25989269017140,9647422924194982967040,
%T A172586 24935177268489106332174087326700,
%U A172586 329566582178898535416815000275618647613440
%N A172586 Number of 5*n X n 0..1 arrays with row sums 2 and column sums 10
%H A172586 R. H. Hardin, <a href="b172586.txt">Table of n, a(n) for n=1..19</a>
%K A172586 nonn,new
%O A172586 1,3
%A A172586 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172585
%S A172585 0,0,0,1,623360743125120,24935177268489106332174087326700,
%T A172585 11251027489082268563385494644966780470231583948800,
%U A172585 54940328391880874295213395391276439149434232556146955107851631952500
%N A172585 Number of 5*n X n 0..1 arrays with row sums 4 and column sums 20
%H A172585 R. H. Hardin, <a href="b172585.txt">Table of n, a(n) for n=1..10</a>
%K A172585 nonn,new
%O A172585 1,5
%A A172585 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172584
%S A172584 1,60871300,2407147216735338000,4847907059804908291247055000000,
%T A172584 189746842198224628153363826062842999921000000,
%U A172584 80261625461932627702056015943137363301590520498062576000000
%N A172584 Number of 4*n X 3*n 0..1 arrays with row sums 3 and column sums 4
%H A172584 R. H. Hardin, <a href="b172584.txt">Table of n, a(n) for n=1..24</a>
%K A172584 nonn,new
%O A172584 1,2
%A A172584 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172583
%S A172583 63063000,7373018003758407000,4847907059804908291247055000000,
%T A172583 6892692735539278753058456514221737762215000,
%U A172583 15195407946323194013765595656097788013666363361791048000
%N A172583 Number of 4*n X 16 0..1 arrays with row sums 4 and column sums n
%H A172583 R. H. Hardin, <a href="b172583.txt">Table of n, a(n) for n=1..12</a>
%K A172583 nonn,new
%O A172583 1,1
%A A172583 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172582
%S A172582 369600,31082452632000,9254768770160124288000,
%T A172582 4847907059804908291247055000000,
%U A172582 3502030404249297872766171625530047385600
%N A172582 Number of 4*n X 12 0..1 arrays with row sums 3 and column sums n
%H A172582 R. H. Hardin, <a href="b172582.txt">Table of n, a(n) for n=1..16</a>
%K A172582 nonn,new
%O A172582 1,1
%A A172582 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172581
%S A172581 11732745024,2144953893641078315315520,
%T A172581 3502030404249297872766171625530047385600,
%U A172581 15195407946323194013765595656097788013666363361791048000
%N A172581 Number of 4*n X 20 0..1 arrays with row sums 5 and column sums n
%H A172581 R. H. Hardin, <a href="b172581.txt">Table of n, a(n) for n=1..10</a>
%K A172581 nonn,new
%O A172581 1,1
%A A172581 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172580
%S A172580 2520,187530840,31082452632000,7373018003758407000,
%T A172580 2144953893641078315315520,712008996110160366168717566400,
%U A172580 259207529217195001892051045386944000
%N A172580 Number of 4*n X 8 0..1 arrays with row sums 2 and column sums n
%H A172580 R. H. Hardin, <a href="b172580.txt">Table of n, a(n) for n=1..24</a>
%K A172580 nonn,new
%O A172580 1,1
%A A172580 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172579
%S A172579 0,1,34650,27770358330,149144498504001000,3776577900841430197548750,
%T A172579 352231966693098849132080247442500,
%U A172579 101028698009849992018229820380077899975000
%N A172579 Number of 4*n X n 0..1 arrays with row sums 2 and column sums 8
%H A172579 R. H. Hardin, <a href="b172579.txt">Table of n, a(n) for n=1..24</a>
%K A172579 nonn,new
%O A172579 1,3
%A A172579 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172578
%S A172578 0,0,0,1,305540235000,3776577900841430197548750,
%T A172578 337799058758556439723480176591336120000,
%U A172578 205853267551222811542072611337854177788999924948228250
%N A172578 Number of 4*n X n 0..1 arrays with row sums 4 and column sums 16
%H A172578 R. H. Hardin, <a href="b172578.txt">Table of n, a(n) for n=1..13</a>
%K A172578 nonn,new
%O A172578 1,5
%A A172578 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172577
%S A172577 0,0,0,0,0,1,66475579247327250000,
%T A172577 101028698009849992018229820380077899975000,
%U A172577 516171760116982949023606058514119493310964000589692899432320000
%N A172577 Number of 4*n X n 0..1 arrays with row sums 6 and column sums 24
%H A172577 R. H. Hardin, <a href="b172577.txt">Table of n, a(n) for n=1..10</a>
%K A172577 nonn,new
%O A172577 1,7
%A A172577 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172576
%S A172576 0,0,0,0,1,3246670537110000,352231966693098849132080247442500,
%T A172576 203269774721184658406475682049947158681837795504000,
%U A172576 746770363520310669802381642326970209622854085785862010798716821780000
%N A172576 Number of 4*n X n 0..1 arrays with row sums 5 and column sums 20
%H A172576 R. H. Hardin, <a href="b172576.txt">Table of n, a(n) for n=1..11</a>
%K A172576 nonn,new
%O A172576 1,6
%A A172576 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172575
%S A172575 0,0,1,63063000,149144498504001000,2834492178580299130305958000,
%T A172575 337799058758556439723480176591336120000,
%U A172575 203269774721184658406475682049947158681837795504000
%N A172575 Number of 4*n X n 0..1 arrays with row sums 3 and column sums 12
%H A172575 R. H. Hardin, <a href="b172575.txt">Table of n, a(n) for n=1..16</a>
%K A172575 nonn,new
%O A172575 1,4
%A A172575 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172574
%S A172574 0,0,1,31082452632000,91880966659006902855367890000,
%T A172574 4623755659802671553551698062433487687144266000,
%U A172574 3253924815644553490177047049527865204808456401213720382274024000
%N A172574 Number of 3*n X 2*n 0..1 arrays with row sums 6 and column sums 9
%H A172574 R. H. Hardin, <a href="b172574.txt">Table of n, a(n) for n=1..10</a>
%K A172574 nonn,new
%O A172574 1,4
%A A172574 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172573
%S A172573 0,1,90291600,729833528645228700,91880966659006902855367890000,
%T A172573 113498154592670786286712327731832894830000,
%U A172573 972197559210817377231037831422299544032898553608000000
%N A172573 Number of 3*n X 2*n 0..1 arrays with row sums 4 and column sums 6
%H A172573 R. H. Hardin, <a href="b172573.txt">Table of n, a(n) for n=1..17</a>
%K A172573 nonn,new
%O A172573 1,3
%A A172573 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172572
%S A172572 90,67950,90291600,154700988750,306407299538340,666569141498660400,
%T A172572 1548539246648239560000,3776577900841430197548750,
%U A172572 9561215418596022668009737500,24935177268489106332174087326700
%N A172572 Number of 3*n X 6 0..1 arrays with row sums 2 and column sums n
%H A172572 R. H. Hardin, <a href="b172572.txt">Table of n, a(n) for n=1..33</a>
%K A172572 nonn,new
%O A172572 1,1
%A A172572 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172571
%S A172571 756756,306407299538340,607807501825156719166080,
%T A172571 2460737690938850798370391240644900,
%U A172571 14962628816774970940772777740084998521738256
%N A172571 Number of 3*n X 15 0..1 arrays with row sums 5 and column sums n
%H A172571 R. H. Hardin, <a href="b172571.txt">Table of n, a(n) for n=1..14</a>
%K A172571 nonn,new
%O A172571 1,1
%A A172571 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172570
%S A172570 34650,154700988750,2407147216735338000,65599839591251908982712750,
%T A172570 2460737690938850798370391240644900,
%U A172570 113498154592670786286712327731832894830000
%N A172570 Number of 3*n X 12 0..1 arrays with row sums 4 and column sums n
%H A172570 R. H. Hardin, <a href="b172570.txt">Table of n, a(n) for n=1..17</a>
%K A172570 nonn,new
%O A172570 1,1
%A A172570 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172569
%S A172569 17153136,666569141498660400,178355054529362612768336870400,
%T A172569 113498154592670786286712327731832894830000,
%U A172569 118362260055245042852951157735613048828754466686430336
%N A172569 Number of 3*n X 18 0..1 arrays with row sums 6 and column sums n
%H A172569 R. H. Hardin, <a href="b172569.txt">Table of n, a(n) for n=1..10</a>
%K A172569 nonn,new
%O A172569 1,1
%A A172569 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172568
%S A172568 1680,90291600,12025780892160,2407147216735338000,
%T A172568 607807501825156719166080,178355054529362612768336870400,
%U A172568 58142871081888953760431694254592000
%N A172568 Number of 3*n X 9 0..1 arrays with row sums 3 and column sums n
%H A172568 R. H. Hardin, <a href="b172568.txt">Table of n, a(n) for n=1..25</a>
%K A172568 nonn,new
%O A172568 1,1
%A A172568 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172567
%S A172567 0,0,0,1,168168000,666569141498660400,12218892645550878420052920000,
%T A172567 959794291357783500093926796260896462300,
%U A172567 289360589460480315988823313632867603848637000544000
%N A172567 Number of 3*n X n 0..1 arrays with row sums 4 and column sums 12
%H A172567 R. H. Hardin, <a href="b172567.txt">Table of n, a(n) for n=1..17</a>
%K A172567 nonn,new
%O A172567 1,5
%A A172567 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172566
%S A172566 0,0,0,0,0,1,182509367040000,712008996110160366168717566400,
%T A172566 7664603814657718197913972357390335773198400000,
%U A172566 299436894513044390420517533660659195248061275496427347927110000
%N A172566 Number of 3*n X n 0..1 arrays with row sums 6 and column sums 18
%H A172566 R. H. Hardin, <a href="b172566.txt">Table of n, a(n) for n=1..12</a>
%K A172566 nonn,new
%O A172566 1,7
%A A172566 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172565
%S A172565 0,0,0,0,0,0,1,369398958888960000,2343976695927269878444049332219968000,
%T A172565 27470008383944006158001488120005460494368265540960000000,
%U A172565 960689698603154580472955204587017348140176811923434228399447157783872000000
%N A172565 Number of 3*n X n 0..1 arrays with row sums 7 and column sums 21
%H A172565 R. H. Hardin, <a href="b172565.txt">Table of n, a(n) for n=1..12</a>
%K A172565 nonn,new
%O A172565 1,8
%A A172565 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172564
%S A172564 0,0,0,0,0,0,0,1,1080491954750208000000,
%T A172564 14896749360215409445279324463510987496360000,
%U A172564 236894392610413599842635477924735827151557418755605148238560000000
%N A172564 Number of 3*n X n 0..1 arrays with row sums 8 and column sums 24
%H A172564 R. H. Hardin, <a href="b172564.txt">Table of n, a(n) for n=1..12</a>
%K A172564 nonn,new
%O A172564 1,9
%A A172564 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172563
%S A172563 0,0,1,369600,2604964362000,89365729945562642000,
%T A172563 12218892645550878420052920000,5638345005774311804302107347272704000,
%U A172563 7664603814657718197913972357390335773198400000
%N A172563 Number of 3*n X n 0..1 arrays with row sums 3 and column sums 9
%H A172563 R. H. Hardin, <a href="b172563.txt">Table of n, a(n) for n=1..23</a>
%K A172563 nonn,new
%O A172563 1,4
%A A172563 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172562
%S A172562 0,1,1680,32496156,2604964362000,666569141498660400,
%T A172562 452489654840076972518400,712008996110160366168717566400,
%U A172562 2343976695927269878444049332219968000
%N A172562 Number of 3*n X n 0..1 arrays with row sums 2 and column sums 6
%H A172562 R. H. Hardin, <a href="b172562.txt">Table of n, a(n) for n=1..33</a>
%K A172562 nonn,new
%O A172562 1,3
%A A172562 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172561
%S A172561 0,0,0,0,1,137225088000,452489654840076972518400,
%T A172561 5638345005774311804302107347272704000,
%U A172561 289360589460480315988823313632867603848637000544000
%N A172561 Number of 3*n X n 0..1 arrays with row sums 5 and column sums 15
%H A172561 R. H. Hardin, <a href="b172561.txt">Table of n, a(n) for n=1..14</a>
%K A172561 nonn,new
%O A172561 1,6
%A A172561 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172560
%S A172560 12870,27770358330,297348703692826500,6637298299496411164868250,
%T A172560 226226183763370772622682749527820,
%U A172560 10156485647319165206190655366818708639900
%N A172560 Number of 2*n X 16 0..1 arrays with row sums 8 and column sums n
%H A172560 R. H. Hardin, <a href="b172560.txt">Table of n, a(n) for n=1..13</a>
%K A172560 nonn,new
%O A172560 1,1
%A A172560 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172559
%S A172559 3432,936369720,1026608414145600,2128370421607383353400,
%T A172559 6368685647203261761403856832,24197157773366540262083203214418624,
%U A172559 108738182111446498614705217754614976371200
%N A172559 Number of 2*n X 14 0..1 arrays with row sums 7 and column sums n
%H A172559 R. H. Hardin, <a href="b172559.txt">Table of n, a(n) for n=1..16</a>
%K A172559 nonn,new
%O A172559 1,1
%A A172559 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172558
%S A172558 924,32496156,3718394156400,729833528645228700,195428754450309450171024,
%T A172558 64051375889927380035549804336,24197157773366540262083203214418624,
%U A172558 10156485647319165206190655366818708639900
%N A172558 Number of 2*n X 12 0..1 arrays with row sums 6 and column sums n
%H A172558 R. H. Hardin, <a href="b172558.txt">Table of n, a(n) for n=1..19</a>
%K A172558 nonn,new
%O A172558 1,1
%A A172558 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172557
%S A172557 252,1172556,14367744720,273957842462220,6736218287430460752,
%T A172557 195428754450309450171024,6368685647203261761403856832,
%U A172557 226226183763370772622682749527820,8588121524476528848476120449733261520
%N A172557 Number of 2*n X 10 0..1 arrays with row sums 5 and column sums n
%H A172557 R. H. Hardin, <a href="b172557.txt">Table of n, a(n) for n=1..24</a>
%K A172557 nonn,new
%O A172557 1,1
%A A172557 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172556
%S A172556 20,1860,297200,60871300,14367744720,3718394156400,1026608414145600,
%T A172556 297348703692826500,89365729945562642000,27658131940039664137360,
%U A172556 8766913970029589057611200,2834492178580299130305958000
%N A172556 Number of 2*n X 6 0..1 arrays with row sums 3 and column sums n
%H A172556 R. H. Hardin, <a href="b172556.txt">Table of n, a(n) for n=1..49</a>
%K A172556 nonn,new
%O A172556 1,1
%A A172556 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172555
%S A172555 70,44730,60871300,116963796250,273957842462220,729833528645228700,
%T A172555 2128370421607383353400,6637298299496411164868250,
%U A172555 21796558191507153941744357500,74580431332898028185458459628980
%N A172555 Number of 2*n X 8 0..1 arrays with row sums 4 and column sums n
%H A172555 R. H. Hardin, <a href="b172555.txt">Table of n, a(n) for n=1..33</a>
%K A172555 nonn,new
%O A172555 1,1
%A A172555 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172554
%S A172554 48620,842090474940,89365729945562642000,21796558191507153941744357500,
%T A172554 8588121524476528848476120449733261520,
%U A172554 4623755659802671553551698062433487687144266000
%N A172554 Number of 2*n X 18 0..1 arrays with row sums 9 and column sums n
%H A172554 R. H. Hardin, <a href="b172554.txt">Table of n, a(n) for n=1..12</a>
%K A172554 nonn,new
%O A172554 1,1
%A A172554 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172553
%S A172553 0,0,1,2520,56586600,3718394156400,617944896216648000,
%T A172553 231528006523766358806400,178355054529362612768336870400,
%U A172553 262180318469859848392495320519360000
%N A172553 Number of 2*n X n 0..1 arrays with row sums 3 and column sums 6
%H A172553 R. H. Hardin, <a href="b172553.txt">Table of n, a(n) for n=1..41</a>
%K A172553 nonn,new
%O A172553 1,4
%A A172553 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172552
%S A172552 0,0,0,0,0,1,681080400,7373018003758407000,
%T A172552 178355054529362612768336870400,
%U A172552 10697695830125272790880922760946584535000
%N A172552 Number of 2*n X n 0..1 arrays with row sums 6 and column sums 12
%H A172552 R. H. Hardin, <a href="b172552.txt">Table of n, a(n) for n=1..17</a>
%K A172552 nonn,new
%O A172552 1,7
%A A172552 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172551
%S A172551 0,0,0,0,0,0,0,1,12504636144000,2532230252503738514963235000,
%T A172551 691778216746194304091147388331566881040000,
%U A172551 358920655858354115406831875940612514848040679965214568750
%N A172551 Number of 2*n X n 0..1 arrays with row sums 8 and column sums 16
%H A172551 R. H. Hardin, <a href="b172551.txt">Table of n, a(n) for n=1..14</a>
%K A172551 nonn,new
%O A172551 1,9
%A A172551 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172550
%S A172550 0,0,0,0,0,0,0,0,1,2375880867360000,86574740102712303011539719750000,
%T A172550 3113598948052994576761792558312393011168670080000,
%U A172550 177115577520155240531744027851206753629735074356474328692651750000
%N A172550 Number of 2*n X n 0..1 arrays with row sums 9 and column sums 18
%H A172550 R. H. Hardin, <a href="b172550.txt">Table of n, a(n) for n=1..14</a>
%K A172550 nonn,new
%O A172550 1,10
%A A172550 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172549
%S A172549 0,0,0,1,113400,154700988750,617944896216648000,
%T A172549 6637298299496411164868250,176264656064665625137392456544800,
%U A172549 10697695830125272790880922760946584535000
%N A172549 Number of 2*n X n 0..1 arrays with row sums 4 and column sums 8
%H A172549 R. H. Hardin, <a href="b172549.txt">Table of n, a(n) for n=1..25</a>
%K A172549 nonn,new
%O A172549 1,5
%A A172549 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172548
%S A172548 0,0,0,0,0,0,1,81729648000,109829050417159537464000,
%T A172548 262180318469859848392495320519360000,
%U A172548 1383686991412953935264882502412625112170448000000
%N A172548 Number of 2*n X n 0..1 arrays with row sums 7 and column sums 14
%H A172548 R. H. Hardin, <a href="b172548.txt">Table of n, a(n) for n=1..15</a>
%K A172548 nonn,new
%O A172548 1,8
%A A172548 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172547
%S A172547 0,0,0,0,1,7484400,807998767676100,231528006523766358806400,
%T A172547 176264656064665625137392456544800,
%U A172547 343706681395653993021794459883981791479056
%N A172547 Number of 2*n X n 0..1 arrays with row sums 5 and column sums 10
%H A172547 R. H. Hardin, <a href="b172547.txt">Table of n, a(n) for n=1..20</a>
%K A172547 nonn,new
%O A172547 1,6
%A A172547 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172546
%S A172546 0,0,0,0,0,0,0,0,0,1,548828480360160000,
%T A172546 4237239732072431006302896746240010000,
%U A172546 22899532804595954935644287742940961780296483414339200000
%N A172546 Number of 2*n X n 0..1 arrays with row sums 10 and column sums 20
%H A172546 R. H. Hardin, <a href="b172546.txt">Table of n, a(n) for n=1..14</a>
%K A172546 nonn,new
%O A172546 1,11
%A A172546 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172544
%S A172544 0,0,0,0,0,1,5040,187530840,12025780892160,1289144584143523800,
%T A172544 226885231700215713535680,64051375889927380035549804336,
%U A172544 28278447454165011203551734584421120
%N A172544 Number of n X n 0..1 arrays with row sums 6 and column sums 6
%H A172544 R. H. Hardin, <a href="b172544.txt">Table of n, a(n) for n=1..28</a>
%K A172544 nonn,new
%O A172544 1,7
%A A172544 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172543
%S A172543 0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,20922789888000,
%T A172543 10268902998771351157327104000,
%U A172543 2767806480542211571651550187279222472704000
%N A172543 Number of n X n 0..1 arrays with row sums 15 and column sums 15
%H A172543 R. H. Hardin, <a href="b172543.txt">Table of n, a(n) for n=1..20</a>
%K A172543 nonn,new
%O A172543 1,16
%A A172543 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172542
%S A172542 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,355687428096000,
%T A172542 3237415247416050491577971184000,
%U A172542 13564406360915457771720399143711430952267776000
%N A172542 Number of n X n 0..1 arrays with row sums 16 and column sums 16
%H A172542 R. H. Hardin, <a href="b172542.txt">Table of n, a(n) for n=1..20</a>
%K A172542 nonn,new
%O A172542 1,17
%A A172542 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172541
%S A172541 0,0,0,0,0,0,1,40320,14398171200,8302816499443200,7722015017013984456000,
%T A172541 11649337108041078980732943360,28278447454165011203551734584421120,
%U A172541 108738182111446498614705217754614976371200
%N A172541 Number of n X n 0..1 arrays with row sums 7 and column sums 7
%H A172541 R. H. Hardin, <a href="b172541.txt">Table of n, a(n) for n=1..23</a>
%K A172541 nonn,new
%O A172541 1,8
%A A172541 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172540
%S A172540 0,0,0,0,0,0,0,0,1,3628800,158815387962000,9254768770160124288000,
%T A172540 769237071909157579108571190000,96986285294151066094112970262797953280,
%U A172540 19092174983817380047229162651397270697765056000
%N A172540 Number of n X n 0..1 arrays with row sums 9 and column sums 9
%H A172540 R. H. Hardin, <a href="b172540.txt">Table of n, a(n) for n=1..20</a>
%K A172540 nonn,new
%O A172540 1,10
%A A172540 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172539
%S A172539 0,0,0,0,0,0,0,0,0,1,39916800,21959547410077200,
%T A172539 14255616537578735986867200,12163525741347497524178307740904300,
%U A172539 14962628816774970940772777740084998521738256
%N A172539 Number of n X n 0..1 arrays with row sums 10 and column sums 10
%H A172539 R. H. Hardin, <a href="b172539.txt">Table of n, a(n) for n=1..19</a>
%K A172539 nonn,new
%O A172539 1,11
%A A172539 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172538
%S A172538 0,0,0,0,0,0,0,0,0,0,0,0,1,87178291200,147320988741542099484000,
%T A172538 190637228506535883540302038364160000,
%U A172538 238871596129285108315684789088803525762942560000
%N A172538 Number of n X n 0..1 arrays with row sums 13 and column sums 13
%H A172538 R. H. Hardin, <a href="b172538.txt">Table of n, a(n) for n=1..19</a>
%K A172538 nonn,new
%O A172538 1,14
%A A172538 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172537
%S A172537 0,0,0,0,0,0,0,0,0,0,0,0,0,1,1307674368000,36574751938491748341360000,
%T A172537 665825560532772251175492202972938240000,
%U A172537 10431401634793817906193873163767479249710797568200000
%N A172537 Number of n X n 0..1 arrays with row sums 14 and column sums 14
%H A172537 R. H. Hardin, <a href="b172537.txt">Table of n, a(n) for n=1..19</a>
%K A172537 nonn,new
%O A172537 1,15
%A A172537 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172536
%S A172536 0,0,0,0,0,0,0,1,362880,1371785398200,7673688777463632000,
%T A172536 65599839591251908982712750,885282776210120715086715619724160,
%U A172536 19040419266278799766631032461849139013040
%N A172536 Number of n X n 0..1 arrays with row sums 8 and column sums 8
%H A172536 R. H. Hardin, <a href="b172536.txt">Table of n, a(n) for n=1..22</a>
%K A172536 nonn,new
%O A172536 1,9
%A A172536 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172535
%S A172535 0,0,0,0,0,0,0,0,0,0,1,479001600,3574340599104475200,
%T A172535 27537152449960680597739468800,254143667822686635850590661555095468000,
%U A172535 3183529624645847695375078143769686741065620316160
%N A172535 Number of n X n 0..1 arrays with row sums 11 and column sums 11
%H A172535 R. H. Hardin, <a href="b172535.txt">Table of n, a(n) for n=1..18</a>
%K A172535 nonn,new
%O A172535 1,12
%A A172535 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172534
%S A172534 0,0,0,0,0,0,0,0,0,0,0,1,6227020800,676508133623135814000,
%T A172534 65662040698002721810659005184000,
%U A172534 6892692735539278753058456514221737762215000
%N A172534 Number of n X n 0..1 arrays with row sums 12 and column sums 12
%H A172534 R. H. Hardin, <a href="b172534.txt">Table of n, a(n) for n=1..18</a>
%K A172534 nonn,new
%O A172534 1,13
%A A172534 Ron Hardin (rhhardin(AT)att.net) Feb 06 2010

%I A172525
%S A172525 111111111,222222222,333333333,444444444,555555555,666666666,777777777,
%T A172525 888888888,999999999,1111111110,1222222221,1333333332,1444444443,
%U A172525 1555555554,1666666665,1777777776,1888888887,1999999998,2111111109
%N A172525 a(n)=9*n*12345679 (with n>0)
%Y A172525 Cf. A021085
%K A172525 nonn,new
%O A172525 1,1
%A A172525 Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Feb 06 2010

%I A173002
%S A173002 10010101,10011101,11171777,11177717,11313331,11333131,11919199,
%T A173002 11919991,13111333,13131133,13131331,13133311,13311313,14441411,
%U A173002 16166611,16616161,17111777,17171177,17171771,17177117,17711717
%N A173002 Primes consisting of two digits only, each digit with frequency f = 4.
%C A173002 2 digits, f = 1: 20 primes p 11 < p < =97: 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97
%C A173002 2 digits, f = 2: no primes as abab has divisor 101, abba and aabb divisor 11
%C A173002 2 digits, f = 3: no primes as sum of digits 3 * (a+b)
%C A173002 2 digits, f = 4: there are 18 possibilities for (a,b):
%C A173002 (1,0), (1,3), (1,4), (1,6), (1,7), (1,9), (2,3), (2,9), (3,4), (3,5), (3,7), (3,8), (4,7), (4,9), (5,9), (6,7), (7,9), (8,9)
%C A173002 Each possibily occurs, 2+9+3+5+13+11+2+6+3+3+10+2+2+5+2+2+6+4 = 90 = 2 * 3^2 * 5 primes
%D A173002 Theo Kempermann, Zahlentheoretische Kostproben, Harri Deutsch, 2. aktualisierte Auflage 2005
%D A173002 Wladyslaw Narkiewicz: The development of prime number theory: from Euclid to Hardy and Littlewood, Springer Monographs in Mathematics, Berlin, New York, 2000
%D A173002 Paulo Ribenboim: The little book of bigger primes, Springer Berlin, New York, 2004
%e A173002 Complete list classified according to the 18 possible "pairs":
%e A173002 10010101, 10011101
%e A173002 11313331, 11333131, 13111333, 13131133, 13131331, 13133311, 13311313, 31133131, 33113131
%e A173002 14441411, 41414411, 44114141
%e A173002 16166611, 16616161, 61116661, 61661161, 66161611
%e A173002 11171777, 11177717, 17111777, 17171177, 17171771, 17177117, 17711717, 17717171, 71117177, 71171717, 71717117, 77111717, 77711171
%e A173002 11919199, 11919991, 19111999, 19199119, 19911919, 19991911, 91919911, 91999111, 99111919, 99119191, 99919111
%e A173002 23223323, 32323223
%e A173002 22929299, 29229929, 29299229, 29992229, 92922299, 99292229
%e A173002 34434343, 44334343, 44343433
%e A173002 35553533, 53355353, 53533553
%e A173002 33373777, 33773737, 37373773, 37377337, 73337377, 73337773, 73373737, 73773373, 77337373, 77733373
%e A173002 38383883, 88838333
%e A173002 47447747, 77474447
%e A173002 44994949, 49444999, 49494499, 49499449, 94449499
%e A173002 55599959, 99555959
%e A173002 67766767, 76767667
%e A173002 77997979, 79779979, 79797997, 79997977, 99977797, 99979777
%e A173002 88989899, 98988899, 98989889, 99898889
%Y A173002 A087511, A087512, A087513, A087514, A087515, A087527, A087528, A087529, A087530, A087531, A087532, A087533, A087534, A087535, A087536, A087537, A087538
%K A173002 base,fini,nonn,new
%O A173002 1,1
%A A173002 Eva-Maria Zschorn (e-m.zschorn(AT)zaschendorf.km3.de), Feb 07 2010

%I A172488
%S A172488 12289,18433,139969,209953,472393
%N A172488 Primes of the form 2^i * 3^j + 1 with i + j = 13
%C A172488 Note that bases 2 = prime(1), 3 = prime(2)
%C A172488 13 = prime(prime(1) * prime(2))
%C A172488 A finite "FUN" sequence with 5 = prime(3) terms
%D A172488 E. I. Ignatjew, Mathematische Spielereien, Urania Verlag Leipzig/Jena/Berlin 1982
%D A172488 Helmut Kracke, Mathe-musische Knobelisken, Duemmler Bonn, 2. Auflage 1983
%e A172488 12289 = 2^12 * 3^1 + 1 = prime(1470) = prime(2 * 3 * 5 * 7^2)
%e A172488 18433 = 2^11 * 3^2 + 1 = prime(2111), index is prime(318)
%e A172488 139969 = 2^6 * 3^7 + 1 = prime(13006), larger of a Prime Twin Couple: PTC(1608)
%e A172488 209953 = 2^5 * 3^8 + 1 = prime(18802)
%e A172488 472393 = 2^3 * 3^10 + 1 = prime(39420)
%Y A172488 A172315, A005105
%K A172488 fini,nonn,new
%O A172488 1,1
%A A172488 Eva-Maria Zschorn (e-m.zschorn(AT)zaschendorf.km3.de), Feb 05 2010

%I A173029
%S A173029 10007,20016,60025,130026,200029,270038,360039,450046,550049,750052,
%T A173029 950061,1250068,1650077,2150086,2850087,3750088,4650095,5650098,6650131,
%U A173029 7650168,9650288,10650387,11650690
%N A173029 Partial sums of naughty primes A164968.
%C A173029 The subsequence of prime partial sums of naughty primes begins: 10007, 200029, 550049, 6650131. The subsubsequence of naughty prime partial sums of naughty primes begins: 10007, and then what? The smallest square in the sequence is 60025 = 5^2 * 7^4.
%F A173029 a(n) = SUM[i=1..n] {p such that p is prime and the number of zeros in the decimal representation of p is greater than the number of all other digits}.
%e A173029 a(24) = 10007 + 10009 + 40009 + 70001 + 70003 + 70009 + 90001 + 90007 + 100003 + 200003 + 200009 + 300007 + 400009 + 500009 + 700001 + 900001 + 900007 + 1000003 + 1000033 + 1000037 + 1000039 + 1000081 + 1000099 + 1000303.
%Y A173029 Cf. A000040, A164968.
%K A173029 base,easy,nonn,new
%O A173029 1,1
%A A173029 Jonathan Vos Post (jvospost3(AT)gmail.com), Feb 07 2010

%I A172966
%S A172966 0,0,0,306,2365,19047,90503,328324,981693,2547955,5933257,12681288,
%T A172966 25284363,47595023,85357395,146879312,243867873,392452803
%N A172966 Number of ways to place 4 nonattacking knights on an n X n cylindrical board
%H A172966 V. Kotesovec, <a href="http://web.telecom.cz/vaclav.kotesovec/math.htm">Number of ways of placing non-attacking queens and kings on boards of various sizes</a>
%F A172966 Explicit formula (Vaclav Kotesovec, 31.1.2010): a(n) = n*(n^7-54n^5+72n^4+1115n^3-2616n^2-8502n+26712)/24, n>=9
%Y A172966 A172531, A172135, A172964, A172965
%K A172966 nonn,new
%O A172966 1,4
%A A172966 Vaclav Kotesovec (kotesovec(AT)chello.cz), Feb 06 2010

%I A172531
%S A172531 0,0,0,228,600,12357,68796,275888,872532,2344025,5580762,12107196,
%T A172531 24392446,46261537,83426400,144157632,240119696,387393921,607715342,
%U A172531 929951100
%N A172531 Number of ways to place 4 nonattacking knights on an n X n toroidal board
%H A172531 V. Kotesovec, <a href="http://web.telecom.cz/vaclav.kotesovec/math.htm">Number of ways of placing non-attacking queens and kings on boards of various sizes</a>
%F A172531 Explicit formula (Vaclav Kotesovec, 31.1.2010): a(n) = n^2*(n^6-54n^4+1115n^2-8934)/24, n>=9
%Y A172531 A172529, A172530, A172135, A172519
%K A172531 nonn,new
%O A172531 1,4
%A A172531 Vaclav Kotesovec (kotesovec(AT)chello.cz), Feb 06 2010

%I A172967
%S A172967 0,0,0,208,3210,58056,458157,2524176,10587591,36576380,109008735,
%T A172967 289450344,700477401,1570789892,3304892985,6586928032,12530769343,
%U A172967 22891446252
%N A172967 Number of ways to place 5 nonattacking knights on an n X n cylindrical board
%H A172967 V. Kotesovec, <a href="http://web.telecom.cz/vaclav.kotesovec/math.htm">Number of ways of placing non-attacking queens and kings on boards of various sizes</a>
%F A172967 Explicit formula (Vaclav Kotesovec, 31.1.2010): a(n) = n*(n^9-90n^7+120n^6+3395n^5-8160n^4-62130n^3+204000n^2+463464n-1888080)/120, n>=10. For any fixed value of k > 1, a(n) = n^(2k)/k! - 9n^(2k-2)/2/(k-2)! + 6n^(2k-3)/(k-2)! + ...
%Y A172967 A172532, A172136, A172964, A172965, A172966
%K A172967 nonn,new
%O A172967 1,4
%A A172967 Vaclav Kotesovec (kotesovec(AT)chello.cz), Feb 06 2010

%I A172532
%S A172532 0,0,0,128,120,30312,283906,1872064,8643186,31702920,98179400,267487920,
%T A172532 659015500,1496908840,3179369070,6382030592,12207535134,22396355496,
%U A172532 39617305308,67860021680
%N A172532 Number of ways to place 5 nonattacking knights on an n X n toroidal board
%H A172532 V. Kotesovec, <a href="http://web.telecom.cz/vaclav.kotesovec/math.htm">Number of ways of placing non-attacking queens and kings on boards of various sizes</a>
%F A172532 Explicit formula (Vaclav Kotesovec, 31.1.2010): a(n) = n^2*(n^8-90n^6+3395n^4-64290n^2+522504)/120, n>=10
%Y A172532 A172529, A172530, A172531, A172136
%K A172532 nonn,new
%O A172532 1,4
%A A172532 Vaclav Kotesovec (kotesovec(AT)chello.cz), Feb 06 2010

%I A172983
%S A172983 113,312,523,746,973,1202,1435,1712,2023,2354,2691,3124,3573,4072,4629,
%T A172983 5206,5805,6466,7143,7916,8727,9604,10485,11368,12255,13166,14143,15140,
%U A172983 16257,18034
%N A172983 Partial sums of near-repdigit primes A056710.
%C A172983 The subsequence of prime partial sums of near-repdigit primes begins 113, 523, 14143. What is the smallest near-repdigit prime partial sums of near-repdigit primes?
%F A172983 a(n) = SUM[i=1..n] A056710 = SUM[i=1..n] {primes in whose base 10 representation all digits are equal except for an end-digit}.
%e A172983 a(3) = 113 + 199 + 211 = 523 is prime. a(20) = 113 + 199 + 211 + 223 + 227 + 229 + 233 + 277 + 311 + 331 + 337 + 433 + 449 + 499 + 557 + 577 + 599 + 661 + 677 + 773.
%Y A172983 Cf. A000040, A056710.
%K A172983 base,easy,nonn,new
%O A172983 1,1
%A A172983 Jonathan Vos Post (jvospost3(AT)gmail.com), Feb 06 2010

%I A172518
%S A172518 0,0,0,0,100,576,2156,7168,17496,41600,82280,161280,280540,486080,
%T A172518 774900,1232896,1844976,2757888,3933456,5606400,7699860,10570560,
%U A172518 14081980,18754560,24365000,31647616,40258296,51204608,63979916
%N A172518 Number of ways to place 3 nonattacking queens on an n X n toroidal board
%H A172518 V. Kotesovec, <a href="http://web.telecom.cz/vaclav.kotesovec/math.htm">Number of ways of placing non-attacking queens and kings on boards of various sizes</a>
%F A172518 Explicit formula (Vaclav Kotesovec, 31.1.2010): a(n) = n^2*(n-2)(n-4)(n^2-6n+12)/6 if n is even and a(n) = n^2*(n-1)(n-3)(n^2-8n+18)/6 if n is odd.
%Y A172518 A047659, A007705, A172517
%K A172518 nonn,new
%O A172518 1,5
%A A172518 Vaclav Kotesovec (kotesovec(AT)chello.cz), Feb 05 2010

%I A171903
%S A171903 99,110,111,112,113,114,115,116,117,118,199,220,221,222,223,224,225,226,
%T A171903 227,228,299,330,331,332,333,334,335,336,337,338,399,440,441,442,443,
%U A171903 444,445,446,447,448,499,550,551,552,553,554,555,556,557,558,599,660
%N A171903 Numbers m such that m and m+1 have at least two identical neighbouring digits in their decimal representations.
%C A171903 a(n) + 1 = A171901(k) for some k;
%C A171903 A171902(a(n)) = 1.
%H A171903 %H A171903 R. Zumkeller, <a href="b171903.txt">Table of n, a(n) for n = 1..5000</a>
%Y A171903 A171904.
%K A171903 base,nonn,new
%O A171903 1,1
%A A171903 Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Feb 07 2010

%I A171904
%S A171904 99,110,110,110,110,110,110,110,110,100,11,88,99,99,99,99,99,99,99,99,
%T A171904 112,11,77,88,88,88,88,88,88,88,88,112,11,66,77,77,77,77,77,77,77,77,
%U A171904 112,11,55,66,66,66,66,66,66,66,66,112,11,44,55,55,55,55,55,55,55,55
%N A171904 Smallest number m such that m and m+n have both at least two identical neighbouring digits in their decimal representations.
%C A171904 a(n) + n = A171901(k) for some k.
%H A171904 R. Zumkeller, <a href="b171904.txt">Table of n, a(n) for n = 1..10000</a>
%Y A171904 A171903.
%K A171904 base,nonn,new
%O A171904 1,1
%A A171904 Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Feb 07 2010

%I A172523
%S A172523 89,498,947,1446,2255,6304,10403,14812,19461,24350,29259,34228,39227,
%T A172523 45316,51765,58234,64923,71792,78691,85640,93649,101718,109807,118416,
%U A172523 127085,135774,144473,153322,162291,171290,180339,189988,199677,209626
%N A172523 Partial sums of primes in which no digit is a prime A061372.
%C A172523 The "Prime Curios" web site calls the underlying sequence "holey primes... primes that have only digits with holes, i.e., 0, 4, 6, 8, or 9." The subsequence of prime partial sums begins: 89, 39227, 78691, 109807, 330233.
%F A172523 a(n) = SUM[i=1..n] A061372(i) = SUM[i=1..n] {Primes in which no digit is a prime} = SUM[i=1..n] {primes having only 0,4,6,8,9 as digits}.
%e A172523 a(3) = 89 + 409 + 449 = 947 is prime. a(13) = 89 + 409 + 449 + 499 + 809 + 4049 + 4099 + 4409 + 4649 + 4889 + 4909 + 4969 + 4999 = 39227 is prime. a(19) = 89 + 409 + 449 + 499 + 809 + 4049 + 4099 + 4409 + 4649 + 4889 + 4909 + 4969 + 4999 + 6089 + 6449 + 6469 + 6689 + 6869 + 6899 = 78691 is prime. a(23) = 89 + 409 + 449 + 499 + 809 + 4049 + 4099 + 4409 + 4649 + 4889 + 4909 + 4969 + 4999 + 6089 + 6449 + 6469 + 6689 + 6869 + 6899 + 6949 + 8009 + 8069 + 8089 = 109807 is prime. a(37) = 89 + 409 + 449 + 499 + 809 + 4049 + 4099 + 4409 + 4649 + 4889 + 4909 + 4969 + 4999 + 6089 + 6449 + 6469 + 6689 + 6869 + 6899 + 6949 + 8009 + 8069 + 8089 + 8609 + 8669 + 8689 + 8699 + 8849 + 8969 + 8999 + 9049 + 9649 + 9689 + 9949 + 40009 + 40099 + 40499 = 330233 is prime.
%Y A172523 Cf. A000040, A061372.
%K A172523 base,easy,nonn,new
%O A172523 1,1
%A A172523 Jonathan Vos Post (jvospost3(AT)gmail.com), Feb 06 2010

%I A170932
%S A170932 1,63,2205,56595,1188495,21630609,353299947,5299499205,74192988870,
%T A170932 980996186170,12360551945742,149450309889426,1743586948709970,
%U A170932 19715944727720430,216875392004924730,2327795874186192102
%N A170932 Binomial[n + 8, 8]*7^n .
%C A170932 With a different offset, number of n-permutations of 8 objects: r, s, t, u, v, z, x, y with repetition allowed, containing exactly eight, (8) u's.
%F A170932 C(n + 8, 8)*7^n, n>=0 .
%t A170932 Table[Binomial[n + 8, 8]*7^n, {n, 0, 20}]
%Y A170932 Cf. A027474, A140107, A139641, A140404, A036226, A050989
%K A170932 nonn,new
%O A170932 0,2
%A A170932 Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Feb 08 2010

%I A172533
%S A172533 0,0,0,56,0,54972,764596,8972896,62560728,322246800,1323868260,
%T A172533 4595943336,14000143196,38413461800,96746410800,226834407552,
%U A172533 500492572112,1048044384360,2096986629308,4031211268200
%N A172533 Number of ways to place 6 nonattacking knights on an n X n toroidal board
%H A172533 V. Kotesovec, <a href="http://web.telecom.cz/vaclav.kotesovec/math.htm">Number of ways of placing non-attacking queens and kings on boards of various sizes</a>
%F A172533 Explicit formula (Vaclav Kotesovec, 31.1.2010): a(n) = n^2*(n^10-135n^8+8005n^6-262665n^4+4816354n^2-39858840)/720, n>=13
%Y A172533 A172529, A172530, A172531, A172532
%K A172533 nonn,new
%O A172533 1,4
%A A172533 Vaclav Kotesovec (kotesovec(AT)chello.cz), Feb 06 2010

%I A172501
%S A172501 1,54,1620,35640,641520,10007712,140107968,1801388160,21616657920,
%T A172501 244988789760,2645878929408,27420927086592,274209270865920,
%U A172501 2657720625315840,25058508752977920,230538280527396864
%N A172501 Binomial(n+8,8)*6^n.
%C A172501 With a different offset, number of n-permutations (n>=8) of 7 objects: r, s, t, u, v, z, x, y with repetition allowed, containing exactlyeight (8) u's.
%t A172501 Table[Binomial[n + 8, 8]*6^n, {n, 0, 20}]
%Y A172501 A081136, A081144, A139626, A036084, A050988, A141407
%K A172501 nonn,new
%O A172501 0,2
%A A172501 Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Feb 05 2010

%I A172519
%S A172519 0,0,0,0,50,288,2450,16384,62208,233600,638880,1755072,3901534,8772176,
%T A172519 17051850,33507328,59175640,105557904,173570244,287904000,447885774,
%U A172519 702042000,1044894554,1565385984,2247132500,3244194304,4519015596
%N A172519 Number of ways to place 4 nonattacking queens on an n X n toroidal board
%H A172519 V. Kotesovec, <a href="http://web.telecom.cz/vaclav.kotesovec/math.htm">Number of ways of placing non-attacking queens and kings on boards of various sizes</a>
%F A172519 Explicit formula (Vaclav Kotesovec, 5.2.2010): a(n) = (n^8/24 - n^7 + 245n^6/24 - 113n^5/2 + 2843n^4/16 - 593n^3/2 + 4757n^2/24) + (n^6/8 - 5n^5/2 + 305n^4/16 - 129n^3/2 + 629n^2/8)*(-1)^n + 8n^2*COS(2*pi*n/3)/3 + 9n^2*COS(pi*n/2)/2
%Y A172519 A061994, A007705, A172517, A172518
%K A172519 nonn,new
%O A172519 1,5
%A A172519 Vaclav Kotesovec (kotesovec(AT)chello.cz), Feb 05 2010

%I A173042
%S A173042 48,71,96,112,128,143,163,176,191,192,208,211,224,244,248,268,288,304,
%T A173042 308,311,312,317,331,336,352,356,376,380,384,422,428,431,432,439,448,
%U A173042 456,460,463,496,512,516,536,544,551,560,568,571,572,599,604,607,608
%N A173042 Numbers n that cannot be decomposed into the sum of up to 4 squares using the following algorithm: If n is not decomposable using the algorithm: [Repeat the following 2 steps 4 times: 1-find the largest square s smaller than n; 2-n=n-s Numbers that can be decomposed yield final values of n=0.] then choose the first square as the second largest square smaller than n and try finding the remaining up to 3 squares using the 2 steps of the algorithm in brackets.
%C A173042 This is a subsequence of A112687.
%H A173042 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/LagrangesFour-SquareTheorem.html">Lagrange's Four-Square Theorem</a>
%e A173042 For n=48: it is not decomposable using the algorithm in brackets, so instead of using the first s=36 we choose s=25 (the second largest). So the attempt to decompose 48 is now 5*5+(up to more 3 squares which will be found using steps 1 and 2 of the algorithm in brackets). This yields 5*5+4*4+2*2+1*1 which does not give 48 hence it is not decomposable using this algorithm.
%Y A173042 Cf. A112687
%K A173042 nonn,new
%O A173042 1,1
%A A173042 Luis F.B.A. Alexandre (lfbaa(AT)di.ubi.pt), Feb 08 2010

%I A173000
%S A173000 1,45,1215,25515,459270,7440174,111602610,1578379770,21308126895,
%T A173000 277005649635,3490271185401,42835146366285,514021756395420,
%U A173000 6049640671423020,70002984912180660,798034027998859524
%N A173000 Binomial[n + 4, 4]*9^n .
%C A173000 Number of n-permutations (n>=4) of 10 objects p, r, q, u, v, w, z, x, y, z with repetition allowed, containing exactly 4 u's.
%t A173000 Table[Binomial[n + 4, 4]*9^n, {n, 0, 20}]
%K A173000 nonn,new
%O A173000 0,2
%A A173000 Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Feb 07 2010

%I A172978
%S A172978 1,44,1056,18304,256256,3075072,32800768,318636032,2867724288,
%T A172978 24216338432,193730707456,1479398129664,10848919617536,76776969601024,
%U A172978 526470648692736,3509804324618240,22813728110018560,144934272698941440
%N A172978 Binomial(n+10, 10)*4^n.
%C A172978 With a different offset, number of n-permutations (n>=10) of 5 objects: u, v, z, x, y with repetition allowed, containing exactly ten, (10) u's.
%t A172978 Table[Binomial[n + 10, 10]*4^n, {n, 0, 20}]
%Y A172978 Cf. A002697, A038845, A038846, A040075, A045543, A054337, A054338, A054339, A054340
%K A172978 nonn,new
%O A172978 0,2
%A A172978 Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Feb 06 2010

%I A175103
%S A175103 0,42,53,117,154,2377,3245,3771,6381,9688,10406,13879,13944,14056,15026,
%T A175103 18884,26521,29565,30735,36362,39310,43337,48808,54415,58308,62228,
%U A175103 63378,69111,77403,81750,86021,93545,94388,103961,119754,152555,162698
%N A175103 Integers n such that 17+30*n are terms in A172456.
%F A175103 a(n)=(A172456(n)-17)/30.
%Y A175103 Cf. A172456.
%K A175103 nonn,new
%O A175103 1,2
%A A175103 Zak Seidov (zakseidov(AT)yahoo.com), Feb 07 2010

%I A172510
%S A172510 1,40,960,17920,286720,4128768,55050240,692060160,8304721920,
%T A172510 95965675520,1074815565824,11725260718080,125069447659520,
%U A172510 1308418837053440,13458022323978240,136374626216312832
%N A172510 Binomial[n + 4, 4]*8^n .
%C A172510 Number of n-permutations (n>=4) of 9 objects p, r, q, u, v, w, z, x, y with repetition allowed, containing exactly 4 u's.
%t A172510 Table[Binomial[n + 4, 4]*8^n, {n, 0, 25}]
%Y A172510 A081138, A140802, A140406, A053107, A141054
%K A172510 nonn,new
%O A172510 0,2
%A A172510 Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Feb 05 2010

%I A172517
%S A172517 0,0,0,32,100,288,588,1152,1944,3200,4840,7200,10140,14112,18900,25088,
%T A172517 32368,41472,51984,64800,79380,96800,116380,139392,165000,194688,227448,
%U A172517 264992,306124,352800,403620,460800,522720,591872,666400,749088,837828
%N A172517 Number of ways to place 2 nonattacking queens on an n X n toroidal board
%H A172517 V. Kotesovec, <a href="http://web.telecom.cz/vaclav.kotesovec/math.htm">Number of ways of placing non-attacking queens and kings on boards of various sizes</a>
%F A172517 a(n) = n^2*(n-2)^2/2 if n is even and a(n) = n^2*(n-1)(n-3)/2 if n is odd.
%Y A172517 A007705, A036464
%K A172517 nonn,new
%O A172517 1,4
%A A172517 Vaclav Kotesovec (kotesovec(AT)chello.cz), Feb 05 2010

%I A172973
%S A172973 19,80,189,388,989,1650,2669,3730,4821,5930,7111,8712,10321,11990,13689,
%T A172973 15500,17401,19400,25411,31502,37603,43802,50421,63771,63771,70462,
%U A172973 77361,84352,94413,104482,114573,125264,136125,147034,158103,169784
%N A172973 Partial sums of A048890.
%C A172973 The underlying sequence is also called invertible primes or strobogrammatic primes. The subsequence of prime partial sum of primes that yield a different prime when rotated by 180 degrees begins: 19, 10321, 17401, 25411, 181693. The subsubsequence of primes that yield a different prime when rotated by 180 degrees which are partial sum of primes that yield a different prime when rotated by 180 degrees begins: 19, and I don't know the next. The smallest values which become a different number (though not a prime) when inverted are: 989, 11990. The subsequence of squares in the sequence begins: 13689 = 3^4 * 13^2.
%e A172973 a(39) = 19 + 61 + 109 + 199 + 601 + 661 + 1019 + 1061 + 1091 + 1109 + 1181 + 1601 + 1609 + 1669 + 1699 + 1811 + 1901 + 1999 + 6011 + 6091 + 6101 + 6199 + 6619 + 6661 + 6689 + 6691 + 6899 + 6991 + 10061 + 10069 + 10091 + 10691 + 10861 + 10909 + 11069 + 10091 + 10691 + 10861 + 10909 = 181693.
%Y A172973 Cf. A000040, A007597, A006567, A046732.
%K A172973 base,easy,nonn,new
%O A172973 1,1
%A A172973 Jonathan Vos Post (jvospost3(AT)gmail.com), Feb 06 2010

%I A172521
%S A172521 17,114,371,708,1589,5286,15943,32504,81801,147338,214315,303356,452413,
%T A172521 1300014,2288431,3434528,5406625,7476866,9999123,12836084,16389861,
%U A172521 20349158,24747735,30133496,37300393,48373610,66027291,98557468
%N A172521 Partial sums of A078902.
%C A172521 It is unknown if this is a finite or infinite sequence. Can it ever have a prime value after a(1) = 17? It can be semiprime, as 371 = 7 * 53; 1589 = 7 * 227; 15943 = 107 * 149; 214315 = 5 * 42863; 2288431 = 23 * 99497; and 16389861 = 3 * 5463287.
%H A172521 Eric W. Weisstein, <a href="http://mathworld.wolfram.com/GeneralizedFermatNumber.html">Generalized Fermat Number.</a> From MathWorld--A Wolfram Web Resource.
%F A172521 SUM[i=1..n] {primes of the form (k+1)^2^m + k^2^m, with m>1.}
%e A172521 a(29) = 17 + 97 + 257 + 337 + 881 + 3697 + 10657 + 16561 + 49297 + 65537 + 66977 + 89041 + 149057 + 847601 + 988417 + 1146097 + 1972097 + 2070241 + 2522257 + 2836961 + 3553777 + 3959297 + 4398577 + 5385761 + 7166897 + 11073217 + 17653681 + 32530177 + 41532497 + 44048497.
%Y A172521 Cf. A000040, A001358, A019434, A077659, A078900, A078901, A078902, A080131, A080208.
%K A172521 nonn,new
%O A172521 1,1
%A A172521 Jonathan Vos Post (jvospost3(AT)gmail.com), Feb 06 2010

%I A173035
%S A173035 15,24,38,32,36,40
%N A173035 Cat years in human years: a(1) = 15, a(2) = 24, a(n) = a(n-1) + 4 for n >= 3.
%K A173035 nonn,new
%O A173035 1,1
%A A173035 Grace Wang (grace_c_wang(AT)yahoo.com), Feb 07 2010

%I A172508
%S A172508 13,29,31,41,47,53,61,79,83,97,163,257,461,479,503,613,631,641,683,853,
%T A172508 863,947
%N A172508 Prime numbers which the differences between any of its two digits is always prime.
%K A172508 nonn,base,new
%O A172508 1,1
%A A172508 Claudio L Meller (claudiomeller(AT)gmail.com), Feb 05 2010

%I A172975
%S A172975 1,13,16,17,18,19,22,23,24,25,26,27,29,30,31,32,33,34,37,38,39,40,41,42,
%T A172975 43,44,46,47,48,49,51,52,53,54,57,58,59,60,61,62,63,64,65,67,68,69,71,
%U A172975 72,73,74,75,76,77,79,80,81,82,83,85,86,87,88,89,90,92,93,94,95,96,97
%N A172975 Numbers which aren't degrees of completeness of Lukasiewicz logics.
%C A172975 Numbers which don't occured in A172974.
%D A172975 Tokarz M, 1977. Degrees of completeness of Lukasiewicz logics. In [Wss^3 jcicki and Malinowski (eds.), 1977, pp. 127-134].
%Y A172975 A172974
%K A172975 nonn,new
%O A172975 1,2
%A A172975 Artur Jasinski (grafix(AT)csl.pl), Feb 06 2010

%I A173036
%S A173036 13,14,16,19,23,28,34,41,49,58,68,79,91,104,118,133,149,166,184,203,223,
%T A173036 244,266,289,313,338,364,391,419,448,478,509,541,574,608,643,679,716,
%U A173036 754,793,833,874,916,959,1003,1048,1094,1141,1189,1238,1288,1339,1391
%N A173036 Triangular numbers+13.
%C A173036 First 15 terms (13,14,16,19,23,28,34,41,49,58,68,79,91,104,118) the same as A017907.
%t A173036 f[n_]:=n*(n+1)/2+13;Table[f[n],{n,0,5!}]
%Y A173036 Cf. A000217
%K A173036 nonn,new
%O A173036 1,1
%A A173036 Vladimir Orlovsky (4vladimir(AT)gmail.com), Feb 07 2010

%I A172506
%S A172506 11,303,123123,6170617,246902469,1929001929,12345671234567,
%T A172506 617283906172839
%N A172506 a(n) = numerator of fraction a / b, where (a, b) = 1, such that its decimal representation has form (1)(2)(3)...(n-1)(n),(1)(2)(3)...(n-1)(n).
%C A172506 Sequence of denominators: 10, 25, 1000, 5000, 20000, 15625, 10000000, 50000000, ... Conjecture: this sequence is not equal to the sequence A078257.
%e A172506 a(6) = 1929001929; 1929001929 / 15625 = 123456,123456.
%K A172506 nonn,new
%O A172506 1,1
%A A172506 Jaroslav Krizek (jaroslav.krizek(AT)atlas.cz), Feb 05 2010

%I A172495
%S A172495 11,53,3123,20617,102469,95679,71234567,406172839,9123456789,
%T A172495 101234567891
%N A172495 a(n) = numerator of fraction a / b, where (a, b) = 1, such that its decimal representation has form (n),(1)(2)(3)...(n-1)(n).
%C A172495 Sequence of denominators: 10, 25, 1000, 5000, 20000, 15625, 10000000, 50000000, 1000000000, 10000000000,... Conjecture: this sequence is not equal to the sequence A078257.
%e A172495 a(6) = 95679; 95679 / 15625 = 6,123456.
%K A172495 nonn,new
%O A172495 1,1
%A A172495 Jaroslav Krizek (jaroslav.krizek(AT)atlas.cz), Feb 05 2010

%I A172526
%S A172526 11,44,100,179,279,403,548,716,906,1119,1354,1612,1892,2194,2519,2866,
%T A172526 3235,3627,4041,4478,4937,5418,5922,6448,6997,7568,8161,8777,9415,10076,
%U A172526 10759,11464,12192,12942,13715,14510,15327,16167,17029,17913,18820
%N A172526 a(n)=floor(3*n^2*(2+sqrt(3)))
%C A172526 Approximate area of the dodecahedron of side n=(1,2,3,4,...,)
%K A172526 nonn,new
%O A172526 1,1
%A A172526 Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Feb 06 2010

%I A171901
%S A171901 11,22,33,44,55,66,77,88,99,100,110,111,112,113,114,115,116,117,118,119,
%T A171901 122,133,144,155,166,177,188,199,200,211,220,221,222,223,224,225,226,
%U A171901 227,228,229,233,244,255,266,277,288,299,300,311,322,330,331,332,333
%N A171901 Numbers with at least two identical neighbouring digits in their decimal representation.
%C A171901 A171902(n) = a(n+1) - a(n) <= 11.
%H A171901 R. Zumkeller, <a href="b171901.txt">Table of n, a(n) for n = 1..10000</a>
%Y A171901 A044821, A171903, A171904.
%K A171901 base,nonn,new
%O A171901 1,1
%A A171901 Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Feb 07 2010

%I A172507
%S A172507 11,11,33,22,11,33,77,44,99,101,1111,303,1313,707,303,404,1717,909,1919,
%T A172507 101,2121,1111,2323,606,101,1313,2727,707,2929,303
%N A172507 a(n) = numerator of fraction a / b, where (a, b) = 1, such that its decimal representation has form (n),(n).
%C A172507 Sequence of denominators: 10, 25, 1000, 5000, 20000, 15625, 10000000, 50000000, ... Conjecture: this sequence is not equal to the sequence A078267.
%e A172507 a(6) = 33; 33 / 5 = 6,6.
%K A172507 nonn,new
%O A172507 1,1
%A A172507 Jaroslav Krizek (jaroslav.krizek(AT)atlas.cz), Feb 05 2010

%I A171902
%S A171902 11,11,11,11,11,11,11,11,1,10,1,1,1,1,1,1,1,1,1,3,11,11,11,11,11,11,11,
%T A171902 1,11,9,1,1,1,1,1,1,1,1,1,4,11,11,11,11,11,11,1,11,11,8,1,1,1,1,1,1,1,1,
%U A171902 1,5,11,11,11,11,11,1,11,11,11,7,1,1,1,1,1,1,1,1,1,6,11,11,11,11,1,11
%N A171902 First differences of consecutive numbers having at least two identical neighbouring digits in decimal representation.
%C A171902 a(n) = A171901(n+1) - A171901(n);
%C A171902 1 <= a(n) <= 11;
%C A171902 a(A171903(n)) = 1.
%H A171902 R. Zumkeller, <a href="b171902.txt">Table of n, a(n) for n = 1..10000</a>
%K A171902 nonn,base,new
%O A171902 1,1
%A A171902 Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Feb 07 2010

%I A173006
%S A173006 1,1,1,1,11,1,1,120,120,1,1,1309,14280,1309,1,1,14279,1699201,1699201,
%T A173006 14279,1,1,155760,202190640,2205562898,202190640,155760,1,1,1699081,
%U A173006 24058986960,2862818956682,2862818956682,24058986960,1699081,1,1
%N A173006 A product triangle sequence based on recursion:a=5; f(n,a)=(2*a+1)*f(n-1,a)+f(n-2,a)
%C A173006 Row sums are:
%C A173006 {1, 2, 13, 242, 16900, 3426962, 2610255700, 5773759285448, 47972252879976100,
%C A173006 1157507562695117906888, 104909162208463229766370000,...}.
%C A173006 a = 1; A034801.
%C A173006 a = 2; A156600.
%C A173006 a = 3; A156602.
%C A173006 This result seems to connect these new recursions directly to q-forms.
%F A173006 a=5; f(n,a)=(2*a+1)*f(n-1,a)+f(n-2,a);
%F A173006 c(n)=If[n == 0, 1, Product[f(i, a), {i, 1, n}]];
%F A173006 t(n,m)=c(n)/(c(m)*c(n-m)
%e A173006 {1},
%e A173006 {1, 1},
%e A173006 {1, 11, 1},
%e A173006 {1, 120, 120, 1},
%e A173006 {1, 1309, 14280, 1309, 1},
%e A173006 {1, 14279, 1699201, 1699201, 14279, 1},
%e A173006 {1, 155760, 202190640, 2205562898, 202190640, 155760, 1},
%e A173006 {1, 1699081, 24058986960, 2862818956682, 2862818956682, 24058986960, 1699081, 1},
%e A173006 {1, 18534131, 2862817257601, 3715936800366098, 40534653607660438, 3715936800366098, 2862817257601, 18534131, 1},
%e A173006 {1, 202176360, 340651194667560, 4823283104057937603, 573930157592104171920, 573930157592104171920, 4823283104057937603, 340651194667560, 202176360, 1},
%e A173006 {1, 2205405829, 40534629348182040, 6260617753130421176727, 8126277060814812179812443, 88644086770258081457215920, 8126277060814812179812443, 6260617753130421176727, 40534629348182040, 2205405829, 1}
%t A173006 Clear[f, c, a, t];
%t A173006 f[0, a_] := 0; f[1, a_] := 1;
%t A173006 f[n_, a_] := f[n, a] = (2*a + 1)*f[n - 1, a] - f[n - 2, a];
%t A173006 c[n_, a_] := If[n == 0, 1, Product[f[i, a], {i, 1, n}]];
%t A173006 t[n_, m_, a_] := c[n, a]/(c[m, a]*c[n - m, a]);
%t A173006 Table[Flatten[Table[Table[t[n, m, a], {m, 0, n}], {n, 0, 10}]], {a, 1, 10}]
%Y A173006 A034801, A156600., A156602.
%K A173006 nonn,tabl,uned,new
%O A173006 0,5
%A A173006 Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Feb 07 2010

%I A172505
%S A172505 10,400,41000,12340000,411500000,41152000000
%N A172505 a(n) = numerator of fraction a / b, where (a, b) = 1, such that its decimal representation has form (1)(2)(3)...(n-1)(n),(1)(2)(3)...(n-1)(n)... with period (1)(2)(3)...(n-1)(n).
%C A172505 Sequence of denominators: 9, 33, 333, 9999, 33333, 333333,... Conjecture: this sequence is not equal to the sequence A172498.
%e A172505 a(6) = 41152000000; 41152000000 / 333333 = 123456,123456123456... (period 123456).
%K A172505 nonn,new
%O A172505 1,1
%A A172505 Jaroslav Krizek (jaroslav.krizek(AT)atlas.cz), Feb 05 2010

%I A172499
%S A172499 10,70,1040,41230,170780,2041150,71234560,90260630,1013717420,
%T A172499 1012345678900,37078189297000
%N A172499 a(n) = numerator of fraction a / b, where (a, b) = 1, such that its decimal representation has form (n),(1)(2)(3)...(n-1)(n)... with period (1)(2)(3)...(n-1)(n).
%C A172499 Sequence of denominators: 9, 33, 333, 9999, 33333, 333333, 9999999, 11111111, 111111111, 99999999999, 3333333333333,... Conjecture: this sequence is not equal to the sequence A172498.
%e A172499 a(10) = 1012345678900; 1012345678900 / 99 999 999 999 = 10,1234567891012345678910... (period 12345678910).
%K A172499 nonn,new
%O A172499 1,1
%A A172499 Jaroslav Krizek (jaroslav.krizek(AT)atlas.cz), Feb 05 2010

%I A172503
%S A172503 10,20,10,40,50,20,70,80,10,1000,100,400,1300,1400,500,1600,1700,200,
%T A172503 1900,2000,700,200,2300,800,2500,2600,300
%N A172503 a(n) = numerator of fraction a / b, where (a, b) = 1, such that its decimal representation has form (n),(n)(n)(n)...with period (n).
%C A172503 Denominators in A172504.
%e A172503 a(10) = 1000; 1000 / 99 = 10,10101010... (period 10). a(9) = 10; 10 / 1 = 9,9999999...
%K A172503 nonn,new
%O A172503 1,1
%A A172503 Jaroslav Krizek (jaroslav.krizek(AT)atlas.cz), Feb 05 2010

%I A173045
%S A173045 1,1,1,1,10,1,1,29,29,1,1,84,6566,84,1,1,247,14348916,14348916,247,1,1,
%T A173045 734,282429536495,150094635296999140,282429536495,734,1,1,2193,
%U A173045 50031545098999727,2503155504993241601315571986085883
%N A173045 A q-form based triangle sequence: q=3;t(n,m,q)=If[m == 0 || m == n, 1, Binomial[n, m] - 1 + q^(n*Binomial[n - 2, m - 1])]
%C A173045 Row sums are:
%C A173045 1, 2, 12, 60, 6736, 28698328, 150095200156073600,
%C A173045 5006311009986483302694234170175608, 21847450052839212627824677102819097563393409763931962119808661357483654308096,
%C A173045 39284862042082643167611399471700774231788053748085474390953577983511620779427 94313740053319169820149039646194575489518861696463621071963747427606467504,
%C A173045 965780214059175804381244203152292843737119463677684309983826005534530244345698 694834272101310974453567376640690192266392960798657510194641855661376180599552 831890693933466731308753442464044840253451882326197449871631461447202060259050 555042483076099181054138570805537221340792626238235748788698176841343604051548 8901814034815484937920
%F A173045 q=3;
%F A173045 t(n,m,q)=If[m == 0 || m == n, 1, Binomial[n, m] - 1 + q^(n*Binomial[n - 2, m - 1])]
%e A173045 {1},
%e A173045 {1, 1},
%e A173045 {1, 10, 1},
%e A173045 {1, 29, 29, 1},
%e A173045 {1, 84, 6566, 84, 1},
%e A173045 {1, 247, 14348916, 14348916, 247, 1},
%e A173045 {1, 734, 282429536495, 150094635296999140, 282429536495, 734, 1},
%e A173045 {1, 2193, 50031545098999727, 2503155504993241601315571986085883, 2503155504993241601315571986085883, 50031545098999727, 2193, 1},
%e A173045 {1, 6568, 79766443076872509863388, 1797010299914431210413179829509605039731475627537851106456, 21847450052839212624230656502990235142567050104912751880812823948662932355270, 1797010299914431210413179829509605039731475627537851106456, 79766443076872509863388, 6568, 1},
%e A173045 {1, 19691, 1144561273430837494885949696462, 149939874158678820041423971072487610193361136600334465711852281855799133432291 9287339806566, 196424310210413215838056997358503871158940268740427371954767739977683945218319 6742630315934708808140908457093943087640906885112545770821459540513711032, 196424310210413215838056997358503871158940268740427371954767739977683945218319 6742630315934708808140908457093943087640906885112545770821459540513711032, 149939874158678820041423971072487610193361136600334465711852281855799133432291 9287339806566, 1144561273430837494885949696462, 19691, 1}, ...
%t A173045 Clear[t, n, m, q];
%t A173045 t[n_, m_, q_] := If[m == 0 || m == n, 1, Binomial[n, m] - 1 + q^(n*Binomial[n - 2, m - 1])];
%t A173045 Table[Table[Table[t[n, m, q], {m, 0, n}], {n, 0, 10}], {q, 1, 10}];
%t A173045 Table[Flatten[Table[Table[t[n, m, q], {m, 0, n}], {n, 0, 10}]], {q, 1, 10}]
%K A173045 nonn,tabl,uned,new
%O A173045 0,5
%A A173045 Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Feb 08 2010

%I A173047
%S A173047 1,1,1,1,10,1,1,29,29,1,1,84,167,84,1,1,247,738,738,247,1,1,734,2930,
%T A173047 4393,2930,734,1,1,2193,10955,21904,21904,10955,2193,1,1,6568,39393,
%U A173047 98470,131289,98470,39393,6568,1,1,19691,137816,413426,689030,689030
%N A173047 A q-form based triangle sequence: q=3;t(n,m,q)=If[m == 0 || m == n, 1, Binomial[n, m] - 1 + (q^n)*Binomial[n - 2, m - 1]]
%C A173047 Row sums are:
%C A173047 {1, 2, 12, 60, 337, 1972, 11723, 70106, 420153, 2519928, 15117559,...}.
%F A173047 q=3;
%F A173047 t(n,m,q)=If[m == 0 || m == n, 1, Binomial[n, m] - 1 + (q^n)*Binomial[n - 2, m - 1]]
%e A173047 {1},
%e A173047 {1, 1},
%e A173047 {1, 10, 1},
%e A173047 {1, 29, 29, 1},
%e A173047 {1, 84, 167, 84, 1},
%e A173047 {1, 247, 738, 738, 247, 1},
%e A173047 {1, 734, 2930, 4393, 2930, 734, 1},
%e A173047 {1, 2193, 10955, 21904, 21904, 10955, 2193, 1},
%e A173047 {1, 6568, 39393, 98470, 131289, 98470, 39393, 6568, 1},
%e A173047 {1, 19691, 137816, 413426, 689030, 689030, 413426, 137816, 19691, 1},
%e A173047 {1, 59058, 472436, 1653491, 3306953, 4133681, 3306953, 1653491, 472436, 59058, 1}
%t A173047 Clear[t, n, m, q];
%t A173047 t[n_, m_, q_] := If[m == 0 || m == n, 1, Binomial[n, m] - 1 + (q^n)*Binomial[n - 2, m - 1]];
%t A173047 Table[Table[Table[t[n, m, q], {m, 0, n}], {n, 0, 10}], {q, 1, 10}];
%t A173047 Table[Flatten[Table[Table[t[n, m, q], {m, 0, n}], {n, 0, 10}]], {q, 1, 10}]
%K A173047 nonn,tabl,uned,new
%O A173047 0,5
%A A173047 Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Feb 08 2010

%I A172498
%S A172498 9,33,333,9999,33333,333333,9999999,11111111,111111111,99999999999,
%T A172498 3333333333333
%N A172498 a(n) = denominator of fraction a / b, where (a, b) = 1, such that its decimal representation has form 0,(1)(2)(3)...(n-1)(n)... with period (1)(2)(3)...(n-1)(n).
%C A172498 Numerators in A172496. Conjecture: a(n) is not equal to the sequence of denominators presented in A172498.
%e A172498 a(10) = 99999999999; 12345678910 / 99999999999 = 0,1234567891012345678910... (period 12345678910).
%K A172498 nonn,new
%O A172498 1,1
%A A172498 Jaroslav Krizek (jaroslav.krizek(AT)atlas.cz), Feb 05 2010

%I A173037
%S A173037 9,15,105,195,825,1485,1875,2085,3255
%N A173037 Odd numbers n such that n-+{2,4} are all primes.
%e A173037 a(1)=9 because 9-4=5=prime, 9-2=7=prime, 9+2=11=prime and 9+4=13=prime.
%Y A173037 Cf. A087679, A164385.
%K A173037 nonn,new
%O A173037 1,1
%A A173037 Juri-Stepan Gerasimov (2stepan(AT)rambler.ru), Feb 07 2010

%I A172502
%S A172502 9,9,3,9,9,3,9,9,1,99,9,33,99,99,33,99,99,11,99,99,33,9,99,33,99,99,11,
%T A172502 99,99,33
%N A172502 a(n) = denominator of fraction a / b, where (a, b) = 1, such that its decimal representation has form 0,(n)(n)(n)...with period (n).
%C A172502 Numerators in A172500.
%e A172502 a(10) = 99; 10 / 99 = 0,10101010... (period 10). a(9) = 1; 1 / 1 = 0,9999999...
%K A172502 nonn,new
%O A172502 1,1
%A A172502 Jaroslav Krizek (jaroslav.krizek(AT)atlas.cz), Feb 05 2010

%I A172504
%S A172504 9,9,3,9,9,3,9,9,1,99,9,33,99,99,33,99,99,11,99,99,33,9,99,33,99,99,11
%N A172504 a(n) = denominator of fraction a / b, where (a, b) = 1, such that its decimal representation has form (n),(n)(n)(n)...with period (n).
%C A172504 Numerators in 172503.
%e A172504 a(10) = 99; 1000 / 99 = 10,10101010... (period 10). a(9) = 1; 10 / 1 = 9,9999999...
%K A172504 nonn,new
%O A172504 1,1
%A A172504 Jaroslav Krizek (jaroslav.krizek(AT)atlas.cz), Feb 05 2010

%I A173005
%S A173005 1,1,1,1,9,1,1,80,80,1,1,711,6320,711,1,1,6319,499201,499201,6319,1,1,
%T A173005 56160,39430560,350439102,39430560,56160,1,1,499121,3114515040,
%U A173005 246007756722,246007756722,3114515040,499121,1,1,4435929,246007257601
%N A173005 A product triangle sequence based on recursion:a=4; f(n,a)=(2*a+1)*f(n-1,a)+f(n-2,a)
%C A173005 Row sums are:
%C A173005 {1, 2, 11, 162, 7744, 1011042, 429412544, 498245541768, 1880728607247424,
%C A173005 19394268001029953928, 650631110504313946320896,...}.
%C A173005 a = 1; A034801.
%C A173005 a = 2; A156600.
%C A173005 a = 3; A156602.
%C A173005 This result seems to connect these new recursions directly to q-forms.
%F A173005 a=4; f(n,a)=(2*a+1)*f(n-1,a)+f(n-2,a);
%F A173005 c(n)=If[n == 0, 1, Product[f(i, a), {i, 1, n}]];
%F A173005 t(n,m)=c(n)/(c(m)*c(n-m)
%e A173005 {1},
%e A173005 {1, 1},
%e A173005 {1, 9, 1},
%e A173005 {1, 80, 80, 1},
%e A173005 {1, 711, 6320, 711, 1},
%e A173005 {1, 6319, 499201, 499201, 6319, 1},
%e A173005 {1, 56160, 39430560, 350439102, 39430560, 56160, 1},
%e A173005 {1, 499121, 3114515040, 246007756722, 246007756722, 3114515040, 499121, 1},
%e A173005 {1, 4435929, 246007257601, 172697094835902, 1534842394188558, 172697094835902, 246007257601, 4435929, 1},
%e A173005 {1, 39424240, 19431458835440, 121233114567545603, 9575881454449171680, 9575881454449171680, 121233114567545603, 19431458835440, 39424240, 1},
%e A173005 {1, 350382231, 1534839240742160, 85105473729326613333, 59743922859711995180563, 530973050767752120484320, 59743922859711995180563, 85105473729326613333, 1534839240742160, 350382231, 1}
%t A173005 Clear[f, c, a, t];
%t A173005 f[0, a_] := 0; f[1, a_] := 1;
%t A173005 f[n_, a_] := f[n, a] = (2*a + 1)*f[n - 1, a] - f[n - 2, a];
%t A173005 c[n_, a_] := If[n == 0, 1, Product[f[i, a], {i, 1, n}]];
%t A173005 t[n_, m_, a_] := c[n, a]/(c[m, a]*c[n - m, a]);
%t A173005 Table[Flatten[Table[Table[t[n, m, a], {m, 0, n}], {n, 0, 10}]], {a, 1, 10}]
%Y A173005 A034801, A156600., A156602.
%K A173005 nonn,tabl,uned,new
%O A173005 0,5
%A A173005 Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Feb 07 2010

%I A172391
%S A172391 1,8,12,0,28,0,264,0,3720,0,63840,0,1232432,0,25731216,0,568130552,0,
%T A172391 13081215840,0,311178567648,0,7597974517056,0,189518147463232,0,
%U A172391 4811962763222784,0,124028853694440640,0,3238304402221646880,0
%N A172391 G.f. satisfies: A(x) = G(x/A(x))^2 and G(x)^2 = A(x*G(x)^2) where G(x) = Sum_{n>=0} C(2n,n)*C(2n+2,n+1)/(n+2)*x^n is the g.f. of A172392.
%F A172391 G.f.: A(x) = x/Series_Reversion(x*G(x)^2)) where G(x) is the g.f. of A172392(n) = A000108(n+1)*A000984(n).
%F A172391 Self-convolution of A172393.
%e A172391 G.f.: A(x) = 1 + 8*x + 12*x^2 + 28*x^4 + 264*x^6 + 3720*x^8 +...
%e A172391 where A(x) = G(x/A(x))^2 where G(x) is the g.f. of A172392:
%e A172391 G(x) = 1 + 4*x + 30*x^2 + 280*x^3 + 2940*x^4 + 33264*x^5 +...+ A172392(n)*x^n +...
%e A172391 G(x) = 1 + 2*2*x + 5*6*x^2 + 14*20*x^3 + 42*70*x^4 + 132*252*x^5 +...
%o A172391 (PARI) {a(n)=local(G=sum(m=0,n,binomial(2*m,m)*binomial(2*m+2,m+1)/(m+2)*x^m)+x*O(x^n));polcoeff(x/serreverse(x*G^2),n)}
%Y A172391 Cf. A172392, A172393, variants: A172390, A168357, A168451, A168452.
%K A172391 nonn,new
%O A172391 0,2
%A A172391 Paul D. Hanna (pauldhanna(AT)juno.com), Feb 05 2010

%I A172490
%S A172490 7,31,43,67,307,367,487,643,1327,1663,2371,3643,3847,4327,4951,6091,
%T A172490 6571,8263,9151,9187,11239,11383,11863,15307,24007,24151,27847,30091,
%U A172490 30643,33619,36871,42187,44171,46279,46591,48787,70843,71887,72103
%N A172490 Primes p of the form 4m+3 for which there are as many primitive roots modulo p in the interval [0,p/2] as in the interval [p/2,p].
%C A172490 The sequence is probably infinite.
%C A172490 Primes of the form 4m+1 allways have as many primitive roots in [0,p/2] as in [p/2,p] (see A172480).
%t A172490 << NumberTheory`NumberTheoryFunctions` m = 2; s = {}; While[m < 10000, m++; p = Prime[m]; If[Mod[p, 4] == 1, , q = (p - 1)/2; g = PrimitiveRoot[p]; se = Select[Range[p - 1], GCD[ #, p - 1] == 1 &]; e = Length[se]; j = 0; t = 0; While[j < e, j++; h = PowerMod[g, se[[j]], p]; If[h <= q, t = t + 1,] ]; If[e == 2t, s = {s, p},] ] ]; s = Flatten[s]
%Y A172490 Cf. A118818, A172480
%K A172490 hard,nonn,unkn,new
%O A172490 1,1
%A A172490 Emmanuel Vantieghem (manuvti(AT)hotmail.com), Feb 05 2010

%I A173024
%S A173024 0,7,14,15,28,30,31,56,60,62,63,112,119,120,124,126,127,224,231,238,239,
%T A173024 240,247,248,252,254,255,448,455,462,463,476,478,479,480,487,494,495,
%U A173024 496,503,504,508,510,511,896,903,910,911,924,926,927,952,956,958,959
%N A173024 Numbers having neither isolated ones nor isolated double ones in their binary representations.
%C A173024 Intersection of A144795 and A173024;
%C A173024 A173021(a(n+1)) = A173021(a(n)) + 1;
%C A173024 if m is a term then also 2*m is a term; if m is an odd term then also 2*m+1 ia term.
%Y A173024 Cf. A005251.
%K A173024 base,nonn,new
%O A173024 1,2
%A A173024 Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Feb 07 2010

%I A172965
%S A172965 0,0,6,240,1010,4056,12068,30000,65628,130480,240856,418968,694200,
%T A172965 1104488,1697820,2533856,3685668,5241600
%N A172965 Number of ways to place 3 nonattacking knights on an n X n cylindrical board
%H A172965 V. Kotesovec, <a href="http://web.telecom.cz/vaclav.kotesovec/math.htm">Number of ways of placing non-attacking queens and kings on boards of various sizes</a>
%F A172965 Explicit formula (Vaclav Kotesovec, 31.1.2010): a(n) = n*(n-3)(n^4+3n^3-18n^2-18n+164)/6, n>=6
%Y A172965 A172530, A172134, A172964
%K A172965 nonn,new
%O A172965 1,3
%A A172965 Vaclav Kotesovec (kotesovec(AT)chello.cz), Feb 06 2010

%I A172530
%S A172530 0,0,6,208,600,3252,10584,27584,61992,125300,233772,409584,682084,
%T A172530 1089172,1678800,2510592,3657584,5208084,7267652,9961200
%N A172530 Number of ways to place 3 nonattacking knights on an n X n toroidal board
%H A172530 V. Kotesovec, <a href="http://web.telecom.cz/vaclav.kotesovec/math.htm">Number of ways of placing non-attacking queens and kings on boards of various sizes</a>
%F A172530 Explicit formula (Vaclav Kotesovec, 31.1.2010): a(n) = n^2*(n^4-27n^2+218)/6, n>=6
%Y A172530 A172529, A172134, A172518
%K A172530 nonn,new
%O A172530 1,3
%A A172530 Vaclav Kotesovec (kotesovec(AT)chello.cz), Feb 06 2010

%I A172489
%S A172489 1,6,36,216,46656,2176782336,4738381338321616896,
%T A172489 22452257707354557240087211123792674816,
%U A172489 504103876157462118901767181449118688686067677834070116931382690099920633856
%N A172489 a(n) = 6^sum_{i=1..n-1}=product_{i=1..n-1} a(0)=1 a(1)=6
%F A172489 Except first term, a(n) = (a(n-1))^2
%K A172489 nonn,new
%O A172489 1,2
%A A172489 Giovanni Teofilatto (g.teofilatto(AT)tiscalinet.it), Feb 05 2010

%I A173031
%S A173031 1,6,24,79,232,632,1633,4058,9788,23063,53332,121452,273089,607534,
%T A173031 1339376,2929951,6366480,13752880,29556545,63232370,134731956,286044711,
%U A173031 605326044,1277246724,2687879137,5642847462,11820387528,24710992303
%N A173031 Sequence whose G.f is given by: 1/(1-z)/(1-2*z)^2/(1-z-z^2).
%F A173031 Recurrence formula: with the first values, a(n+5):=6*a(n+4)-12*a(n+3)+7*a(n+2)+4*a(n+1)-4*a(n). a(n)= (38/5*5^(1/2)+17)*((1+sqrt(5))/2)^n+(-38/5*5^(1/2)+17)*((1-sqrt(5))/2)^n-32*2^n-1+16*2^(n-1)*n. a(n)=F(n+8)+2^(n+3)*(n-4)-1, where (F(n)) is the Fibonacci sequence for which F(0)=F(1)=1, F(2)=2, aso (linked with A000045).
%e A173031 from f(z)=1+6z+24z^2+79z^3+232z^4+632z^5+1633z^6+4058z^7+9788z^8+23063z^9+53332z^10+121452z^11+273089z^12+607534z^13+... comes a(0)=1, a(6)=1633 for examples.
%p A173031 c(0):=1:c(1):=6:c(2):=24:c(3):=79:c(4):=232:for n from 0 to 30 do : c(n+5):=6*c(n+4)-12*c(n+3)+7*c(n+2)+4*c(n+1)-4*c(n): od :seq(c(n),n=0..30); taylor((-1/(-1+z)/(-1+2*z)^2/(1-z-z^2)),z=0,30); for n from 0 to 30 do a(n):=simplify((38/5*5^(1/2)+17)*((1+sqrt(5))/2)^n+(-38/5*5^(1/2)+17)*((1-sqrt(5))/2)^n-32*2^n-1+16*2^(n-1)*n):od:seq(a(n),n=0..30);
%K A173031 easy,nonn,new
%O A173031 0,2
%A A173031 Richard Choulet (richardchoulet(AT)yahoo.fr), Feb 07 2010

%I A172522
%S A172522 6,18,36,56,77,101,131,167,207,249,297,351,411,474,540,612,690,770,854,
%T A172522 944,1040,1140,1242,1347,1455,1565,1679,1799,1925,2057,2193,2331,2471,
%U A172522 2615,2762,2912,3067,3223,3383,3545
%N A172522 Partial sums of A049094.
%C A172522 The subsequence of primes in this sequence begins: 101, 131, 167, 3067. The subsequence of primes in this sequence begins: 36.
%F A172522 a(n) = SUM[i=1..n] {i such that 2^i - 1 is divisible by a square}.
%e A172522 a(20) = 6 + 12 + 18 + 20 + 21 + 24 + 30 + 36 + 40 + 42 + 48 + 54 + 60 + 63 + 66 + 72 + 78 + 80 + 84 + 90.
%Y A172522 Cf. A000040, A014491, A049093, A049096, A049095, A001220.
%K A172522 more,nonn,new
%O A172522 1,1
%A A172522 Jonathan Vos Post (jvospost3(AT)gmail.com), Feb 06 2010

%I A173041
%S A173041 5,23,203,2003,20003,200003,2000003,20000003,200000003,2000000003,
%T A173041 20000000003,200000000003,2000000000003,20000000000003,200000000000003,
%U A173041 2000000000000003,20000000000000003,200000000000000003
%N A173041 a(n)=2*10^n+3
%K A173041 nonn,new
%O A173041 0,1
%A A173041 Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Feb 08 2010

%I A173034
%S A173034 1,5,20,52,116,244,500,1012,2036,4084,8180,16372,32756,65524,131060,
%T A173034 262132,524276,1048564,2097140,4194292,8388596,16777204,33554420,
%U A173034 67108852,134217716,268435444,536870900,1073741812,2147483636
%N A173034 Sequence whose G.f is f such that: f(z)=8/(1-2*z)-12/(1-z)+z+5.
%C A173034 The Granvik array of A170119 is here written in "square": 1 :: 1 :: 1 :: 1 :: 1 :: 1 :: 1 :: 1 :: 1 :: 1 // 1 :: 2 :: 2 :: 2 :: 2 :: 2 :: 2 :: 2 :: 2 :: 2 // 1 :: 3 :: 4 :: 4 :: 4 :: 4 :: 4 :: 4 :: 4 :: 4 // 1 :: 4 :: 7 :: 8 :: 8 :: 8 :: 8 :: 8 :: 8 :: 8 // This sequence gives the third diagonal under the main diagonal.
%e A173034 From 1+5*z+20*z^2+52*z^3+116*z^4+244*z^5+500*z^6+1012*z^7+2036*z^8+4084*z^9+8180*z^10+16372*z^11+32756*z^12+65524*z^13+131060*z^14+262132*z^15+524276*z^16+1048564*z^17+2097140*z^18+4194292*z^19+8388596*z^20+16777204*z^21+33554420*z^22+67108852*z^23+134217716*z^24+268435444*z^25+536870900*z^26+1073741812*z^27+2147483636*z^28+4294967284*z^29+O(z^30) we get a(6)=500.
%Y A173034 C.f A172119, A173033.
%K A173034 easy,nonn,new
%O A173034 0,2
%A A173034 Richard Choulet (richardchoulet(AT)yahoo.fr), Feb 07 2010

%I A172996
%S A172996 5,17,1097,304133393856678854559841996309430657,
%T A172996 6747294337140546901864078698261942458701251147015050957033352683510961004453136793029129177,
%U A172996 145166236803607666075029365382581885219152159697093242310434615656917871505314679334525305616146366580565745544908139736351707393151902400605236799289167195530782480090041796692270130317057
%N A172996 Primes of form (3^n+7)/2.
%Y A172996 Cf.172995
%K A172996 nonn,new
%O A172996 1,1
%A A172996 Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Feb 07 2010

%I A172988
%S A172988 5,11,13,17,19,23,29,37,43,47,53,59,67,73,79,89,103,107,109,113,137,157,
%T A172988 163,173,179,197,199,229,233,239,257,263,269,277,283,313,317,337,353,
%U A172988 359,373,379,389,439,463,467,509,547,563,569,577,593,599,607,613,619
%N A172988 Primes p such that either p-9/2-+3/2 is prime.
%F A172988 a(n+1)=A067831(n+1).
%e A172988 a(1)=5 because 5-9/2-3/2=-1=nonprime and 5-9/2+3/2=2=prime; a(2)=11 because 11-9/2-3/2=5=prime and 11-9/2+3/2=8=nonprime.
%Y A172988 Cf. A000040, A046117, A067831.
%K A172988 nonn,new
%O A172988 1,1
%A A172988 Juri-Stepan Gerasimov (2stepan(AT)rambler.ru), Feb 07 2010

%I A173043
%S A173043 1,1,1,1,5,1,1,10,10,1,1,19,261,19,1,1,36,32777,32777,36,1,1,69,
%T A173043 16777230,68719476755,16777230,69,1,1,134,34359738388,
%U A173043 1180591620717411303458,1180591620717411303458,34359738388,134,1,1,263
%N A173043 A q-form based triangle sequence: q=2;t(n,m,q)=If[m == 0 || m == n, 1, Binomial[n, m] - 1 + q^(n*Binomial[n - 2, m - 1])]
%C A173043 Row sums are:
%C A173043 {1, 2, 7, 22, 301, 65628, 68753031355, 2361183241503542083962,
%C A173043 1461501637333561374195254664462091196726446654201,
%C A173043 133499189745056880149688856635597007164238308081137468312649047844690649465334 069981624848614904,
%C A173043 526013590154837350724098988288012866555034735074802350258098538617789413083486 265595610736915844520363958346010450300201072360838897194363212103954632575602 4682836497111382552055971384380062912111180219292847095,...}.
%F A173043 q=2;
%F A173043 t(n,m,q)=If[m == 0 || m == n, 1, Binomial[n, m] - 1 + q^(n*Binomial[n - 2, m - 1])]
%e A173043 {1},
%e A173043 {1, 1},
%e A173043 {1, 5, 1},
%e A173043 {1, 10, 10, 1},
%e A173043 {1, 19, 261, 19, 1},
%e A173043 {1, 36, 32777, 32777, 36, 1},
%e A173043 {1, 69, 16777230, 68719476755, 16777230, 69, 1},
%e A173043 {1, 134, 34359738388, 1180591620717411303458, 1180591620717411303458, 34359738388, 134, 1},
%e A173043 {1, 263, 281474976710683, 1329227995784915872903807060280344631, 1461501637330902918203684832716283019655932543045, 1329227995784915872903807060280344631, 281474976710683, 263, 1},
%e A173043 {1, 520, 9223372036854775843, 784637716923335095479473677900958302012794430558004314195, 667495948725284400748444283177985035813345163236453990608450502444443664306450 17188217565216893, 667495948725284400748444283177985035813345163236453990608450502444443664306450 17188217565216893, 784637716923335095479473677900958302012794430558004314195, 9223372036854775843, 520, 1},
%e A173043 {1, 1033, 1208925819614629174706220, 194266889222572907091946190682351890664240683905213952125181240973890428520520 8498295, 377396242482154135224155458098826889091692122041644042837620630024562 416239214885208612672517765876754146837503076384489977058462992479263256143425 1432696043649395327185, 526013590154837350724098988288012866555033980282317385949828090306873215429708 082211366653627758845122698296885617821771301943225018380386312781477065188084 9955223671128444598191663757884322717271293251735781627, 377396242482154135224155458098826889091692122041644042837620630024562416239214 885208612672517765876754146837503076384489977058462992479263256143425143269604 3649395327185, 194266889222572907091946190682351890664240683905213952125181240973890428520520 8498295, 1208925819614629174706220, 1033, 1}
%t A173043 Clear[t, n, m, q];
%t A173043 t[n_, m_, q_] := If[m == 0 || m == n, 1, Binomial[n, m] - 1 + q^(n*Binomial[n - 2, m - 1])];
%t A173043 Table[Table[Table[t[n, m, q], {m, 0, n}], {n, 0, 10}], {q, 1, 10}];
%t A173043 Table[Flatten[Table[Table[t[n, m, q], {m, 0, n}], {n, 0, 10}]], {q, 1, 10}]
%K A173043 nonn,tabl,uned,new
%O A173043 0,5
%A A173043 Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Feb 08 2010

%I A173046
%S A173046 1,1,1,1,5,1,1,10,10,1,1,19,37,19,1,1,36,105,105,36,1,1,69,270,403,270,
%T A173046 69,1,1,134,660,1314,1314,660,134,1,1,263,1563,3895,5189,3895,1563,263,
%U A173046 1,1,520,3619,10835,18045,18045,10835,3619,520,1,1,1033,8236,28791
%N A173046 A q-form based triangle sequence: q=2;t(n,m,q)=If[m == 0 || m == n, 1, Binomial[n, m] - 1 + (q^n)*Binomial[n - 2, m - 1]]
%C A173046 Row sums are:
%C A173046 {1, 2, 7, 22, 77, 284, 1083, 4218, 16633, 66040, 263159,...}.
%F A173046 q=2;
%F A173046 t(n,m,q)=If[m == 0 || m == n, 1, Binomial[n, m] - 1 + (q^n)*Binomial[n - 2, m - 1]]
%e A173046 {1},
%e A173046 {1, 1},
%e A173046 {1, 5, 1},
%e A173046 {1, 10, 10, 1},
%e A173046 {1, 19, 37, 19, 1},
%e A173046 {1, 36, 105, 105, 36, 1},
%e A173046 {1, 69, 270, 403, 270, 69, 1},
%e A173046 {1, 134, 660, 1314, 1314, 660, 134, 1},
%e A173046 {1, 263, 1563, 3895, 5189, 3895, 1563, 263, 1},
%e A173046 {1, 520, 3619, 10835, 18045, 18045, 10835, 3619, 520, 1},
%e A173046 {1, 1033, 8236, 28791, 57553, 71931, 57553, 28791, 8236, 1033, 1}
%t A173046 Clear[t, n, m, q];
%t A173046 t[n_, m_, q_] := If[m == 0 || m == n, 1, Binomial[n, m] - 1 + (q^n)*Binomial[n - 2, m - 1]];
%t A173046 Table[Table[Table[t[n, m, q], {m, 0, n}], {n, 0, 10}], {q, 1, 10}];
%t A173046 Table[Flatten[Table[Table[t[n, m, q], {m, 0, n}], {n, 0, 10}]], {q, 1, 10}]
%K A173046 nonn,tabl,uned,new
%O A173046 0,5
%A A173046 Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Feb 08 2010

%I A173038
%S A173038 0,0,0,4,49,481,4681,47881,524161,6168961,78019201,1057795201,
%T A173038 15328051201,236626790401,3879433958401,67345229952001,1234444603392001,
%U A173038 23831057682432001,483379214782464001,10279010984546304001
%N A173038 Numbers of the form (1/4)*(n^2 - 5 n + 2)*(n - 2)! + 1
%C A173038 Genera of curves B whose function field K(B) is Galois closure [Galois closed, perhaps? - N. J. A. Sloane, Feb 08 2010].
%D A173038 Nils Bruin and Noam D. Elkies, Trinomials ax^7+bx+c and ax^8+bx+c with Galois Groups of Order 168 and 8*168, in Claus Fieker, David R. Kohel (Eds.): Algorithmic Number Theory, 5th International Symposium, ANTS-V, Lecture Notes in Computer Science 2369 Springer 2002, pp. 172 - 188.
%t A173038 Table[(1/4) (n^2 - 5 n + 2) (n - 2)! + 1, {n, 2, 30}]
%K A173038 nonn,new
%O A173038 2,4
%A A173038 Artur Jasinski (grafix(AT)csl.pl), Feb 08 2010

%I A172496
%S A172496 1,4,41,1234,4115,41152,1234567,1371742,13717421,12345678910,
%T A172496 411522630337
%N A172496 a(n) = numerator of fraction a / b, where (a, b) = 1, such that its decimal representation has form 0,(1)(2)(3)...(n-1)(n)... with period (1)(2)(3)...(n-1)(n).
%C A172496 Denominators in A172497.
%e A172496 a(10) = 12345678910; 12345678910 / 99999999999 = 0,1234567891012345678910... (period 12345678910).
%K A172496 nonn,new
%O A172496 1,2
%A A172496 Jaroslav Krizek (jaroslav.krizek(AT)atlas.cz), Feb 05 2010

%I A172392
%S A172392 1,4,30,280,2940,33264,396396,4907760,62573940,816621520,10861066216,
%T A172392 146738321184,2008917492400,27815780664000,388924218927000,
%U A172392 5484594083378400,77926940934668100,1114620641232714000
%N A172392 a(n) = C(2n,n)*C(2n+2,n+1)/(n+2).
%F A172392 G.f. A(X) satisfies: A(x)^2 = G(x*A(x)^2) and G(x) = A(x/G(x))^2 = g.f. of A172391.
%F A172392 G.f. A(X) satisfies: A(x) = G(x*A(x)^2) and G(x) = A(x/G(x)^2) = g.f. of A172393.
%F A172392 a(n) = (n+1)*A005568(n) = A000108(n+1)*A000984(n), where A000108 is the Catalan numbers and A000984 is the central binomial coefficients.
%e A172392 G.f.: A(x) = 1 + 4*x + 30*x^2 + 280*x^3 + 2940*x^4 + 33264*x^5 +...
%e A172392 A(x) = 1 + 2*2*x + 5*6*x^2 + 14*20*x^3 + 42*70*x^4 + 132*252*x^5 +...
%e A172392 where A(x)^2 = G(x*A(x)^2) and G(x) = A(x/G(x))^2 = g.f. of A172391:
%e A172392 A172391=[1,8,12,0,28,0,264,0,3720,0,63840,0,1232432,0,25731216,0,...].
%o A172392 (PARI) {a(n)=binomial(2*n,n)*binomial(2*n+2,n+1)/(n+2)}
%Y A172392 Cf. A172391, A172393, A005568.
%K A172392 nonn,new
%O A172392 0,2
%A A172392 Paul D. Hanna (pauldhanna(AT)juno.com), Feb 05 2010

%I A172964
%S A172964 0,4,18,92,230,522,1022,1808,2970,4610,6842,9792,13598,18410,24390,
%T A172964 31712,40562,51138
%N A172964 Number of ways to place 2 nonattacking knights on an n X n cylindrical board
%H A172964 V. Kotesovec, <a href="http://web.telecom.cz/vaclav.kotesovec/math.htm">Number of ways of placing non-attacking queens and kings on boards of various sizes</a>
%F A172964 Explicit formula (Vaclav Kotesovec, 31.1.2010): a(n) = n*(n^3-9n+12)/2, n>=5
%Y A172964 A172529, A172132
%K A172964 nonn,new
%O A172964 1,2
%A A172964 Vaclav Kotesovec (kotesovec(AT)chello.cz), Feb 06 2010

%I A171851
%S A171851 0,0,0,1,4,14,46,140,412,1186,3354,9368,25920,71182,194322,527927,
%T A171851 1428530,3852594,10360700,27795561,74414408,198862280,530590812,
%U A171851 1413712094,3762056094,10000260036,26556402534,70459947925,186796151768
%N A171851 The difference between the area under a peakless Motzkin path and the number of its U-steps, summed over all peakless Motzkin paths of length n (n>=0).
%C A171851 a(n)=Sum(k*A171850(n,k), k>=0).
%D A171851 R. Willenbring, RNA structure, permutations, and statistics, Discrete Appl. Math., 157, 2009, 1607-1614.
%F A171851 G.f. = z^3*g^2(1 - 2z + 2zg)/(1 - z + z^2 - 2z^2*g)^2, where g=g(z) satisfies g = 1 + zg + z^2*g(g - 1).
%e A171851 a(4)=4 because for the 4 (=A004148(4)) peakless Motzkin paths of length 4, namely, HHHH, HUHD, UHHD, UHDH, the areas under the paths are 0,2,3,2 and the number of U-steps are 0,1,1,1; now, (0-0) + (2-1) + (3-1) + (2-1) = 0 + 1 + 2 + 1 = 4.
%p A171851 g := ((1-z+z^2-sqrt(1-2*z-z^2-2*z^3+z^4))*1/2)/z^2: G := z^3*g^2*(1-2*z+2*z*g)/(1-z+z^2-2*z^2*g)^2: Gser := series(G, z = 0, 35): seq(coeff(Gser, z, n), n = 0 .. 30);
%Y A171851 A004148, A171850
%K A171851 nonn,new
%O A171851 0,5
%A A171851 Emeric Deutsch (deutsch(AT)duke.poly.edu), Feb 08 2010

%I A171849
%S A171849 0,0,0,1,4,12,36,104,292,810,2224,6058,16408,44240,118848,318339,850608,
%T A171849 2268206,6037892,16048945,42604344,112974302,299284044,792164740,
%U A171849 2095161996,5537651796,14627504340,38616930931,101899265656
%N A171849 Total area under all the level steps in all peakless Motzkin paths of length n (n>=0).
%C A171849 a(n)=Sum(k*A171848(n,k), k>=0).
%D A171849 R. Willenbring, RNA structure, permutations, and statistics, Discrete Appl. Math., 157, 2009, 1607-1614.
%F A171849 G.f. = z^3*g^2/[(1+z+z^2)(1-3z+z^2)], where g=g(z) satisfies g = 1 + zg + z^2*g(g - 1).
%e A171849 a(4)=4 because the areas under the level steps of the paths HHHH, HUHD, UHHD, UHDH are 0, 1, 2, 1, respectively.
%p A171849 g := ((1-z+z^2-sqrt(1-2*z-z^2-2*z^3+z^4))*1/2)/z^2: G := z^3*g^2/((1+z+z^2)*(1-3*z+z^2)): Gser := series(G, z = 0, 35): seq(coeff(Gser, z, n), n = 0 .. 30);
%Y A171849 A004148, A171848
%K A171849 nonn,new
%O A171849 0,5
%A A171849 Emeric Deutsch (deutsch(AT)duke.poly.edu), Feb 08 2010

%I A173033
%S A173033 1,4,12,28,60,124,252,508,1020,2044,4092,8188,16380,32764,65532,131068,
%T A173033 262140,524284,1048572,2097148,4194300,8388604,16777212,33554428,
%U A173033 67108860,134217724,268435452,536870908,1073741820,2147483644
%N A173033 Second diagonal under the main diagonal in A172119 written in a square (see comment)
%C A173033 The Granvik array of A170119 is here written in "square": 1 :: 1 :: 1 :: 1 :: 1 :: 1 :: 1 :: 1 :: 1 :: 1 // 1 :: 2 :: 2 :: 2 :: 2 :: 2 :: 2 :: 2 :: 2 :: 2 // 1 :: 3 :: 4 :: 4 :: 4 :: 4 :: 4 :: 4 :: 4 :: 4 // 1 :: 4 :: 7 :: 8 :: 8 :: 8 :: 8 :: 8 :: 8 :: 8 // 1 :: 5 :: 12 :: 15 :: 16 :: 16 :: 16 :: 16 :: 16 :: 16 // 1 :: 6 :: 20 :: 28 :: 31 :: 32 :: 32 :: 32 :: 32 :: 32 // 1 :: 7 :: 33 :: 52 :: 60 :: 63 :: 64 :: 64 :: 64 :: 64 // 1 :: 8 :: 54 :: 96 :: 116 :: 124 :: 127 :: 128 :: 128 :: 128 // 1 :: 9 :: 88 :: 177 :: 224 :: 244 :: 252 :: 255 :: 256 :: 256 //
%F A173033 G.f: f such that f(z)=4/(1-2*z)-4/(1-z)+1. a(n)=2^(n+2)-4 for n>=1, a(0)=1.
%e A173033 a(3)=2^5-4=32-4=28.
%p A173033 taylor(4/(1-2*z)-4/(1-z)+1,z=0,31);
%K A173033 easy,nonn,new
%O A173033 0,2
%A A173033 Richard Choulet (richardchoulet(AT)yahoo.fr), Feb 07 2010

%I A172513
%S A172513 1,4,7,11,14,17,20,24,27,30,33,37,40,43,46,50,53,56,59,63,66,69,73,76,
%T A172513 79,82,86,89,92,95,99,102,105,108,112,115,118,121,125,128,131,134,138,
%U A172513 141,144,147,151,154,157,161,164,167,170,174,177,180,183,187,190,193
%N A172513 Complement of A167389
%p A172513 sort(convert((convert([seq(n, n = 1 .. 1000)], set) minus convert([seq(round(evalf((argument(exp(-(ln(2)+LambertW(n, -(1/2)*ln(2)))/ln(2)))*ln(2)+Im(LambertW(n, -(1/2)*ln(2))))/(2*Pi*ln(2)))), n = 1 .. 1000)], set)), list))
%Y A172513 A167389
%K A172513 hard,nonn,new
%O A172513 1,2
%A A172513 Stephen Crowley (crow(AT)crowlogic.net), Feb 05 2010

%I A173049
%S A173049 1,4,4,28,24,28,730,390,390,730,59050,29280,7020,29280,59050,14348908,
%T A173049 7145292,914760,914760,7145292,14348908,10460353204,5223003240,
%U A173049 650485836,49397040,650485836,5223003240,10460353204,22876792454962
%N A173049 A triangle of polynomial coefficients:q=3;p(x,n,q)=If[n == 0, 1, Product[x + q^i, {i, 1, n}] + Product[x*q^i + 1, {i, 1, n}]]
%C A173049 Row sums are:
%C A173049 {1, 8, 80, 2240, 183680, 44817920, 32717081600, 71584974540800,
%C A173049 469740602936729600, 9246374028206585446400, 545998386365598870609920000,...}.
%F A173049 q=3;
%F A173049 p(x,n,q)=If[n == 0, 1, Product[x + q^i, {i, 1, n}] + Product[x*q^i + 1, {i, 1, n}]]
%e A173049 {1},
%e A173049 {4, 4},
%e A173049 {28, 24, 28},
%e A173049 {730, 390, 390, 730},
%e A173049 {59050, 29280, 7020, 29280, 59050},
%e A173049 {14348908, 7145292, 914760, 914760, 7145292, 14348908},
%e A173049 {10460353204, 5223003240, 650485836, 49397040, 650485836, 5223003240, 10460353204},
%e A173049 {22876792454962, 11433166054158, 1427188022442, 55340738838, 55340738838, 1427188022442, 11433166054158, 22876792454962},
%e A173049 {150094635296999122, 75035879252281920, 9375196185919800, 360108133318080, 8965199691756, 360108133318080, 9375196185919800, 75035879252281920, 150094635296999122},
%e A173049 {2954312706550833698644, 1477081305957768379284, 184607021930620264560, 7097023989026400240, 88955675043980472, 88955675043980472, 7097023989026400240, 184607021930620264560, 1477081305957768379284, 2954312706550833698644},
%e A173049 {174449211009120179071170508, 87223128348206814118824504, 10902337119274290150769308, 419256747357452492954400, 5238342611663783986584, 43232458071374509392, 5238342611663783986584, 419256747357452492954400, 10902337119274290150769308, 87223128348206814118824504, 174449211009120179071170508}
%t A173049 Clear[p, x, n, q];
%t A173049 p[x_, n_, q_] = If[n == 0, 1, Product[x + q^i, {i, 1, n}] + Product[x*q^i + 1, {i, 1, n}]];
%t A173049 Table[Table[CoefficientList[p[x, n, q], x], {n, 0, 10}], {q, 1, 10}];
%t A173049 Table[Flatten[Table[CoefficientList[p[x, n, q], x], {n, 0, 10}]], {q, 1, 10}]
%K A173049 nonn,tabl,uned,new
%O A173049 0,2
%A A173049 Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Feb 08 2010

%I A172985
%S A172985 1,1,1,1,1,1,1,1,1,1,1,4,4,4,1,1,1,4,4,1,1,1,6,6,24,6,6,1,1,1,6,6,6,6,1,
%T A172985 1,1,8,8,48,12,48,8,8,1,1,9,72,72,108,108,72,72,9,1,1,10,90,720,180,
%U A172985 1080,180,720,90,10,1
%N A172985 A composite number product triangle sequence: c(n)=Product[If[i is Prime,1,i],{i,0,n}];t(n,m)=c(n)/(c(m)*c(n-m))
%C A172985 Row sums are:
%C A172985 {1, 2, 3, 4, 14, 12, 50, 28, 142, 524, 3082,...}.
%C A172985 The composite product is:
%C A172985 {1, 1, 1, 1, 4, 4, 24, 24, 192, 1728, 17280, 17280, 207360, 207360, 2903040,
%C A172985 43545600, 696729600, 696729600, 12541132800, 12541132800, 250822656000,...}.
%F A172985 c(n)=Product[If[i is Prime,1,i],{i,0,n}];
%F A172985 t(n,m)=c(n)/(c(m)*c(n-m))
%e A172985 {1},
%e A172985 {1, 1},
%e A172985 {1, 1, 1},
%e A172985 {1, 1, 1, 1},
%e A172985 {1, 4, 4, 4, 1},
%e A172985 {1, 1, 4, 4, 1, 1},
%e A172985 {1, 6, 6, 24, 6, 6, 1},
%e A172985 {1, 1, 6, 6, 6, 6, 1, 1},
%e A172985 {1, 8, 8, 48, 12, 48, 8, 8, 1},
%e A172985 {1, 9, 72, 72, 108, 108, 72, 72, 9, 1},
%e A172985 {1, 10, 90, 720, 180, 1080, 180, 720, 90, 10, 1}
%t A172985 c[n_] := Product[If[i == 0, 1, If[PrimeQ[i], 1, i]], {i, 0, n}];
%t A172985 t[n_, m_] := c[n]/(c[m]*c[n - m]);
%t A172985 Table[Table[t[n, m], {m, 0, n}], {n, 0, 10}];
%t A172985 Flatten[%]
%K A172985 nonn,tabl,uned,new
%O A172985 0,12
%A A172985 Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Feb 06 2010

%I A172393
%S A172393 1,4,2,8,20,96,324,1648,6348,33200,137848,732640,3193296,17148608,
%T A172393 77335400,418289696,1934677436,10518803376,49611450120,270796872160,
%U A172393 1297234193744,7102371571840,34458382484976,189117499963840
%V A172393 1,4,-2,8,-20,96,-324,1648,-6348,33200,-137848,732640,-3193296,17148608,
%W A172393 -77335400,418289696,-1934677436,10518803376,-49611450120,270796872160,
%X A172393 -1297234193744,7102371571840,-34458382484976,189117499963840
%N A172393 G.f. satisfies: A(x) = G(x/A(x)^2) and G(x) = A(x*G(x)^2) = Sum_{n>=0} C(2n,n)*C(2n+2,n+1)/(n+2)*x^n is the g.f. of A172392.
%F A172393 G.f. satisfies: A(x) = Sum_{n>=0} A000108(n+1)*A000984(n)*x^n/A(x)^(2n), where A000108 is the Catalan numbers and A000984 is the central binomial coefficients.
%F A172393 Self-convolution equals A172391.
%e A172393 G.f.: A(x) = 1 + 4*x - 2*x^2 + 8*x^3 - 20*x^4 + 96*x^5 - 324*x^6 +...
%e A172393 A(x)^2 = 1 + 8*x + 12*x^2 + 28*x^4 + 264*x^6 + 3720*x^8 +...
%e A172393 where A(x)^2 equals the g.f. of A172391:
%e A172393 A172391=[1,8,12,0,28,0,264,0,3720,0,63840,0,1232432,0,25731216,0,...].
%e A172393 Let G(x) = A(x*G(x)^2) = Sum_{n>=0} C(2n+2,n+1)/(n+2)*C(2n,n)*x^n:
%e A172393 G(x) = 1 + 2*2*x + 5*6*x^2 + 14*20*x^3 + 42*70*x^4 + 132*252*x^5 +...
%o A172393 (PARI) {a(n)=local(G=sum(m=0,n,binomial(2*m,m)*binomial(2*m+2,m+1)/(m+2)*x^m)+x*O(x^n));polcoeff((x/serreverse(x*G^2))^(1/2),n)}
%Y A172393 Cf. A172391, A172392.
%K A172393 sign,new
%O A172393 0,2
%A A172393 Paul D. Hanna (pauldhanna(AT)juno.com), Feb 05 2010

%I A173008
%S A173008 1,4,1,64,20,1,4096,1344,84,1,1048576,348160,22848,340,1,1073741824,
%T A173008 357564416,23744512,371008,1364,1,4398046511104,1465657589760,
%U A173008 97615085568,1543393280,5957952,5460,1,72057594037927936
%N A173008 A triangle of polynomial coefficients of:q=4; p(x,n,q)=If[n == 0, 1, Product[x + q^i, {i, 1, n}]]
%C A173008 Row sums are:
%C A173008 {1, 5, 85, 5525, 1419925, 1455423125, 5962868543125, 97701601079103125,
%C A173008 6403069829921181503125, 1678532740564688125136703125,
%C A173008 1760070825503098980191468752703125,...}.
%C A173008 a = 2; A108084.
%F A173008 q=4;
%F A173008 p(x,n,q)=If[n == 0, 1, Product[x + q^i, {i, 1, n}]]
%e A173008 {1},
%e A173008 {4, 1},
%e A173008 {64, 20, 1},
%e A173008 {4096, 1344, 84, 1},
%e A173008 {1048576, 348160, 22848, 340, 1},
%e A173008 {1073741824, 357564416, 23744512, 371008, 1364, 1},
%e A173008 {4398046511104, 1465657589760, 97615085568, 1543393280, 5957952, 5460, 1},
%e A173008 {72057594037927936, 24017731997138944, 1600791219535872, 25384570585088, 99158478848, 95414592, 21844, 1},
%e A173008 {4722366482869645213696, 1574098141758535761920, 104933471095500046336, 1665204009083863040, 6523834640367616, 6352249180160, 1526982976, 87380, 1},
%e A173008 {1237940039285380274899124224, 412645105639632468417970176, 27509253945000522682466304, 436628173228375692804096, 1711849311973612191744, 1671727843724230656, 406641674440704, 24433125696, 349524, 1},
%e A173008 {1298074214633706907132624082305024, 432690992231222540584116394393600, 28845956109738507704758205153280, 457865332625062270980430233600, 1795440732325270753262960640, 1754645544772952496537600, 428066428262059868160, 26026630884249600, 390935603520, 1398100, 1}
%t A173008 Clear[p, x, n, q]
%t A173008 p[x_, n_, q_] = If[n == 0, 1, Product[x + q^i, {i, 1, n}]];
%t A173008 Table[Table[CoefficientList[p[x, n, q], x], {n, 0, 10}], {q, 2, 10}];
%t A173008 Table[Flatten[Table[CoefficientList[p[x, n, q], x], {n, 0, 10}]], {q, 2, 10}]
%Y A173008 A108084
%K A173008 nonn,tabl,uned,new
%O A173008 0,2
%A A173008 Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Feb 07 2010

%I A172395
%S A172395 1,1,1,0,1,0,4,0,27,0,248,0,2830,0,38232,0,593859,0,10401712,0,
%T A172395 202601898,0,4342263000,0,101551822350,0,2573779506192,0,70282204726396,
%U A172395 0,2057490936366320,0,64291032462761955,0,2136017303903513184,0
%N A172395 G.f. satisfies: A(x) = G(x/A(x)) where o.g.f. G(x) = A(x*G(x)) = Sum_{n>=0} A000085(n)*x^n.
%C A172395 The e.g.f. of A000085 is exp(x+x^2/2) = Sum_{n>=0} A000085(n)*x^n/n!, where A000085(n) is the number of self-inverse permutations on n letters.
%F A172395 a(2n-2) = A000699(n), the number of irreducible diagrams with 2n nodes, for n>=1.
%F A172395 a(2n-1) = 0 for n>=2, with a(1)=1.
%e A172395 G.f.: A(x) = 1 + x + x^2 + x^4 + 4*x^6 + 27*x^8 + 248*x^10 +...
%e A172395 where G(x) = A(x*G(x)) is the o.g.f. of A000085:
%e A172395 G(x) = 1 + x + 2*x^2 + 4*x^3 + 10*x^4 + 26*x^5 + 76*x^6 + 232*x^7 +...
%e A172395 while the e.g.f. of A000085 is given by:
%e A172395 exp(x+x^2/2) = 1 + x + 2*x^2/2! + 4*x^3/3! + 10*x^4/4! + 26*x^5/5! +...
%o A172395 (PARI) {a(n)=local(G=sum(m=0,n,m!*polcoeff(exp(x+x^2/2+x*O(x^m)),m)*x^m)+x*O(x^n));polcoeff(x/serreverse(x*G),n)}
%Y A172395 Cf. A000085, A000699, A172394 (variant).
%K A172395 nonn,new
%O A172395 0,7
%A A172395 Paul D. Hanna (pauldhanna(AT)juno.com), Feb 06 2010

%I A172394
%S A172394 1,1,1,0,1,0,4,0,27,0,248,0,2830,0,38232,0,593859,0,10401712,0,
%T A172394 202601898,0,4342263000,0,101551822350,0,2573779506192,0,
%U A172394 70282204726396,0,2057490936366320,0,64291032462761955,0
%V A172394 1,-1,-1,0,1,0,-4,0,27,0,-248,0,2830,0,-38232,0,593859,0,-10401712,0,
%W A172394 202601898,0,-4342263000,0,101551822350,0,-2573779506192,0,
%X A172394 70282204726396,0,-2057490936366320,0,64291032462761955,0
%N A172394 G.f. satisfies: A(x) = G(x/A(x)) where o.g.f. G(x) = A(x*G(x)) = Sum_{n>=0} A001464(n)*x^n.
%C A172394 The e.g.f. of A001464 is exp(-x-x^2/2) = Sum_{n>=0} A001464(n)*x^n/n!.
%F A172394 a(2n-2) = (-1)^(n-1)*A000699(n), where A000699(n) is the number of irreducible diagrams with 2n nodes, for n>=1.
%F A172394 a(2n-1) = 0 for n>=2, with a(1) = -1.
%e A172394 G.f.: A(x) = 1 - x - x^2 + x^4 - 4*x^6 + 27*x^8 - 248*x^10 +...
%e A172394 where G(x) = A(x*G(x)) is the o.g.f. of A001464:
%e A172394 G(x) = 1 - x + 2*x^3 - 2*x^4 - 6*x^5 + 16*x^6 + 20*x^7 - 132*x^8 +...
%e A172394 while the e.g.f. of A001464 is given by:
%e A172394 exp(-x-x^2/2) = 1 - x + 2*x^3/3! - 2*x^4/4! - 6*x^5/5! + 16*x^6/6! +...
%o A172394 (PARI) {a(n)=local(G=sum(m=0,n,m!*polcoeff(exp(-x-x^2/2+x*O(x^m)),m)*x^m)+x*O(x^n));polcoeff(x/serreverse(x*G),n)}
%Y A172394 Cf. A001464, A000699, A172395 (variant).
%K A172394 sign,new
%O A172394 0,7
%A A172394 Paul D. Hanna (pauldhanna(AT)juno.com), Feb 06 2010

%I A172494
%S A172494 1,3,87,195,243,297,408,495,522,528,573,600,798,885,903,957,1038,1053,
%T A172494 1110,1200,1233,1293,1302,1308,1368,1473,1482,1578,1623,1797,1953,2028,
%U A172494 2142,2238,2370,2772,2868,2973,3033,3393,3483,3582,3777,3822,3840,3912
%N A172494 Numbers n with (p,p+2) = ((2*n)^3/2 - 1,(2*n)^3/2 + 1) is a twin prime pair with cube sum (2*n)^3 (see A172271)
%C A172494 For k > 1: n = a(k) is necessarily a multiple of 3
%D A172494 G. H. Hardy, E. M. Wright, E. M., An Introduction to the Theory of Numbers (Fifth edition), Oxford University Press, 1980
%D A172494 N. J. A. Sloane, S. Plouffe: The Encyclopedia of Integer Sequences, Academic Press, 1995
%e A172494 3 = (2 * 1)^3/2-1 = prime(2), 3+2 = 5 = (2 * 1)^3/2+1, (3,5) is 1st prime twin pair (PTC) => a(1) = 1
%e A172494 107 = (2 * 3)^3/2-1 = prime(28), 107+2 = 109 = (2 * 3)^3/2+1, (107,109) is PTC(10) => a(2) = 3
%Y A172494 A001359, A061308, A069496, A061308, A119859, A172271
%K A172494 base,nonn,new
%O A172494 1,2
%A A172494 Ulrich Krug (leuchtfeuer37(AT)gmx.de), Feb 05 2010

%I A171853
%S A171853 0,0,0,1,3,8,20,49,119,291,715,1768,4396,10983,27551,69351,175081,
%T A171853 443119,1123963,2856383,7271377,18538391,47327615,120972510,309555666,
%U A171853 792917565,2032905981,5216436109,13395813003,34425270629,88527064337
%N A171853 Sum of the trapezoid weights of all peakless Motzkin paths of length n (n>=0).
%C A171853 A trapezoid in a peakless Motzkin path is a factor of the form U^i H^j D^i (i, j>=1), i being the height of the trapezoid and U=(1,1), H=(1,0), D=(1,-1). A trapezoid in a peakless Motzkin path w is maximal if, as a factor in w, it is not immediately preceded by a U and immediately followed by a D. The trapezoid weight of a peakless Motzkin path is the sum of the heights of its maximal trapezoids. For example, in the peakless Motzkin path w=UH(UHD)D(UUHHDD) we have two maximal trapezoids (shown between parentheses) of heights 1 and 2, respectively. The trapezoid weight of w is 1+2=3. This concept is analogous to the concept of pyramid weight in a Dyck path (see the Denise-Simion paper).
%C A171853 a(n)=Sum(k*A171852(n,k), k>=0).
%D A171853 A. Denise and R. Simion, Two combinatorial statistics on Dyck paths, Discrete Math., 137, 1995, 155-176).
%F A171853 G.f.=z^3*g/[(1 + z)(1 - z + z^2 - 2z^2*g)(1 - z)^2], where g=g(z) satisfies g=1+zg+z^2*g(g-1).
%e A171853 a(4)=3 because the 4 (=A004148(4)) peakless Motzkin paths of length 4, namely HHHH, HUHD, UHHD, and UHDH have trapezoid weights 0, 1, 1, and 1, respectively.
%p A171853 eqg := g = 1+z*g+z^2*g*(g-1): g := RootOf(eqg, g): G := z^3*g/((1+z)*(1-z)^2*(1-z+z^2-2*z^2*g)): Gser := series(G, z = 0, 38): seq(coeff(Gser, z, n), n = 0 .. 35);
%Y A171853 A004148, A171852
%K A171853 nonn,new
%O A171853 0,5
%A A171853 Emeric Deutsch (deutsch(AT)duke.poly.edu), Feb 08 2010

%I A172995
%S A172995 1,3,7,75,191,395
%N A172995 Numbers n such that (3^n+7)/2 is prime.
%K A172995 nonn,new
%O A172995 1,2
%A A172995 Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Feb 07 2010

%I A172514
%S A172514 3,7,19,97,823,3499,2777,6827,2437,21523,300299,446273,339769,1168523,
%T A172514 14117417,29227421,14160061,78521987,161187707,1200085823,2125209127,
%U A172514 1369430897
%N A172514 First prime not the middle of a prime two digits longer in base n=2,3,4,...
%K A172514 nonn,base,new
%O A172514 2,1
%A A172514 James G. Merickel (merk7(AT)verizon.net), Feb 05 2010

%I A171854
%S A171854 0,0,0,1,3,7,19,50,129,334,862,2220,5715,14706,37836,97353,250535,
%T A171854 644905,1660558,4277165,11020698,28406449,73245390,188928736,487492213,
%U A171854 1258305122,3248994414,8391747865,21681628237,56035444491,144864062529
%N A171854 Number of ladders in all peakless Motzkin paths of length n (n>=0). A string of consecutive up steps U_1, U_2, ..., U_m and their matching down steps D_1, D_2, ..., D_m are said to form a ladder if (i) D_1, D_2, ..., D_m are consecutive steps and (ii) the sequence of pairs (U_j, D_j) (j=1,2,...,m) is maximal. For example, in the path (UU)[U]H[D]H(DD), where U=(1,1), H=(1,0), D=(1,-1), we have 2 ladders, shown between parentheses and square brackets, respectively (can be easily expressed also in RNA secondary structure terminology).
%C A171854 a(n)=Sum(k*A098093(n,k), k>=0).
%D A171854 I. L. Hofacker, P. Schuster and P. F. Stadler, Combinatorics of RNA secondary structures, Discrete Appl. Math., 88, 1998, 207-237.
%F A171854 G.f.=z^2*(1-z^2)*g^2*(g-1)/(1-z^2*g^2), where g=g(z) is the g.f. of the number of peakless Motzkin paths (A004148), defined by g=1+z*g+z^2*g*(g-1). See also eq. (65) in the Hofacker et al. reference.
%e A171854 a(5)=7 because in the eight (=A004148(5)) peakless Motzkin paths of length 5, i.e. HHHHH, HH(U)H(D), H(U)HH(D), H(U)H(D)H, (U)H(D)HH, (U)HH(D)H, (U)HHH(D), and (UU)H(DD), each path, with the exception of the first, has 1 ladder (shown between parentheses).
%p A171854 eq := g = 1+z*g+z^2*g*(g-1): g := RootOf(eq, g): G := z^2*(1-z^2)*g^2*(g-1)/(1-z^2*g^2): Gser := series(G, z = 0, 35): seq(coeff(Gser, z, n), n = 0 .. 32);
%Y A171854 A004148, A098093
%K A171854 nonn,new
%O A171854 0,5
%A A171854 Emeric Deutsch (deutsch(AT)duke.poly.edu), Feb 08 2010

%I A173010
%S A173010 0,0,1,3,7,14,30,62,126,253,509,1021,2045,4092,8188,16380,32764,65531,
%T A173010 131067,262139,524283,1048570,2097146,4194298
%N A173010 The variance v(n) = sum((k-m(n))^2*p(n,k), k = 0 .. 2^n-n-1) of the distribution function p(n,k) := binomial(2^n-n-1, k)/2^(2^n-n-1) with m(n) its mean value is 0., 0.25, 1., 2.75, 6.5, 14.25, 30., 61.75, 125.5, 253.25, 509., 1020.75, 2044.5, 4092.25, 8188... We set A173010(n)= round(v(n)).
%H A173010 Thomas Wieder, <a href="http://www.thomas-wieder.privat.t-online.de/default.html">Home Page</a>.
%H A173010 Thomas Wieder, <a href="http://homepages.tu-darmstadt.de/~wieder/Welcome.html">(Old) Home Page</a>
%F A173010 a(n)=-1/4+(1/4)*2^n-(1/4)*n.
%F A173010 G.f.: (x^2-x^5+x^6)/(-1+3*x-2*x^2+2*x^6+x^4-3*x^5).
%F A173010 v(n)=(1/8)*2^n-1/4+v(-1+n) with v(1)=0 and a(n)=round(v(n)).
%Y A173010 A173009.
%K A173010 nonn,new
%O A173010 1,4
%A A173010 Thomas Wieder (thomas.wieder(AT)t-online.de), Feb 07 2010

%I A172306
%S A172306 0,1,3,7,11,15,23,31,35,43,55,67,83,95
%N A172306 a(n) = A172304(n)/2.
%H A172306 David Applegate, Omar E. Pol and N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/tooth.pdf">The Toothpick Sequence and Other Sequences from Cellular Automata</a>
%H A172306 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/toothlist.html"> Catalog of Toothpick and Cellular Automata Sequences in the OEIS</a>
%Y A172306 Cf. A139250, A172304, A172305, A172307, A172308, A172309, A172310, A172311, A172312, A172313.
%K A172306 more,nonn,new
%O A172306 0,3
%A A172306 Omar E. Pol (info(AT)polprimos.com), Feb 06 2010

%I A172987
%S A172987 1,3,7,9,13,27,31,37,49,51,63,73,79,91,93,97,99,127,129,139,141,157,163,
%T A172987 169,171,181,183,213,217,231,347,253,259,267,273,283,297,301,321,339,
%U A172987 343,349,357,363,379,387,391,399,409,421,447,457,469,477,493,499,513
%N A172987 Odd numbers n such that n+10^{1,2} are both primes.
%e A172987 a(1)=1 because 1+10^1=11=prime and 1+10^2=101=prime. a(6)=27 because 27+10^1=37=prime and 27+10^2=127=prime.
%Y A172987 Cf. A000040, A005408.
%K A172987 nonn,new
%O A172987 1,2
%A A172987 Juri-Stepan Gerasimov (2stepan(AT)rambler.ru), Feb 07 2010

%I A172308
%S A172308 0,1,3,5,7,11,15,17,21,27,33,41,47,59,69,75,79
%N A172308 L-toothpick sequence in the first quadrant.
%C A172308 The same as A172310 and A172304, but starting from half L-toothpick in the first quadrant.
%C A172308 Note that if n is odd then we add the small L-toothpicks to the structure, otherwise we add the large L-toothpicks to the structure.
%C A172308 We start at stage 0 with half L-toothpick: A segment from (0,0) to (1,1).
%C A172308 At stage 1 we place a small L-toothpick at the exposed toothpick end.
%C A172308 At stage 2 we place two large L-toothpicks.
%C A172308 At stage 3 we place two small L-toothpicks.
%C A172308 At stage 4 we place two large L-toothpicks.
%C A172308 And so on...
%C A172308 The sequence gives the number of L-toothpicks after n stages. A172309 (the first differences) gives the number of L-toothpicks added at the n-th stage.
%H A172308 David Applegate, Omar E. Pol and N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/tooth.pdf">The Toothpick Sequence and Other Sequences from Cellular Automata</a>
%H A172308 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/toothlist.html"> Catalog of Toothpick and Cellular Automata Sequences in the OEIS</a>
%Y A172308 Cf. A139250, A153000, A160120, A160170, A160172, A161206, A161328, A172304, A172305, A172306, A172307, A172309, A172310, A172311, A172312, A172313.
%K A172308 more,nonn,new
%O A172308 0,3
%A A172308 Omar E. Pol (info(AT)polprimos.com), Feb 06 2010

%I A173027
%S A173027 1,3,4,5,16,19,22,25,28,31,34,97,105,113,121,129,137,145,153,161,169,
%T A173027 177,185,193,201,209,217,225,233,631,652,673,694,715,736,757,778,799,
%U A173027 820,841,862,883,904,925,946,967,988,1009,1030,1051,1072,1093,1114,1135
%N A173027 Numbers of rows R of the Wythoff array such that R is a multiple of a tail of the Fibonacci sequence.
%C A173027 Row 1 of the array A173028.
%e A173027 Referring to rows of the Wythoff array (A035513),
%e A173027 Row 1: (1,2,3,5,...) = 1*(1,2,3,...)
%e A173027 Row 3: (6,10,16,...) = 2*(3,5,8,...)
%e A173027 Row 4: (9,15,24,...) = 3*(3,5,8,...)
%e A173027 Row 5: (12,20,32,...) = 4*(3,5,8,...)
%e A173027 Row 16: (40,65,105...) = 8*(5,13,21,...).
%Y A173027 Cf. A000045, A035513, A173028.
%K A173027 nonn,new
%O A173027 1,2
%A A173027 Clark Kimberling (ck6(AT)evansville.edu), Feb 07 2010

%I A173001
%S A173001 1,3,4,5,7,8,11,12,13,15,16,17,19,20,21,22,24,25,32,35,42,43,44,48,49,
%T A173001 50,52,55,60,61,63,69,70,71,73,74,81,82,84,85,86,87,88,92,93,94,98,101,
%U A173001 106,107,108,109,110,112,115,117,120,123,126,127,131,132,135,137
%N A173001 Values of n such that either {0,5}*n^2+5*n+1 is a prime.
%e A173001 a(1)=1 because 0*1^2+5*1+1=6=nonprime and 5*1^2+5*1+1=11=prime. a(6)=8 because 0*8^2+5*8+1=41=prime and 5*8^2+5*8+1=361=nonprime.
%Y A173001 Cf. A000040, A090563.
%K A173001 nonn,new
%O A173001 1,2
%A A173001 Juri-Stepan Gerasimov (2stepan(AT)rambler.ru), Feb 07 2010

%I A172990
%S A172990 3,4,3,12,17,6,9,10,9,30,5,54,33,14,3,24,11,168,81,20,9,60,17,18,3,80,9,
%T A172990 18,73,192,75,14,63,54,7,54,255,38,303,42,11,114,63,4,33,180,5,30,93,28,
%U A172990 21,84,115,18,15,40,9,228,61,318,171,4,93,42,5,24,9,70,51,72,49,444,3
%N A172990 Prime Symmetricity of cubes; least n such that cube +- n are primes.
%F A172990 2^3+-3=primes,3^3+-4=primes,4^3+-3=primes,5^3+-12=primes,..
%t A172990 f[n_]:=Block[{k},If[OddQ[n],k=2,k=1];While[ !PrimeQ[n-k]||!PrimeQ[n+k],k+=2];k];Table[f[n^3],{n,2,40}]
%Y A172990 Cf. A172989
%K A172990 nonn,new
%O A172990 1,1
%A A172990 Vladimir Orlovsky (4vladimir(AT)gmail.com), Feb 07 2010

%I A173048
%S A173048 1,3,3,9,12,9,65,70,70,65,1025,990,560,990,1025,32769,31806,11160,11160,
%T A173048 31806,32769,2097153,2064510,671832,178560,671832,2064510,2097153,
%U A173048 268435457,266338558,87413592,12850368,12850368,87413592,266338558
%N A173048 A triangle of polynomial coefficients:q=2;p(x,n,q)=If[n == 0, 1, Product[x + q^i, {i, 1, n}] + Product[x*q^i + 1, {i, 1, n}]]
%C A173048 Row sums are:
%C A173048 {1, 6, 30, 270, 4590, 151470, 9845550, 1270075950, 326409519150,
%C A173048 167448083323950, 171634285407048750,...}.
%F A173048 q=2;
%F A173048 p(x,n,q)=If[n == 0, 1, Product[x + q^i, {i, 1, n}] + Product[x*q^i + 1, {i, 1, n}]]
%e A173048 {1},
%e A173048 {3, 3},
%e A173048 {9, 12, 9},
%e A173048 {65, 70, 70, 65},
%e A173048 {1025, 990, 560, 990, 1025},
%e A173048 {32769, 31806, 11160, 11160, 31806, 32769},
%e A173048 {2097153, 2064510, 671832, 178560, 671832, 2064510, 2097153},
%e A173048 {268435457, 266338558, 87413592, 12850368, 12850368, 87413592, 266338558, 268435457},
%e A173048 {68719476737, 68451041790, 22638842200, 3189792960, 411211776, 3189792960, 22638842200, 68451041790, 68719476737},
%e A173048 {35184372088833, 35115652613118, 11659494378840, 1652679610560, 111842970624, 111842970624, 1652679610560, 11659494378840, 35115652613118, 35184372088833},
%e A173048 {36028797018963969, 35993612646877182, 11974437542118744, 1703952176836800, 112764343667712, 7157950119936, 112764343667712, 1703952176836800, 11974437542118744, 35993612646877182, 36028797018963969}
%t A173048 Clear[p, x, n, q];
%t A173048 p[x_, n_, q_] = If[n == 0, 1, Product[x + q^i, {i, 1, n}] + Product[x*q^i + 1, {i, 1, n}]];
%t A173048 Table[Table[CoefficientList[p[x, n, q], x], {n, 0, 10}], {q, 1, 10}];
%t A173048 Table[Flatten[Table[CoefficientList[p[x, n, q], x], {n, 0, 10}]], {q, 1, 10}]
%K A173048 nonn,tabl,uned,new
%O A173048 0,2
%A A173048 Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Feb 08 2010

%I A172515
%S A172515 3,3,4,3,3,3,4,3,3,3,4,3,3,3,4,3,3,3,4,3,3,4,3,3,3,4,3,3,3,4,3,3,3,4,3,
%T A172515 3,3,4,3,3,3,4,3,3,3,4,3,3,4,3,3,3,4,3,3,3,4,3,3,3,4,3,3,3,4,3,3,3,4,3,
%U A172515 3,3,4,3,3,3,4,3,3,4,3,3,3,4,3,3,3,4,3,3,3,4,3,3,3,4,3,3,3,4,3,3,3,4,3
%N A172515 First differences of A172513
%Y A172515 A172513
%K A172515 hard,nonn,new
%O A172515 1,1
%A A172515 Stephen Crowley (crow(AT)crowlogic.net), Feb 05 2010

%I A172991
%S A172991 3,2,5,11,17,23,29,41,47,53,59,71,83,89,101,107,113,131,137,149,167,173,
%T A172991 179,191,197,227,233,239,251,257,263,269,281,293,311,317,347,353,359,
%U A172991 383,389,401,419,431,443,449,461,467,479,491,503,509,521,557,563,569
%N A172991 Primes of the form 3*n+1-+2.
%C A172991 3 together with primes of form 3n-1.
%F A172991 a(n+1)=A003627(m).
%e A172991 If 3*0+1-2=-1=nonprime and 3*0+1+2=3=prime, then a(1)=3. If 3*1+1-2=2=prime and 3*1+1+2=6=nonprime, then a(2)=2. If 3*2+1-2=5=prime and 3*2+1+2=9=nonprime, then a(3)=5.
%Y A172991 Cf. A003627(primes of form 3n-1).
%K A172991 nonn,new
%O A172991 1,1
%A A172991 Juri-Stepan Gerasimov (2stepan(AT)rambler.ru), Feb 07 2010

%I A173028
%S A173028 1,3,2,4,9,6,5,13,29,7,16,45,43,35,8,19,56,57,52,15,10,22,67,186,181,58,
%T A173028 51,11,25,78,225,226,77,199,55,12,28,89,260,271,96,265,82,61,14,31,262,
%U A173028 297,316,115,331,109,91,71,17,34,291,334,361,351,397,136,314,106,87,18
%N A173028 Partition of the row numbers of the Wythoff array W: two numbers are in the same row if and only if their rows in W have (essentially) a common divisor greater than 1.
%C A173028 (Row 1) = A173027. Every positive integer occurs exactly once, so
%C A173028 that, as a sequence, this is a permutation of the natural numbers.
%F A173028 Let R(n,k) be the number in row n, column k. After Row 1 (A173027),
%F A173028 inductively, R(n,1) is the least positive integer not in the first n-1
%F A173028 rows, and the rest of row n consists of the numbers of rows X of the
%F A173028 Wythoff array W for X a multiple of a tail of row R(n,1) of W.
%e A173028 First four rows of R:
%e A173028 1...3....4....5.....16....19....22...25...28...
%e A173028 2...9....13...45....56....67....78...89...262..
%e A173028 6...29...43...57....186...225...260..297..334...
%e A173028 7...35...52...181...226...271...316..361..1063...
%e A173028 For example, row 3 begins with 6, which is the least positive
%e A173028 integer not in rows 1 and 2. Row 6 of W is (14,23,37,60,...)
%e A173028 Row 29 of W is (27,120,194,...) = 2*(37,60,97...).
%e A173028 Row 43 of W is (111,180,291,...) = 3*(37,60,97,...).
%e A173028 So row 3 of R begins with (6,29,43...) as there are no other rows
%e A173028 of W numbered <43 which are multiples of row 6 of W.
%Y A173028 Cf. A000045, A035513, A173027.
%K A173028 nonn,tabl,new
%O A173028 1,2
%A A173028 Clark Kimberling (ck6(AT)evansville.edu), Feb 07 2010

%I A173007
%S A173007 1,3,1,27,12,1,729,351,39,1,59049,29160,3510,120,1,14348907,7144929,
%T A173007 882090,32670,363,1,10460353203,5223002148,650188539,24698520,297297,
%U A173007 1092,1,22876792454961,11433166050879,1427185336941,54665851779
%N A173007 A triangle of polynomial coefficients of:q=3; p(x,n,q)=If[n == 0, 1, Product[x + q^i, {i, 1, n}]]
%C A173007 Row sums are:
%C A173007 {1, 4, 40, 1120, 91840, 22408960, 16358540800, 35792487270400,
%C A173007 234870301468364800, 4623187014103292723200, 272999193182799435304960000,...}.
%C A173007 a = 2; A108084.
%F A173007 q=3;
%F A173007 p(x,n,q)=If[n == 0, 1, Product[x + q^i, {i, 1, n}]]
%e A173007 {1},
%e A173007 {3, 1},
%e A173007 {27, 12, 1},
%e A173007 {729, 351, 39, 1},
%e A173007 {59049, 29160, 3510, 120, 1},
%e A173007 {14348907, 7144929, 882090, 32670, 363, 1},
%e A173007 {10460353203, 5223002148, 650188539, 24698520, 297297, 1092, 1},
%e A173007 {22876792454961, 11433166050879, 1427185336941, 54665851779, 674887059, 2685501, 3279, 1},
%e A173007 {150094635296999121, 75035879252272080, 9375196161720780, 360089838858960, 4482599845878, 18294459120, 24199020, 9840, 1},
%e A173007 {2954312706550833698643, 1477081305957768349761, 184607021930402384820, 7097023494422630460, 88591102605275634, 364572438704838, 494603769780, 217879740, 29523, 1},
%e A173007 {174449211009120179071170507, 87223128348206814118735932, 10902337119274288189585941, 419256747344092308417360, 5238313041233343542526, 21616229035687254696, 29570430440444058, 13360184537040, 1961183367, 88572, 1}
%t A173007 Clear[p, x, n, q]
%t A173007 p[x_, n_, q_] = If[n == 0, 1, Product[x + q^i, {i, 1, n}]];
%t A173007 Table[Table[CoefficientList[p[x, n, q], x], {n, 0, 10}], {q, 2, 10}];
%t A173007 Table[Flatten[Table[CoefficientList[p[x, n, q], x], {n, 0, 10}]], {q, 2, 10}]
%Y A173007 A108084
%K A173007 nonn,tabl,uned,new
%O A173007 0,2
%A A173007 Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Feb 07 2010

%I A171852
%S A171852 1,1,1,1,1,1,3,1,6,1,1,12,4,1,24,11,1,1,48,28,5,1,96,70,17,1,1,192,173,
%T A171852 51,6,1,384,421,147,24,1,1,768,1010,415,82,7,1,1536,2392,1147,264,32,1,
%U A171852 1,3072,5600,3107,825,122,8,1,6144,12976,8265,2513,431,41,1,1,12288
%N A171852 Triangle read by rows: T(n,k) is the number of peakless Motzkin paths of length n having trapezoid weight k (n>=0, k>=0).
%C A171852 A trapezoid in a peakless Motzkin path is a factor of the form U^i H^j D^i (i, j>=1), i being the height of the trapezoid and U=(1,1), H=(1,0), D=(1,-1). A trapezoid in a peakless Motzkin path w is maximal if, as a factor in w, it is not immediately preceded by a U and immediately followed by a D. The trapezoid weight of a peakless Motzkin path is the sum of the heights of its maximal trapezoids. For example, in the peakless Motzkin path w=UH(UHD)D(UUHHDD) we have two maximal trapezoids (shown between parentheses) of heights 1 and 2, respectively. The trapezoid weight of w is 1+2=3. This concept is analogous to the concept of pyramid weight in a Dyck path (see the Denise-Simion paper).
%C A171852 The number of terms in rows 0,1,2,3,4,5,6,7,... is 1,1,1,2,2,3,3,4,4,5,5,... , respectively.
%C A171852 The sum of the entries in row n is A004148(n) (the secondary structure numbers).
%C A171852 Sum(k*T(n,k), k>=0)=A171853(n).
%D A171852 A. Denise and R. Simion, Two combinatorial statistics on Dyck paths, Discrete Math., 137, 1995, 155-176).
%F A171852 The g.f. G=G(t,z) satisfies G = 1 + zG + z^2*G*(G - 1 + z*(t - 1)/[(1 - z)(1 - t*z^2)].
%e A171852 T(4,1)=3 because each of the paths HUHD, UHHD, and UHDH has trapezoid weight 1.
%e A171852 Triangle starts:
%e A171852 1;
%e A171852 1;
%e A171852 1;
%e A171852 1,1;
%e A171852 1,3;
%e A171852 1,6,1;
%e A171852 1,12,4;
%e A171852 1,24,11,1
%p A171852 eq := G = 1+z*G+z^2*G*(G-1+z*(t-1)/((1-z)*(1-t*z^2))): G := RootOf(eq, G): Gser := simplify(series(G, z = 0, 20)): for n from 0 to 16 do P[n] := sort(coeff(Gser, z, n)) end do: 1; for n to 16 do seq(coeff(P[n], t, k), k = 0 .. floor(((n^2-1)*1/2)/n)) end do; # yields sequence in triangular form
%Y A171852 A004148, A171853
%K A171852 nonn,tabf,new
%O A171852 0,7
%A A171852 Emeric Deutsch (deutsch(AT)duke.poly.edu), Feb 08 2010

%I A172972
%S A172972 1,1,1,1,3,1,1,1,1,1,1,0,2,0,1,1,1,2,2,1,1,1,1,1,2,1,
%T A172972 1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,1,1,1,1,1,1,1,1,0,0,1,1,1,
%U A172972 1,1,1,1,1,0,0,0,1,1,1,1
%V A172972 -1,-1,-1,-1,-3,-1,-1,-1,-1,-1,-1,0,2,0,-1,-1,-1,2,2,-1,-1,-1,-1,1,2,1,
%W A172972 -1,-1,-1,-1,1,1,1,1,-1,-1,-1,-1,1,1,0,1,1,-1,-1,-1,-1,1,1,0,0,1,1,-1,
%X A172972 -1,-1,-1,1,1,0,0,0,1,1,-1,-1
%N A172972 Subtraction triangle based on A029826: c(n)=Product[A029826(i),{i,0,n)];t(n,m)=c(n)-c(m)-c(n-m)
%C A172972 Row sums are:
%C A172972 {-1, -2, -5, -4, 0, 0, 0, 0, 0, 0, 0,...}.
%F A172972 c(n)=Product[A029826(i),{i,0,n)];
%F A172972 t(n,m)=c(n)-c(m)-c(n-m)
%e A172972 {-1},
%e A172972 {-1, -1},
%e A172972 {-1, -3, -1},
%e A172972 {-1, -1, -1, -1},
%e A172972 {-1, 0, 2, 0, -1},
%e A172972 {-1, -1, 2, 2, -1, -1},
%e A172972 {-1, -1, 1, 2, 1, -1, -1},
%e A172972 {-1, -1, 1, 1, 1, 1, -1, -1},
%e A172972 {-1, -1, 1, 1, 0, 1, 1, -1, -1},
%e A172972 {-1, -1, 1, 1, 0, 0, 1, 1, -1, -1},
%e A172972 {-1, -1, 1, 1, 0, 0, 0, 1, 1, -1, -1}
%t A172972 (*A029826 Inverse of Salem polynomial : 1/(x^10 + x^9 - x^7 - x^6 - x^5 - x^4 - x^3 + x + 1).*)
%t A172972 p[x_] = (x^(10) + x^9 - x^7 - x^6 - x^5 - x^4 - x^3 + x + 1); q[ x_] = Expand[x^10*p[1/x]]; a = Table[SeriesCoefficient[Series[1/ q[x], {x, 0, 100}], n], {n, 0, 100}];
%t A172972 c[n_] := Product[a[[m]], {m, 1, n}];
%t A172972 t[n_, m_] := c[n] - (c[m] + c[n - m]);
%t A172972 Table[Table[t[n, m], {m, 0, n}], {n, 0, 10}];
%t A172972 Flatten[%]
%Y A172972 A029826
%K A172972 sign,tabl,uned,new
%O A172972 0,5
%A A172972 Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Feb 06 2010

%I A172396
%S A172396 1,1,1,0,3,0,38,0,947,0,37394,0,2120190,0,162980012,0,16330173251,0,
%T A172396 2070201641498,0,324240251016266,0,61525045423103316,0,
%U A172396 13913915097436287598,0,3698477457114061621492,0
%N A172396 G.f. satisfies: A(x) = G(x/A(x)) where o.g.f. G(x) = A(x*G(x)) = Sum_{n>=0} A003701(n)*x^n.
%C A172396 The e.g.f. of A003701 is exp(x)/cos(x) = Sum_{n>=0} A003701(n)*x^n/n!.
%C A172396 Compare to A157308 and A157310.
%F A172396 a(n) = |A157308(n)| = |A157310(n)| for n>=0.
%F A172396 a(2n) = A158119(n) for n>=0; a(2n-1) = 0 for n>=2, with a(1)=1.
%e A172396 G.f.: A(x) = 1 + x + x^2 + 3*x^4 + 38*x^6 + 947*x^8 + 37394*x^10 +...
%e A172396 where G(x) = A(x*G(x)) is the o.g.f. of A003701:
%e A172396 G(x) = 1 + x + 2*x^2 + 4*x^3 + 12*x^4 + 36*x^5 + 152*x^6 + 624*x^7 +...
%e A172396 while the e.g.f. of A003701 is given by:
%e A172396 exp(x)/cos(x) = 1 + x + 2*x^2/2! + 4*x^3/3! + 12*x^4/4! + 36*x^5/5! +...
%o A172396 (PARI) {a(n)=local(X=x+x*O(x^n),G=sum(m=0,n,m!*polcoeff(exp(X)/cos(X),m)*x^m)+x*O(x^n)); polcoeff(x/serreverse(x*G),n)}
%Y A172396 Cf. A003701, A158119, A157308, A157310.
%K A172396 nonn,new
%O A172396 0,5
%A A172396 Paul D. Hanna (pauldhanna(AT)juno.com), Feb 07 2010

%I A172994
%S A172994 2,460724,610357585,2096681555
%N A172994 First number outperforming x=10 with simultaneous primality of {x^2k+x^k-1} for n values, starting with n=4, occurring with a smaller number than the nth term of A096594.
%C A172994 This sequence is related to the remarkable occurrence of primes in the
%C A172994 sequence 109, 10099, 1000999, etc. Credit for second and third terms
%C A172994 goes to Jens Kruse Andersen.
%e A172994 The first three terms are primes for all k from 1 through 4, 5, and 6; the
%e A172994 fourth term is prime for k=1 to 6 and k=8, while x=10 requires k=9 to get
%e A172994 the seventh prime.
%Y A172994 A096594
%K A172994 more,nonn,new
%O A172994 4,1
%A A172994 James G. Merickel (merk7(AT)verizon.net), Feb 07 2010

%I A173026
%S A173026 1,2,38,80
%N A173026 n+nth non-Cuban prime=nth Cuban prime.
%C A173026 Numbers n such that n+A003627(n)=A007645(n).
%e A173026 a(1)=1 because 1+A003627(1)=1+2=3+A007645(1); a(2)=2 because 2+A003627(2)=2+5=7=A007645(2); a(3)=38 because 38+A003627(38)=38+359=397=A007645(38); a(4)=80 because 80+A003627(80)=80+911=991=A007645(80).
%Y A173026 Cf. A000027, A003627, A007645.
%K A173026 nonn,new
%O A173026 1,2
%A A173026 Juri-Stepan Gerasimov (2stepan(AT)rambler.ru), Feb 07 2010

%I A172492
%S A172492 1,2,24,864,69120,10368000,2612736000,1024192512000,589934886912000,
%T A172492 477847258398720000,525631984238592000000,763217641114435584000000,
%U A172492 1428743424166223413248000000,3380406941577284595744768000000
%N A172492 a(n)=(n!)^2*(n+1)!, n=0,1... .
%C A172492 Asymptotics: a(n)->(1/16)*Pi^(3/2)*sqrt(2)*(32*n^2+40*n+9)*exp(-3*n)*(n)^(1/2+3*n),
%C A172492 n->infinity.
%F A172492 Generating function of hypergeometric type , in Maple notation:
%F A172492 sum(a(n)*x^n/(n!)^3,n=0..infinity)=1/(1-x)^2.
%F A172492 Integral representation as n-th moment of a positive function on a positive
%F A172492 halfaxis (solution of the Stieltjes moment problem), in Maple notation:
%F A172492 a(n)=int(x^n*MeijerG([[],[]],[[0,0,1],[]],x),x=0..infinity), n=0,1... .
%F A172492 The MeijerG function above cannot be represented by any other known special function.
%F A172492 This solution of the Stieltjes moment problem is not unique.
%K A172492 nonn,new
%O A172492 0,2
%A A172492 Karol A. Penson (penson(AT)lptl.jussieu.fr), Feb 05 2010

%I A172529
%S A172529 0,2,18,88,200,486,980,1760,2916,4550,6776,9720,13520,18326,24300,31616,
%T A172529 40460,51030,63536,78200
%N A172529 Number of ways to place 2 nonattacking knights on an n X n toroidal board
%H A172529 V. Kotesovec, <a href="http://web.telecom.cz/vaclav.kotesovec/math.htm">Number of ways of placing non-attacking queens and kings on boards of various sizes</a>
%F A172529 Explicit formula (Vaclav Kotesovec, 31.1.2010): a(n) = n^2*(n+3)(n-3)/2, n>=5
%Y A172529 A172132, A172517
%K A172529 nonn,new
%O A172529 1,2
%A A172529 Vaclav Kotesovec (kotesovec(AT)chello.cz), Feb 06 2010

%I A172968
%S A172968 1,2,13,89,610,4181,28657,196418,1346269,9227465,63245986,433494437,
%T A172968 2971215073,20365011074,139583862445,956722026041,6557470319842,
%U A172968 44945570212853,308061521170129,2111485077978050,14472334024676221
%N A172968 Linear recurrence a(n)=7a(n-1)-a(n-2) a(0)=1; a(1)=2
%F A172968 1) a(n)=(1/10)((5+Sqrt[5])((7-3Sqrt[5])/2)^n+((5-Sqrt[5])((7+3Sqrt[5])/2)^n)
%F A172968 2) a(n) = Sqrt[1-2Fibonacci[2n+1]Fibonacci[2n+2]+5(Fibonacci[2n+1]Fibonacci[2n+2])^2]=Sqrt[1-2*A081016(n)+5*(A081016(n))^2
%F A172968 a(n) = A033891(n-1), n>0. [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Feb 08 2010]
%t A172968 Table[Sqrt[1 - 2 m + 5 m^2] /. m -> Fibonacci[2 n + 1] Fibonacci[2 n + 2], {n, -1, 20}] (*Artur Jasinski*)
%Y A172968 A081016, A172969.
%K A172968 nonn,new
%O A172968 0,2
%A A172968 Artur Jasinski (grafix(AT)csl.pl), Feb 06 2010

%I A171847
%S A171847 0,0,0,2,7,22,68,198,563,1578,4367,11980,32648,88500,238886,642598,
%T A171847 1723629,4612170,12316357,32832302,87390763,232305470,616812557,
%U A171847 1636084020,4335770052,11480937084,30379110906,80332372838,212300488377
%N A171847 Total area under all peakless Motzkin paths of length n (n>=0).
%C A171847 a(n)=Sum(k*A171846(n,k), k>=0).
%F A171847 G.f. = z^2*(g^2 -1)/[(1+z+z^2)(1-3z+z^2)], where g=g(z) satisfies g = 1 + zg + z^2*g(g - 1).
%e A171847 a(4)=7 because the areas under the paths HHHH, HUHD, UHHD, and UHDH are 0, 2, 3, and 2, respectively.
%p A171847 g := ((1-z+z^2-sqrt(1-2*z-z^2-2*z^3+z^4))*1/2)/z^2: G := z^2*(g^2-1)/((1+z+z^2)*(1-3*z+z^2)): Gser := series(G, z = 0, 35): seq(coeff(Gser, z, n), n = 0 .. 30);
%Y A171847 A004148, A171846
%K A171847 nonn,new
%O A171847 0,4
%A A171847 Emeric Deutsch (deutsch(AT)duke.poly.edu), Feb 08 2010

%I A172304
%S A172304 0,2,6,14,22,30,46,62,70,86,110,134,166,190
%N A172304 L-toothpick sequence starting with two opposite L-toothpicks.
%C A172304 The same as A172310 but starting with two L-toothpicks.
%C A172304 We start at stage 0 with no L-toothpicks.
%C A172304 At stage 1 we place two large L-toothpicks in the horizontal direction, as a "X", anywhere in the plane.
%C A172304 At stage 2 we place four small L-toothpicks.
%C A172304 At stage 3 we place eight large L-toothpicks.
%C A172304 At stage 4 we place eight small L-toothpicks.
%C A172304 And so on...
%C A172304 The sequence gives the number of L-toothpicks after n stages. A172305 (the first differences) gives the number of L-toothpicks added at the n-th stage.
%H A172304 David Applegate, Omar E. Pol and N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/tooth.pdf">The Toothpick Sequence and Other Sequences from Cellular Automata</a>
%H A172304 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/toothlist.html"> Catalog of Toothpick and Cellular Automata Sequences in the OEIS</a>
%Y A172304 Cf. A139250, A160120, A160170, A160172, A161206, A161328, A172305, A172306, A172307, A172308, A172309, A172310, A172311, A172312, A172313.
%K A172304 more,nonn,new
%O A172304 0,2
%A A172304 Omar E. Pol (info(AT)polprimos.com), Feb 06 2010

%I A173009
%S A173009 0,1,2,6,13,29,60,124,251,507,1018,2042,4089,8185,16376,32760,65527,
%T A173009 131063,262134,524278,1048565,2097141,4194292,8388596
%N A173009 The mean value m(n) = sum(k*p(n,k), k = 0 .. 2^n-n-1) of the distribution function p(n,k) := binomial(2^n-n-1, k)/2^(2^n-n-1) is 0., 0.5, 2., 5.5, 13., 28.5, 60., 123.5, 251., 506.5, 1018., 2041.5, 4089., 8184.5... We set A173009(n)=round(m(n)).
%C A173009 The half-integer sequence h(n)= 0, 1/2, 2, 11/2, 13, 57/2, 60, 247/2, 251,
%C A173009 1013/2, 1018, 4083/2, 4089, 16369/2, 16376, 65519/2, 65527
%C A173009 is the BINOMIALi transform of 0, 1/2, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
%H A173009 Thomas Wieder, <a href="http://www.thomas-wieder.privat.t-online.de/default.html">Home Page</a>.
%H A173009 Thomas Wieder, <a href="http://homepages.tu-darmstadt.de/~wieder/Welcome.html">(Old) Home Page</a>
%F A173009 a(n)=(1/2)*2^n-(1/2)*n-1/2.
%F A173009 G.f.:(x-x^2+x^3)/(2*x^4-3*x^3-x^2+3*x-1).
%F A173009 m(n)=(1/4)*2^n-1/2+m(-1+n) with m(1)=0 and a(n)=round(m(n)).
%Y A173009 Cf. A173010, A016031, A000295.
%K A173009 nonn,new
%O A173009 1,3
%A A173009 Thomas Wieder (thomas.wieder(AT)t-online.de), Feb 07 2010

%I A172405
%S A172405 1,1,2,5,15,47,153,515,1782,6293,22576,82043,301417,1117693,4177687,
%T A172405 15723545,59538258,226656336,866983080,3330496250,12843380569,
%U A172405 49700905874,192942347560,751191150163,2932439491143,11475503589091
%N A172405 G.f. satisfies: A(x) = G(x*A(x)) where G(x) = A(x/G(x)) = Sum_{n>=0} x^(n(n+1)/2)*(1+x)^n.
%F A172405 G.f.: A(x) = Sum_{n>=0} [x*A(x)]^(n(n+1)/2) * (1 + x*A(x))^n.
%e A172405 G.f.: A(x) = 1 + x + 2*x^2 + 5*x^3 + 15*x^4 + 47*x^5 + 153*x^6 +...
%e A172405 where G(x) = A(x/G(x)) = Sum_{n>=0} x^(n(n+1)/2)*(1+x)^n is illustrated by:
%e A172405 G(x) = 1 + x*(1+x) + x^3*(1+x)^2 + x^6*(1+x)^3 + x^10*(1+x)^4 +...
%e A172405 which has the binomial coefficients of the flattened Pascal's triangle:
%e A172405 G(x) = 1 + (x + x^2) + (x^3 + 2*x^4 + x^5) + (x^6 + 3*x^7 + 3*x^8 + x^9) +...
%o A172405 (PARI) {a(n)=local(G=sum(m=0,(sqrtint(8*n+1)+1)\2,x^(m*(m+1)/2)*(1+x)^m)+x*O(x^n));polcoeff(G^(n+1)/(n+1),n)}
%K A172405 nonn,new
%O A172405 0,3
%A A172405 Paul D. Hanna (pauldhanna(AT)juno.com), Feb 06 2010

%I A173032
%S A173032 2,5,10,17,28,129,260,411,592,783,1096,1449,1822,2205,2932,3689,4476,
%T A173032 5273,6192,7121,17422,27923,38524,49835,61246,73667,86388,99209,112540,
%U A173032 126371,140302,154643,169384,184835,200386,216447,232808,249369,266030
%N A173032 Partial sums of palindromic primes A002385.
%C A173032 The subsequence of prime partial sum of palindromic primes begins: 2, 5, 17, 5273, 7121, 154643, 283501. What is the smallest nontrivial (i.e. multidigit) palindromic prime partial sum of palindromic primes?
%F A173032 a(n) = SUM[i=1..n] A002385(i) = SUM[i=1..n] {p prime and R(p) = p, i.e. primes whose decimal expansion is a palindrome}.
%e A173032 a(42) = 2 + 3 + 5 + 7 + 11 + 101 + 131 + 151 + 181 + 191 + 313 + 353 + 373 + 383 + 727 + 757 + 787 + 797 + 919 + 929 + 10301 + 10501 + 10601 + 11311 + 11411 + 12421 + 12721 + 12821 + 13331 + 13831 + 13931 + 14341 + 14741 + 15451 + 15551 + 16061 + 16361 + 16561 + 16661 + 17471 + 17971 + 18181.
%Y A173032 Cf. A000040, A002385, A007500, A006567, A016041, A029732, A117697.
%K A173032 base,easy,nonn,new
%O A173032 1,1
%A A173032 Jonathan Vos Post (jvospost3(AT)gmail.com), Feb 07 2010

%I A172982
%S A172982 2,5,10,17,28,47,88,149,238,647,1096,1595,2476,3467,9936,16885,25886,
%T A172982 34935,44584,54533,115182,781831,1728500,61728549,127728598,194328647
%N A172982 Partial sums of minimal set of prime-strings in base 10 (A071062).
%C A172982 The subsequence of primes in this partial sum is: 2, 5, 47, 149, 647, 3467, of which only 2 and 5 are in the original sequence.
%e A172982 a(26) = 194328647 = 2 + 3 + 5 + 7 + 11 + 19 + 41 + 61 + 89 + 409 + 449 + 499 + 881 + 991 + 6469 + 6949 + 9001 + 9049 + 9649 + 9949 + 60649 + 666649 + 946669 + 60000049 + 66000049 + 66600049.
%Y A172982 Cf. A000040, A071062, A071070, A071071, A071072, A071073.
%K A172982 base,fini,full,nonn,new
%O A172982 1,1
%A A172982 Jonathan Vos Post (jvospost3(AT)gmail.com), Feb 06 2010

%I A172512
%S A172512 2,5,10,17,28,47,76,123,196,275,388,539,696,859,1026,1265,1506,1789,
%T A172512 2142,2509,2888,3345,4342,5709,8750,18891,33590,61119,110326,187617,
%U A172512 272854,379547,539970,743759,1108048,2100009
%N A172512 Partial sums of Gaussian-Mersenne primes A057429.
%C A172512 The subsequence of prime partial sums of Gaussian-Mersenne primes begins 2, 5, 17, 47, 859, 1789; no more are known. The subsubsequence of Gaussian-Mersenne prime partial sums of Gaussian-Mersenne primes begins 2, 5, 47. The only known square in the sequence is 196.
%e A172512 a(36) = 2100009 = 2 + 3 + 5 + 7 + 11 + 19 + 29 + 47 + 73 + 79 + 113 + 151 + 157 + 163 + 167 + 239 + 241 + 283 + 353 + 367 + 379 + 457 + 997 + 1367 + 3041 + 10141 + 14699 + 27529 + 49207 + 77291 + 85237 + 106693 + 160423 + 203789 + 364289 + 991961.
%Y A172512 Cf. A000040, A000043, A057429, A066408, A007670, A007671, A027206, A027206, A103329
%K A172512 nonn,new
%O A172512 1,1
%A A172512 Jonathan Vos Post (jvospost3(AT)gmail.com), Feb 05 2010

%I A172981
%S A172981 2,5,7,11,23,29,31,53,71,79,83,89,97,107,127,131,139,151,167,191,197,
%T A172981 211,227,229,239,241,257,269,271,277,281,283,347,349,359,379,383,409,
%U A172981 433,449,461,499,521,587,647,673,677,709,739,743,769,787,811,823
%N A172981 Primes p such that 210*p+41 are booth prime
%Y A172981 Cf. A140848
%K A172981 nonn,new
%O A172981 1,1
%A A172981 Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Feb 06 2010

%I A172977
%S A172977 1,2,4,8,37,46,50,75,106,145,151,163,168,169,207,226,232,260
%N A172977 Numbers n such that {1,3}*prime(n)*2-1 are both primes.
%e A172977 a(1)=1 because 1*2*2-1=3=prime and 3*2*2-1=13=prime; a(2)=2 because 1*3*2-1=7=prime and 3*3*2-1=17=prime; a(3)=4 because 1*7*2-1=13=prime and 3*7*2-1=41=prime; a(4)=8 because 1*19*2-1=37=prime and 3*19*2-1=113=prime.
%Y A172977 Cf. A005383(primes of the form 2n-1), A007528(primes of the form 6n-1).
%K A172977 nonn,new
%O A172977 1,2
%A A172977 Juri-Stepan Gerasimov (2stepan(AT)rambler.ru), Feb 06 2010

%I A172305
%S A172305 0,2,4,8,8,8,16,16,8,16,24,24,32,24
%N A172305 Number of L-toothpicks added to the L-toothpick structure of A172304 at the n-th stage.
%H A172305 David Applegate, Omar E. Pol and N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/tooth.pdf">The Toothpick Sequence and Other Sequences from Cellular Automata</a>
%H A172305 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/toothlist.html"> Catalog of Toothpick and Cellular Automata Sequences in the OEIS</a>
%Y A172305 Cf. First differences of A172304.
%Y A172305 Cf. A139250, A139251, A172304, A172306, A172307, A172308, A172309, A172310, A172311, A172312, A172313.
%K A172305 more,nonn,new
%O A172305 0,2
%A A172305 Omar E. Pol (info(AT)polprimos.com), Feb 06 2010

%I A172524
%S A172524 0,1,2,4,7,12,20,33,72,196,710,1546,2599,6738,19553,80688,185625,978142,
%T A172524 2432840,12112678,29466988,39202128,40962878,41948928,42570288,42684103,
%U A172524 43265540,44518036,52194742,65214030,159581828,337649208
%N A172524 Partial sums of Iccanobif numbers A001129.
%C A172524 The only primes in this sequence are: 2, 7, and 19553. The squares in this sequence begin: 0, 1, 4, 196.
%F A172524 a(n) = SUM[i=0..n] A001129(i) = SUM[i=0..n] {a(0) = 0, a(1) = 1, a(i+2) = R(a(i)) + R(a(i+1))} = SUM[i=0..n] A001129(i) = SUM[i=1..n] {a(0) = 0, a(1) = 1, a(i+2) = A004086(a(i)) + A004086(a(i+1))}.
%e A172524 a(14) = 0 + 1 + 1 + 2 + 3 + 5 + 8 + 13 + 39 + 124 + 514 + 836 + 1053 + 4139 + 12815 = 19553 is prime. a(31) = 0 + 1 + 1 + 2 + 3 + 5 + 8 + 13 + 39 + 124 + 514 + 836 + 1053 + 4139 + 12815 + 61135 + 104937 + 792517 + 1454698 + 9679838 + 17354310 + 9735140 + 1760750 + 986050 + 621360 + 113815 + 581437 + 1252496 + 7676706 + 13019288 + 94367798 + 178067380.
%Y A172524 Cf. A000040, A000045, A001129, A004086, A014258-A014260.
%K A172524 base,easy,nonn,new
%O A172524 0,3
%A A172524 Jonathan Vos Post (jvospost3(AT)gmail.com), Feb 06 2010

%I A173025
%S A173025 0,1,2,4,5,7,8,9,10,14,15,16,17,18,20,21,23,28,29,30,31,32,33,34,36,37,
%T A173025 39,40,41,42,46,47,56,57,58,60,61,62,63,64,65,66,68,69,71,72,73,74,78,
%U A173025 79,80,81,82,84,85,87,92,93,94,95,112,113,114,116,117,119,120,121,122
%N A173025 Numbers having no isolated digits "11" in their binary representations.
%C A173025 A173021(a(n+1)) = A173021(a(n)) + 1;
%C A173025 A173024 is a subsequence.
%H A173025 R. Zumkeller, <a href="b173025.txt">Table of n, a(n) for n = 1..1000</a>
%Y A173025 Cf. A144795.
%K A173025 base,nonn,new
%O A173025 1,3
%A A173025 Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Feb 07 2010

%I A172307
%S A172307 0,1,2,4,4,4,8,8,4,8,12,12,16,12
%N A172307 a(n) = A172305(n)/2.
%H A172307 David Applegate, Omar E. Pol and N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/tooth.pdf">The Toothpick Sequence and Other Sequences from Cellular Automata</a>
%H A172307 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/toothlist.html"> Catalog of Toothpick and Cellular Automata Sequences in the OEIS</a>
%Y A172307 Cf. A139250, A172304, A172305, A172306, A172308, A172309, A172310, A172311, A172312, A172313.
%K A172307 more,nonn,new
%O A172307 0,3
%A A172307 Omar E. Pol (info(AT)polprimos.com), Feb 06 2010

%I A173004
%S A173004 0,0,1,0,1,1,0,1,2,4,0,1,3,7,8,0,1,4,12,22,28,0,1,5,19,48,79,76,0,1,6,
%T A173004 28,92,204,290,272,0,1,7,39,160,463,900,1133,880,0,1,8,52,258,940,2404,
%U A173004 4128,4586,3328
%N A173004 Antidiagonal triangle sequence based on recursion: f(n,a)=a*f(n-1,a)+n*f(n-2,a)
%C A173004 Row sums are:
%C A173004 {0, 1, 2, 7, 19, 67, 228, 893, 3583, 15705,...}.
%F A173004 f(n,a)=a*f(n-1,a)+n*f(n-2,a);
%F A173004 t(n,m)=antidiagonal(f(n,a))
%e A173004 {0},
%e A173004 {0, 1},
%e A173004 {0, 1, 1},
%e A173004 {0, 1, 2, 4},
%e A173004 {0, 1, 3, 7, 8},
%e A173004 {0, 1, 4, 12, 22, 28},
%e A173004 {0, 1, 5, 19, 48, 79, 76},
%e A173004 {0, 1, 6, 28, 92, 204, 290, 272},
%e A173004 {0, 1, 7, 39, 160, 463, 900, 1133, 880},
%e A173004 {0, 1, 8, 52, 258, 940, 2404, 4128, 4586, 3328}
%t A173004 f[0, a_] := 0; f[1, a_] := 1;
%t A173004 f[n_, a_] := f[n, a] = a*f[n - 1, a] + n*f[n - 2, a];
%t A173004 m1 = Table[f[n, a], {n, 0, 10}, {a, 1, 11}];
%t A173004 Table[Table[m1[[m, n - m + 1]], {m, 1, n}], {n, 1, 10}];
%t A173004 Flatten[%]
%K A173004 nonn,tabl,uned,new
%O A173004 0,9
%A A173004 Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Feb 07 2010

%I A172409
%S A172409 0,1,2,3,11,12,17,18,23,24,27,28,29,30,35,36,37,38,41,42,46,47,48,51,52,
%T A172409 53,54,57,58,59,60,65,66,67,68,71,72,77,78,79,80,83,84,88,89,90,93,94,
%U A172409 95,96,97,98,101,102,107,108,113,114,117,118,119,120,121,122,123,124
%N A172409 Numbers n such that neither n-trivial prime is prime.
%C A172409 The trivial or frontier primes: primes p such that either p-+1 is prime.
%e A172409 a(1)=0 because 0-2=-2=nonprime and 0-3=-1=nonprime; a(2)=1 because 1-2=-1=nonprime and 1-3=-2=nonprime; a(3)=2 because 2-2=0=nonprime and 2-3=-1=nonprime.
%Y A172409 Cf. A000027, A000040, A169606.
%K A172409 nonn,new
%O A172409 1,3
%A A172409 Juri-Stepan Gerasimov (2stepan(AT)rambler.ru), Feb 06 2010

%I A172976
%S A172976 2,3,7,19,157,199,299,379,439,577,829,877,967,997,1009,1279,1429,1459,
%T A172976 1657
%N A172976 Primes p such that {1,3}*p*2-1 are both primes.
%e A172976 a(1)=2 because 1*2*2-1=3=prime and 3*2*2-1=13=prime; a(2)=3 because 1*3*2-1=7=prime and 3*3*2-1=17=prime; a(3)=7 because 1*7*2-1=13=prime and 3*7*2-1=41=prime; a(4)=19 because 1*19*2-1=37=prime and 3*19*2-1=113=prime.
%Y A172976 Cf. A005383(primes of the form 2n-1), A007528(primes of the form 6n-1).
%K A172976 nonn,new
%O A172976 1,1
%A A172976 Juri-Stepan Gerasimov (2stepan(AT)rambler.ru), Feb 06 2010

%I A172516
%S A172516 2,3,6,10,18,30,60,108,180,360,720,1260,2520,5040,9240,17640,35280,
%T A172516 65520,131040,257040,498960,982800,1884960,3603600,7207200,14414400,
%U A172516 28274400,56548800,110270160,220540320,428828400,845404560,1690809120
%N A172516 Least number k such that sigma(k) >= 2^n.
%C A172516 For n-bit arithmetic, m=a(n)-1 is the largest number for which sigma(m) can be computed without overflow. This is a subsequence of the highly abundant numbers, A002093, which is very useful for computing this sequence. a(63) is 1454751268447276800.
%F A172516 a(n) <= 2 * a(n-1)
%t A172516 k=1; Table[While[DivisorSigma[1,k]<2^n, k++ ]; k, {n,20}]
%Y A172516 A141847 (least number k such that sigma2(k) >= 2^n)
%K A172516 nonn,new
%O A172516 1,1
%A A172516 T. D. Noe (noe(AT)sspectra.com), Feb 05 2010

%I A172989
%S A172989 1,2,3,6,5,12,3,2,3,18,5,12,3,2,15,18,7,12,21,2,63,42,55,6,15,10,27,12,
%T A172989 19,78,15,2,93,12,5,78,15,10,21,12,23,18,57,14,27,30,7,120,117,8,15,42,
%U A172989 37,24,27,58,93,18,7,12,75,38,3,6,7,132,27,28,69,18,5,102,27,34,75,78,5
%N A172989 Prime Symmetricity of squares; least n such that square +- n are primes.
%F A172989 2^2+-1=primes,3^2+-2=primes,4^2+-3=primes,5^2+-6=primes,..
%t A172989 f[n_]:=Block[{k},If[OddQ[n],k=2,k=1];While[ !PrimeQ[n-k]||!PrimeQ[n+k],k+=2];k];Table[f[n^2],{n,2,40}]
%K A172989 nonn,new
%O A172989 1,2
%A A172989 Vladimir Orlovsky (4vladimir(AT)gmail.com), Feb 07 2010

%I A172491
%S A172491 1,2,3,5,7,10,13,19,25,33,42,54,68,85
%N A172491 Number of partitions of n, where n has to be a sum of positive integer multiples of Fibonacci numbers of any non-intermittent subsequence of Fibonacci numbers, starting with the first Fibonacci number. The twice occuring "1" at the beginning of the sequence of Fibonacci numbers must be considered as being differrent.
%C A172491 You can think of two "different ones" as two different slots, each valued 1 or as two different colored stamps (i.e. red and green) , both with the same value 1.
%e A172491 a(4) = 5
%e A172491 Considering only the first Fibonacci number (a "red" 1) there is only one such partition:
%e A172491 4 = 4 * 1(red)
%e A172491 Considering partions using the first two Fibonacci numbers (a "red" 1 and a "green" 1), there are three partitions:
%e A172491 4 = 3 * 1(red) + 1 * 1(green)
%e A172491 4 = 2 * 1(red) + 2 * 1(green)
%e A172491 4 = 1 * 1(red) + 3 * 1(green)
%e A172491 Considering partitions using the first three Fibonacci numbers, there is just one partition:
%e A172491 4 = 1 * 1(red) + 1 * 1(green) + 1 * 2
%e A172491 Altogether these are 5 different partitions.
%e A172491 Note that every Fibonacci number between the first and the last one of any considered subsequence must occur at least once. There is no valid partition of the number 4 which includes the Fibonacci number 3 as 4 = 1(red) + 3 would omit 1(green) as well as 2.
%Y A172491 Cf. A000045. Fibonacci numbers
%K A172491 nonn,new
%O A172491 1,2
%A A172491 Frank Schwellinger (nummer_eins(AT)web.de), Feb 05 2010

%I A172984
%S A172984 1,1,2,3,5,3,3,1,4,5,4,4,3,2,5,2,2,4,2
%N A172984 Sequence congruent to the Fibonacci sequence modulo 5, with 1 added to the last term. Seen on "Mathnet"
%C A172984 This sequence was used as a puzzle in the "Mathnet" portion of the children's mathematics television show Square One TV. In the series "Case of the Willing Parrot" (Episodes 201-205), the sequence was found on a tile wall. The mystery was solved by the discovery of the sequence and a key behind the final term.
%D A172984 Schneider, Joel, et al., Square One TV: Season Two Content Analysis and Show Rundowns. 21 Jul 1988, Children's Television Workshop; New York.
%H A172984 Children's Television Workshop, <a href="http://www.youtube.com/watch?v=vLhQfcZ-BWk">Mathnet - Case of the Willing Parrot (Recap & Finale) Pt.2 </a>
%H A172984 Schneider, Joel, et al, <a href="http://www.squareonetv.org/default.asp?ID=17">Mathnet Guide</a>
%F A172984 F(n) = (F(n-1) + F(n-2)) mod 5 for n=/= 18, F(19)=(F(n-1) + F(n-2)+1) mod 5 , F(0) = 0, F(1) = 1, F(2) = 1
%Y A172984 A000045
%K A172984 fini,full,nonn,new
%O A172984 0,3
%A A172984 Jon Suen (jsuen(AT)ece.ucsb.edu), Feb 06 2010

%I A172998
%S A172998 1,2,3,4,372
%N A172998 k such that 10^4k+10^3k-10^k-3 are prime.
%C A172998 Corresponds to the numbers 10987,100999897,1000999998997,10000999999989997,...
%K A172998 nonn,new
%O A172998 1,2
%A A172998 James G. Merickel (merk7(AT)verizon.net), Feb 07 2010

%I A172366
%S A172366 1,2,3,4,5,8,9,10,11,14,15,16,17,20,21,26,27,28,29,34,35,38,39,40,41,44,
%T A172366 45,50,51,56,57,58,59,64,65,68,69,70,71,76,77,80,81,86,87,94,95,98,99,
%U A172366 100,101,104,105,106,107,110,111,124,125,128,129,134,135,136,137,146
%N A172366 Numbers n such that either n+trivial prime is prime.
%C A172366 The trivial or frontier primes: primes p such that either p-+1 is prime.
%F A172366 a(n+1)=A169606(m).
%e A172366 a(1)=1 because 1+2=3=prime and 1+3=4=nonprime; a(2)=2 because 2+2=4=nonprime and 2+3=5=prime; a(3)=3 because 3+2=5=prime and 3+3=6=nonprime.
%Y A172366 Cf. A000027, A000040, A169606.
%K A172366 nonn,new
%O A172366 1,2
%A A172366 Juri-Stepan Gerasimov (2stepan(AT)rambler.ru), Feb 06 2010

%I A172974
%S A172974 2,3,4,5,6,7,8,9,10,11,12,14,15,20,21,28,35,36,45,50,55,56,66,70,78,84,
%T A172974 91
%N A172974 Degrees of completeness of Lukasiewicz logics.
%D A172974 Tokarz M, 1977. Degrees of completeness of Lukasiewicz logics. In [Wss^3 jcicki and Malinowski (eds.), 1977, pp. 127-134].
%K A172974 nonn,new
%O A172974 0,1
%A A172974 Artur Jasinski (grafix(AT)csl.pl), Feb 06 2010

%I A173023
%S A173023 1,2,3,3,4,5,5,6,7,8,9,9,9,9,10,11,12,13,14,14,15,16,16,17,17,17,17,17,
%T A173023 18,19,20,21,22,23,24,24,25,26,26,27,28,29,30,30,30,30,31,32,32,32,32,
%U A173023 32,32,32,32,32,33,34,35,35,36,37,38,39,40,41,42,42,43,44,44,45,46,47
%N A173023 Number of numbers <= n having no isolated digits "11" in their binary representations.
%C A173023 a(A173025(n+1)) = a(A173025(n)) + 1;
%C A173023 A173021(n) <= A173022(n) <= a(n).
%H A173023 R. Zumkeller, <a href="b173023.txt">Table of n, a(n) for n = 0..10000</a>
%e A173023 a(20) = #{0,1,2,4,5,7,8,9,10,14,15,16,17,18,20} = #{0,1,10,100,101,111,1000,1001,1010,1110,1111,10000,10001,10010,10100} = 15.
%K A173023 base,nonn,new
%O A173023 0,2
%A A173023 Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Feb 07 2010

%I A172528
%S A172528 1,1,1,1,2,3,3,1,3,8,19,38,60,60,1,4,15,53,175,535,1490,3675,7700,12600,
%T A172528 12600,1,5,24,111,494,2111,8634,33635,123998,428820,1373820,4003230,
%U A172528 10325700,22522500,37837800,37837800
%N A172528 Triangular array T(n,k) n,k>=0 is the number of k letter words formed using at most 1a,2b's,3c's,...,n#'s
%C A172528 The row lengths are n(n+1)/2 +1
%F A172528 E.g.f. for row n is Product_m=0...n[Sum_i=0...m[x^i/i! ]]
%e A172528 T(3,2) = 8 because there are 8 two letter words that can be formed using the letters a,b,b,c,c,c: {a, b}, {a, c}, {b, a}, {b, b}, {b, c}, {c, a}, {c, b}, {c, c}. Triangle Begins 1; 1,1; 1,2,3,3; 1,3,8,19,38,60,60; 1,4,15,53,175,535,1490,3675,7700,12600,12600;
%t A172528 Table[CoefficientList[Series[Product[Sum[x^i/i!, {i, 0, n}], {n, 0, m}], {x,0, (m^2 + m)/2}], x]*Table[n!, {n, 0, (m^2 + m)/2}], {m, 0,5}] // Grid
%Y A172528 The last entry in row n is A022915(n)
%K A172528 nonn,new
%O A172528 0,5
%A A172528 Geoffrey Critzer (critzer.geoffrey(AT)usd443.org), Feb 06 2010

%I A172520
%S A172520 1,2,3,2,4,5,3,4,6,7,2,5,6,8,9,4,5,7,8,10,11,2,5,7,9,10,12,13,4,6,8,9,
%T A172520 11,12,14,15,3,6,7,10,11,13,14,16,17,4,5,9,10,12,13,15,16,18,19,2,7,8,
%U A172520 10,12,14,15,17,18,20,21,6,7,9,11,13,14,16,17,19,20,22,23,2,5,8,11,12
%N A172520 Triangle in which each row gives the number of divisors of numbers in the range n to n+k for k=0..n-1.
%C A172520 Row n begins with the number of divisors of n and ends with 2n-1. Observe that the reverse of row n starts 2n-1, 2n-2, 2n-4, 2n-5,...; that is, 2n-r where r is in A001651, numbers not divisible by 3. Why?
%e A172520 For n=5, we have 2 numbers that divide 5 (namely, 1 and 5), 5 numbers that divide numbers in the range [5,6] (namely, 1, 2, 3, 5, and 6), 6 divisors that divide numbers in the range [5,7] (namely, 1, 2, 3, 5, 6, and 7), 8 divisors that divide numbers in the range [5,8] (namely, all numbers from 1 to 8), and 9 divisors that divide numbers in the range [5,9] (namely, all numbers from 1 to 9). Hence row 5 is 2, 5, 6, 8, 9.
%t A172520 Flatten[Table[Length[Union[Flatten[Divisors[Range[n,n+k]]]]], {n,50}, {k,0,n-1}]]
%K A172520 nonn,tabl,new
%O A172520 1,2
%A A172520 T. D. Noe (noe(AT)sspectra.com), Feb 06 2010

%I A173050
%S A173050 1,0,1,0,1,1,0,2,3,1,0,5,10,6,1,0,14,36,31,10,1,0,42,135,156,77,15,1,0,
%T A173050 132,518,771,534,169,21,1,0,429,2015,3745,3451,1610,345,28,1,0,1430,
%U A173050 7906,17897,21094,13569,4537,676,36,1,0,4862,31195,84278,123203,103986
%N A173050 Triangle, read by rows, given by [0,1,1,1,1,1,1,1,...] DELTA [1,0,1,0,2,0,3,0,4,0,5,0,6,0,...] where DELTA is the operator defined in A084938.
%F A173050 Sum_{k, 0<=k<=n}T(n,k) = A074664(n).
%e A173050 Triangle begins : 1 ; 0,1 ; 0,1,1 ; 0,2,3,1 ; 0,5,10,6,1 ; 0,14,36,31,10,1 ; ...
%Y A173050 Cf. A000108
%K A173050 nonn,tabl,new
%O A173050 0,8
%A A173050 Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Feb 08 2010

%I A172993
%S A172993 0,0,0,1,1,1,2,2,3,5,6,8,9,11,13,15,17,18,20,23
%N A172993 Maximal number of 4-tree rows in n-tree orchard problem in which the eascoordinate of trees may be complex numbers.
%H A172993 Du, Zhao Hui, <a href="http://zdu.spaces.live.com/blog/cns!C95152CB25EF2037!122.entry">Detailed information about the problem</a>
%e A172993 such as 12 trees in 8 rows: example for 12 trees with 8 rows: AEFG,AHIJ,BEHK,BFIL,CEJL,CGIK,DFJK,DGHL. Let t^2-t+1=0, coordinate in project plane: A(1,0,0),B(0,1),C(-t,1),D(1-t,1),E(0,0),F(1,0),G(1-t,0),H(0,1,0),I(1,-1,0),J(1,t-1,0),K(0,1-t),L(1-t,t)
%Y A172993 A006065
%K A172993 hard,nonn,new
%O A172993 1,7
%A A172993 Du, Zhao Hui (zhao.hui.du(AT)gmail.com), Feb 07 2010

%I A172992
%S A172992 0,0,0,1,1,1,2,2,3,5,6,7,9,10,12,14,15,17,20,23
%N A172992 Maximal number of 4-tree rows in n-tree orchard problem in which the coordinate of trees must be integers.
%H A172992 Du, Zhao Hui, <a href="http://zdu.spaces.live.com/blog/cns!C95152CB25EF2037!122.entry">More detailed results</a>
%Y A172992 A006065
%K A172992 hard,nonn,new
%O A172992 1,7
%A A172992 Du, Zhao Hui (zhao.hui.du(AT)gmail.com), Feb 07 2010

%I A172309
%S A172309 0,1,2,2,2,4,4,2,4,6,6,8,6,12,10,6,4
%N A172309 Number of L-toothpicks added to the L-toothpick structure of A172308 (First quadrant) at the n-th stage.
%H A172309 David Applegate, Omar E. Pol and N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/tooth.pdf">The Toothpick Sequence and Other Sequences from Cellular Automata</a>
%H A172309 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/toothlist.html"> Catalog of Toothpick and Cellular Automata Sequences in the OEIS</a>
%F A172309 a(0)=0: a(n)=A172305(n+1)/4, for n>=1.
%Y A172309 Cf. First differences of A172308.
%Y A172309 Cf. A139250, A139251, A152978, A172304, A172305, A172306, A172307, A172308, A172310, A172311, A172312, A172313.
%K A172309 more,nonn,new
%O A172309 0,3
%A A172309 Omar E. Pol (info(AT)polprimos.com), Feb 06 2010

%I A173022
%S A173022 1,1,1,2,2,2,3,4,4,4,4,4,5,5,6,7,7,7,7,7,7,7,7,7,8,8,8,9,10,10,11,12,12,
%T A173022 12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,13,13,13,14,14,14,15,16,
%U A173022 17,17,17,18,19,19,20,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21
%N A173022 Number of numbers <= n having no isolated ones in their binary representations.
%C A173022 a(A144795(n+1)) = a(A144795(n)) + 1;
%C A173022 a(2^n - 1)) = A005251(n+1);
%C A173022 A173021(n) <= a(n) <= A173023(n).
%H A173022 R. Zumkeller, <a href="b173022.txt">Table of n, a(n) for n = 0..10000</a>
%e A173022 a(20)=#{0,3,6,7,12,14,15}=#{0,11,110,111,1100,1110,1111}=7.
%K A173022 base,nonn,new
%O A173022 0,4
%A A173022 Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Feb 07 2010

%I A175102
%S A175102 1,1,2,2,2,2,2,3,3,3,2,3,2,3,4,4,2,4,2,4,4
%N A175102 a(n) = number of "divisor islands" of n. A divisor island is any set of consecutive divisors of a number where no pairs of consecutive divisors in the set are separated by 2 or more.
%e A175102 The divisors of 30 are 1,2,3,5,6,10,15,30. The divisor islands are (1,2,3), (5,6), (10), (15), (30). (Note that the differences between consecutive divisors 5-3, 10-6, 15-10, and 30-15 are all > 1.) There are 5 such islands, so a(30)=5.
%K A175102 more,nonn,new
%O A175102 1,3
%A A175102 Leroy Quet (q1qq2qqq3qqqq(AT)yahoo.com), Feb 07 2010

%I A173021
%S A173021 1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6,7,7,7,7,
%T A173021 7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,9,9,10,11,11,11,11,
%U A173021 11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11
%N A173021 Number of numbers <= n having in binary representation neither isolated ones nor isolated double ones.
%C A173021 a(A173024(n+1)) = a(A173024(n)) + 1;
%C A173021 a(2^n - 1)) = A005252(n+1);
%C A173021 a(n) <= A173022(n) <= A173023(n).
%H A173021 R. Zumkeller, <a href="b173021.txt">Table of n, a(n) for n = 0..10000</a>
%Y A173021 a(30)=#{0, 7, 14, 15, 28, 30}=#{0, 111, 1110, 1111, 11100, 11110}=6.
%K A173021 base,nonn,new
%O A173021 0,8
%A A173021 Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Feb 07 2010

%I A172497
%S A172497 1,1,1,1,1,1,1,1,1,1,1,2,2,2,1,1,1,2,2,1,1,1,2,2,4,2,2,1,1,3,6,6,6,6,3,
%T A172497 1,1,2,6,12,6,12,6,2,1,1,4,8,24,24,24,24,8,4,1,1,3,12,24,36,72,36,24,12,
%U A172497 3,1
%N A172497 A product triangle based on Leymer's polynomial A029826 :c(n)=Product[A029826(n+10);t(n,m)=c(n)/(c(m)*c(n-m))
%C A172497 Row sums are:
%C A172497 {1, 2, 3, 4, 8, 8, 14, 32, 48, 122, 224,...}.
%C A172497 An offset of 10 is used to avoid zeros in the sequence.
%F A172497 c(n)=Product[A029826(n+10);
%F A172497 t(n,m)=c(n)/(c(m)*c(n-m))
%e A172497 {1},
%e A172497 {1, 1},
%e A172497 {1, 1, 1},
%e A172497 {1, 1, 1, 1},
%e A172497 {1, 2, 2, 2, 1},
%e A172497 {1, 1, 2, 2, 1, 1},
%e A172497 {1, 2, 2, 4, 2, 2, 1},
%e A172497 {1, 3, 6, 6, 6, 6, 3, 1},
%e A172497 {1, 2, 6, 12, 6, 12, 6, 2, 1},
%e A172497 {1, 4, 8, 24, 24, 24, 24, 8, 4, 1},
%e A172497 {1, 3, 12, 24, 36, 72, 36, 24, 12, 3, 1}
%t A172497 (*A029826 Inverse of Salem polynomial : 1/(x^10 + x^9 - x^7 - x^6 - x^5 - x^4 - x^3 + x + 1).*)
%t A172497 p[x_] = (x^(10) + x^9 - x^7 - x^6 - x^5 - x^4 - x^3 + x + 1);
%t A172497 q[x_] = Expand[x^10*p[1/x]];
%t A172497 a = Table[SeriesCoefficient[Series[1/ q[x], {x, 0, 100}], n], {n, 0, 100}];
%t A172497 c[n_] := Product[a[[m + 10]], {m, 1, n}];
%t A172497 t[n_, m_] := c[n]/(c[m]*c[n - m]);
%t A172497 Table[Table[t[n, m], {m, 0, n}], {n, 0, 10}];
%t A172497 Flatten[%]
%Y A172497 A029826
%K A172497 nonn,tabl,uned,new
%O A172497 0,12
%A A172497 Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Feb 05 2010

%I A172313
%S A172313 1,1,1,2,2,1,2,3,3,4,3,6,5,3,2
%N A172313 a(n) = A172309(n+1)/2.
%H A172313 David Applegate, Omar E. Pol and N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/tooth.pdf">The Toothpick Sequence and Other Sequences from Cellular Automata</a>
%H A172313 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/toothlist.html"> Catalog of Toothpick and Cellular Automata Sequences in the OEIS</a>
%Y A172313 Cf. A139250, A172304, A172305, A172306, A172307, A172308, A172309, A172310, A172311, A172312.
%K A172313 more,nonn,new
%O A172313 1,4
%A A172313 Omar E. Pol (info(AT)polprimos.com), Feb 06 2010

%I A172500
%S A172500 1,2,1,4,5,2,7,8,1,10,1,4,13,14,5,16,17,2,19,20,7,2,23,8,25,26,3,28,29,
%T A172500 10
%N A172500 a(n) = numerator of fraction a / b, where (a, b) = 1, such that its decimal representation has form 0,(n)(n)(n)... with period (n).
%C A172500 Denominators in A172501.
%e A172500 a(10) = 10; 10 / 99 = 0,10101010... (period 10). a(9) = 1; 1 / 1 = 0,9999999...
%K A172500 nonn,new
%O A172500 1,2
%A A172500 Jaroslav Krizek (jaroslav.krizek(AT)atlas.cz), Feb 05 2010

%I A172986
%S A172986 0,1,1,1,0,0,1,0,1,0,1,1,1,1,2,1,2,3,2,4,3,3,5,4,4,5,4,5,4,5,5,5,5,6,5,
%T A172986 6,7,6,8,7,8,6,8,7,7,8,7,8,7,8,8,8,8,9,8,9,10,9,11,10,11,9,11,10,10,11,
%U A172986 10,11,10,11,11,11,11,12,11,12,13,12,14,13,14,12,14,13,13,14,13,14,13
%V A172986 0,1,-1,1,0,0,1,0,1,0,1,1,1,1,2,1,2,3,2,4,3,3,5,4,4,5,4,5,4,5,5,5,5,6,5,
%W A172986 6,7,6,8,7,8,6,8,7,7,8,7,8,7,8,8,8,8,9,8,9,10,9,11,10,11,9,11,10,10,11,
%X A172986 10,11,10,11,11,11,11,12,11,12,13,12,14,13,14,12,14,13,13,14,13,14,13
%N A172986 A periodic chaotic sequence based on A029826: a(n)=If[n==0,0,If[n <= 20, A029826(n+1), a(n - 1 - Mod[n, 20]) + A029826(2 + Mod[n, 20])]
%C A172986 The idea was to use the chaotic initial behavior of A029826 as a slow moving pattern for the new sequence.
%F A172986 a(n)=If[n==0,0,If[n <= 20, A029826(n+1), a(n - 1 - Mod[n, 20]) + A029826(2 + Mod[n, 20])]
%t A172986 Clear[a, b, c, p, q]
%t A172986 p[x_] = (x^(10) + x^9 - x^7 - x^6 - x^5 - x^4 - x^3 + x + 1);
%t A172986 q[x_] = Expand[x^10*p[1/x]];
%t A172986 a = Table[SeriesCoefficient[Series[1/q[x], {x, 0, 50}], n], {n, 0, 20}];
%t A172986 b[0] := 0;
%t A172986 b[n_] := b[n] = If[n <= 20, a[[n]], b[n - 1 - Mod[n, 20]] + a[[1 + Mod[n, 20]]]];
%t A172986 c = Table[b[n], {n, 0, 100}]
%Y A172986 A029826
%K A172986 sign,uned,new
%O A172986 0,15
%A A172986 Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Feb 06 2010

%I A171848
%S A171848 1,1,1,1,1,1,2,1,1,3,3,1,1,4,6,4,2,1,5,10,10,7,3,1,1,6,15,20,18,12,7,2,
%T A171848 1,1,7,21,35,39,33,24,14,7,3,1,1,8,28,56,75,76,65,48,32,18,10,4,2,1,9,
%U A171848 36,84,132,156,153,131,102,72,47,28,16,7,3,1,1,10,45,120,217,294,326
%N A171848 Triangle read by rows: T(n,k) is the number of peakless Motzkin paths of length n for which the area below the level steps (i.e. the sum of the altitudes of the level steps) is k (n>=0, k>=0).
%C A171848 The considered statistic (area below level steps) in RNA secondary structure terminology is called concentration (see the Willenbring reference, p. 1610).
%C A171848 Row n has 1+sum(floor(k/4),k=0..n+1) entries.
%C A171848 Sum of entries in row n = A004148(n) (the secondary structure numbers).
%C A171848 Sum(k*T(n,k), k>=0) = A171849(n).
%D A171848 R. Willenbring, RNA structure, permutations, and statistics, Discrete Appl. Math., 157, 2009, 1607-1614.
%F A171848 The trivariate g.f. G=G(t,u,z), where z marks length, t marks area below the level steps, and u marks number of level steps, satisfies G(t,u,z)=1+uzG(t,u,z)+z^2*(G(t,tu,z) - 1)G(t,u,z).
%e A171848 T(4,2)=1 because we have UHHD, where U=(1,1), H=(1,0), D=(1,-1).
%e A171848 Triangle starts:
%e A171848 1;
%e A171848 1;
%e A171848 1;
%e A171848 1,1;
%e A171848 1,2,1;
%e A171848 1,3,3,1;
%e A171848 1,4,6,4,2;
%e A171848 1,5,10,10,7,3,1
%p A171848 g[0] := 1/(1-u*z+z^2-z^2*g[1]): for n to 15 do g[n] := subs({u = t*u, g[n] = g[n+1]}, g[n-1]) end do: G := subs({u = 1, g[16] = 0}, g[0]): Gser := simplify(series(G, z = 0, 15)): for n from 0 to 12 do P[n] := sort(coeff(Gser, z, n)) end do: for n from 0 to 12 do seq(coeff(P[n], t, k), k = 0 .. sum(floor((1/4)*j), j = 0 .. n+1)) end do; # yields sequence in triangular form
%Y A171848 A004148, A171849
%K A171848 nonn,new
%O A171848 0,7
%A A171848 Emeric Deutsch (deutsch(AT)duke.poly.edu), Feb 08 2010

%I A171850
%S A171850 1,1,1,1,1,1,2,1,1,3,2,1,1,1,4,4,2,3,2,1,1,5,7,5,5,5,5,2,1,1,1,6,11,10,
%T A171850 10,10,10,8,6,4,3,2,1,1,7,16,18,18,21,21,17,16,14,11,9,7,5,2,1,1,1,8,22,
%U A171850 30,32,38,43,40,34,32,32,26,23,20,14,10,8,4,3,2,1,1,9,29,47,55,67,79,83
%N A171850 Triangle read by rows: T(n,k) is the number of peakless Motzkin paths of length n for which the area below the path minus the number of U-steps is k (n>=0, k>=0).
%C A171850 The considered statistic (area below the path minus number of U-steps) in RNA secondary structure terminology is called density (see the Willenbring reference, p. 1611).
%C A171850 Number of entries in row n is 1+floor((n-1)^2/4).
%C A171850 Sum of entries in row n = A004148(n) (the secondary structure numbers).
%C A171850 Sum(k*T(n,k), k>=0) = A171851(n).
%D A171850 R. Willenbring, RNA structure, permutations, and statistics, Discrete Appl. Math., 157, 2009, 1607-1614.
%F A171850 The trivariate g.f. G=G(t,u,z), where z marks length, t marks the area below the path, and x marks number of U-steps, satisfies G(t,x,z)=1+zG(t,x,z)+txz^2*(G(t,x,tz) - 1)G(t,x,z) (yielding a continued fraction expression for G(t,1/t,z)).
%e A171850 T(4,2)=1 because we have UHHD, where U=(1,1), H=(1,0), D=(1,-1).
%e A171850 Triangle starts:
%e A171850 1;
%e A171850 1;
%e A171850 1;
%e A171850 1,1;
%e A171850 1,2,1;
%e A171850 1,3,2,1,1;
%e A171850 1,4,4,2,3,2,1;
%e A171850 1,5,7,5,5,5,5,2,1,1;
%e A171850 1,6,11,10,10,10,10,8,6,4,3,2,1;
%p A171850 g[0] := 1/(1-z+z^2-z^2*g[1]): for n to 12 do g[n] := subs({z = t*z, g[n] = g[n+1]}, g[n-1]) end do: G := subs(g[16] = 0, g[0]): Gser := simplify(series(G, z = 0, 15)): for n from 0 to 11 do P[n] := sort(coeff(Gser, z, n)) end do: for n from 0 to 11 do seq(coeff(P[n], t, k), k = 0 .. floor((1/4)*(n-1)^2)) end do; # yields sequence in triangular form
%Y A171850 A004148, A171851
%K A171850 nonn,tabf,new
%O A171850 0,7
%A A171850 Emeric Deutsch (deutsch(AT)duke.poly.edu), Feb 08 2010

%I A172971
%S A172971 1,1,1,1,1,1,1,0,0,1,1,1,2,1,1,1,7,9,9,7,1,1,35,43,44,43,
%T A172971 35,1,1,191,227,234,234,227,191,1,1,1199,1391,1426,1432,1426,1391,
%U A172971 1199,1,1,10079,11279,11470,11504,11504,11470,11279,10079,1,1
%V A172971 -1,-1,-1,-1,-1,-1,-1,0,0,-1,-1,1,2,1,-1,-1,7,9,9,7,-1,-1,35,43,44,43,
%W A172971 35,-1,-1,191,227,234,234,227,191,-1,-1,1199,1391,1426,1432,1426,1391,
%X A172971 1199,-1,-1,10079,11279,11470,11504,11504,11470,11279,10079,-1,-1
%N A172971 Subtraction triangle based on Q partitions: c(n)=Product[PartitionsQ[i],{i,0,n)];t(n,m)=c(n)-c(m)-c(n-m)
%C A172971 Row sums are:
%C A172971 {-1, -2, -3, -2, 2, 30, 198, 1302, 9462, 88662, 1010262,...}.
%F A172971 c(n)=Product[PartitionsQ[i],{i,0,n)];
%F A172971 t(n,m)=c(n)-c(m)-c(n-m)
%e A172971 {-1},
%e A172971 {-1, -1},
%e A172971 {-1, -1, -1},
%e A172971 {-1, 0, 0, -1},
%e A172971 {-1, 1, 2, 1, -1},
%e A172971 {-1, 7, 9, 9, 7, -1},
%e A172971 {-1, 35, 43, 44, 43, 35, -1},
%e A172971 {-1, 191, 227, 234, 234, 227, 191, -1},
%e A172971 {-1, 1199, 1391, 1426, 1432, 1426, 1391, 1199, -1},
%e A172971 {-1, 10079, 11279, 11470, 11504, 11504, 11470, 11279, 10079, -1},
%e A172971 {-1, 103679, 113759, 114958, 115148, 115176, 115148, 114958, 113759, 103679, -1}
%t A172971 c[n_] := Product[PartitionsQ[m], {m, 1, n}];
%t A172971 t[n_, m_] := c[n] - (c[m] + c[n - m]);
%t A172971 Table[Table[t[n, m], {m, 0, n}], {n, 0, 10}];
%t A172971 Flatten[%]
%K A172971 sign,tabl,uned,new
%O A172971 0,13
%A A172971 Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Feb 06 2010

%I A172970
%S A172970 1,1,1,1,1,1,1,0,0,1,1,1,2,1,1,1,7,9,9,7,1,1,35,43,44,43,
%T A172970 35,1,1,143,179,186,186,179,143,1,1,575,719,754,760,754,719,575,1,
%U A172970 1,3071,3647,3790,3824,3824,3790,3647,3071,1,1,19199,22271,22846
%V A172970 -1,-1,-1,-1,-1,-1,-1,0,0,-1,-1,1,2,1,-1,-1,7,9,9,7,-1,-1,35,43,44,43,
%W A172970 35,-1,-1,143,179,186,186,179,143,-1,-1,575,719,754,760,754,719,575,-1,
%X A172970 -1,3071,3647,3790,3824,3824,3790,3647,3071,-1,-1,19199,22271,22846
%N A172970 Subtraction triangle based on A004001: c(n)=Product[A004001(i),{i,0,n)];t(n,m)=c(n)-c(m)-c(n-m)
%C A172970 Row sums are:
%C A172970 {-1, -2, -3, -2, 2, 30, 198, 1014, 4854, 28662, 197622,...}.
%F A172970 c(n)=Product[A004001(i),{i,0,n)];
%F A172970 t(n,m)=c(n)-c(m)-c(n-m)
%e A172970 {-1},
%e A172970 {-1, -1},
%e A172970 {-1, -1, -1},
%e A172970 {-1, 0, 0, -1},
%e A172970 {-1, 1, 2, 1, -1},
%e A172970 {-1, 7, 9, 9, 7, -1},
%e A172970 {-1, 35, 43, 44, 43, 35, -1},
%e A172970 {-1, 143, 179, 186, 186, 179, 143, -1},
%e A172970 {-1, 575, 719, 754, 760, 754, 719, 575, -1},
%e A172970 {-1, 3071, 3647, 3790, 3824, 3824, 3790, 3647, 3071, -1},
%e A172970 {-1, 19199, 22271, 22846, 22988, 23016, 22988, 22846, 22271, 19199, -1}
%t A172970 (*A004001*)
%t A172970 f[0] = 0; f[1] = 1; f[2] = 1;
%t A172970 f[n_] := f[n] = f[f[n - 1]] + f[n - f[n - 1]];
%t A172970 c[n_] := If[n == 0, 1, Product[f[m], {m, 1, n}]];
%t A172970 t[n_, m_] := c[n] - (c[m] + c[n - m]);
%t A172970 Table[Table[t[n, m], {m, 0, n}], {n, 0, 10}];
%t A172970 Flatten[%]
%Y A172970 A004001
%K A172970 sign,tabl,uned,new
%O A172970 0,13
%A A172970 Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Feb 06 2010

%I A171846
%S A171846 1,1,1,1,0,1,1,0,2,1,1,0,3,2,1,0,1,1,0,4,3,3,1,2,2,1,1,0,5,4,6,4,4,4,5,
%T A171846 2,1,0,1,1,0,6,5,10,9,9,7,11,8,5,3,3,2,2,1,1,0,7,6,15,16,18,14,20,20,16,
%U A171846 10,11,8,8,6,5,2,1,0,1,1,0,8,7,21,25,32,28,36,39,41,29,27,24,25,20,17
%N A171846 Triangle read by rows: T(n,k) is the number of peakless Motzkin paths of length n for which the area below the path is k (n>=0, k>=0).
%C A171846 Row 2n (n>0) has n^2 entries; row 2n+1 has n^2 + n + 1 entries.
%C A171846 Sum of entries in row n = A004148(n) (the secondary structure numbers).
%C A171846 Sum(k*T(n,k), k>=0) = A171847(n).
%F A171846 G.f. G=G(t,z) satisfies: G(t,z)=1/[1 - z + tz^2 - tz^2*G(t,tz)] (yielding a continued-fraction expression for G(t,z)).
%e A171846 T(4,2)=2 because we have HUHD and UHDH, where U=(1,1), H=(1,0), D=(1,-1).
%e A171846 Triangle starts:
%e A171846 1;
%e A171846 1;
%e A171846 1;
%e A171846 1,0,1;
%e A171846 1,0,2,1;
%e A171846 1,0,3,2,1,0,1;
%e A171846 1,0,4,3,3,1,2,2,1;
%p A171846 g[0] := 1/(1-z+t*z^2-t*z^2*g[1]): for n to 15 do g[n] := subs({z = t*z, g[n] = g[n+1]}, g[n-1]) end do: G := subs(g[16] = 0, g[0]): Gser := simplify(series(G, z = 0, 15)): for n from 0 to 10 do P[n] := sort(coeff(Gser, z, n)) end do: d := proc (n) if n = 0 then 0 elif `mod`(n, 2) = 0 then (1/4)*(n-2)*(n+2) else (1/4)*(n-1)*(n+1) end if end proc: for n from 0 to 10 do seq(coeff(P[n], t, k), k = 0 .. d(n)) end do; # yields sequence in triangular form
%Y A171846 A004148, A171847
%K A171846 nonn,new
%O A171846 0,9
%A A171846 Emeric Deutsch (deutsch(AT)duke.poly.edu), Feb 08 2010

%I A173003
%S A173003 0,0,1,0,1,2,0,1,4,7,0,1,6,25,30,0,1,8,55,204,157,0,1,10,97,666,2065,
%T A173003 972,0,1,12,151,1560,10045,24984,6961,0,1,14,217,3030,31297,181476,
%U A173003 351841,56660,0,1,16,295,5220,75901,752688,3821041,5654440,516901
%N A173003 Antidiagonal triangle sequence based on recursion: f(n,a)=a*n*f(n-1,a)+f(n-2,a)
%C A173003 Row sums are:
%C A173003 {0, 1, 3, 12, 62, 425, 3811, 43714, 624536, 10826503,...}.
%F A173003 f(n,a)=a*n*f(n-1,a)+f(n-2,a);
%F A173003 t(n,m)=antidiagonal(f(n,a))
%e A173003 {0},
%e A173003 {0, 1},
%e A173003 {0, 1, 2},
%e A173003 {0, 1, 4, 7},
%e A173003 {0, 1, 6, 25, 30},
%e A173003 {0, 1, 8, 55, 204, 157},
%e A173003 {0, 1, 10, 97, 666, 2065, 972},
%e A173003 {0, 1, 12, 151, 1560, 10045, 24984, 6961},
%e A173003 {0, 1, 14, 217, 3030, 31297, 181476, 351841, 56660},
%e A173003 {0, 1, 16, 295, 5220, 75901, 752688, 3821041, 5654440, 516901}
%t A173003 f[0, a_] := 0; f[1, a_] := 1;
%t A173003 f[n_, a_] := f[n, a] = a*n*f[n - 1, a] + f[n - 2, a];
%t A173003 m1 = Table[f[n, a], {n, 0, 10}, {a, 1, 11}];
%t A173003 Table[Table[m1[[m, n - m + 1]], {m, 1, n}], {n, 1, 10}];
%t A173003 Flatten[%]
%K A173003 nonn,tabl,uned,new
%O A173003 0,6
%A A173003 Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Feb 07 2010

%I A160018
%S A160018 0,0,0,1,1,0,0,2,3,0,0,2,2,0,0,4,7,0,0,2,2,0,0,4,6,0,0,4,4,0,0,8,15,0,0,2,
%T A160018 2,0,0,4,6,0,0,4,4,0,0,8,14,0,0,4,4,0,0,8,12,0,0,8,8,0,0,16,31,0,0,2,2,
%U A160018 0,0,4,6,0,0,4,4,0,0,8,14,0,0,4,4,0,0,8,12,0,0,8,8,0,0,16,30,0,0,4,4,0
%N A160018 A175099 with a(3) changed from 0 to 1.
%H A160018 David Applegate, Omar E. Pol and N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/tooth.pdf">The Toothpick Sequence and Other Sequences from Cellular Automata</a>
%H A160018 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/toothlist.html">Catalog of Toothpick and Cellular Automata Sequences in the OEIS</a>
%F A160018 a(n) = a(1) = 0; for k >= 1, a(2^k) = 2^(k-1)-1, a(2^k+i) = 2*a(i) for 1 <= i < 2^k.
%Y A160018 Cf. A160125, A175098, A175099.
%K A160018 nonn,new
%O A160018 0,8
%A A160018 N. J. A. Sloane, Feb 07 2010

%I A108578
%S A108578 0,0,0,0,8,24,32,56,80,104,136,176,208,256,304,352,408,472,528,600,672,744,824,
%T A108578 912,992,1088,1184,1280,1384,1496,1600,1720,1840,1960,2088,2224,2352,
%U A108578 2496,2640,2784,2936,3096,3248,3416,3584,3752,3928,4112,4288
%N A108578 Number of 3 X 3 magic squares with magic sum 3n.
%H A108578 M. Beck and T. Zaslavsky, <a href="http://arXiv.org/abs/math.CO/0506315">An enumerative geometry for magic and magilatin labellings</a>, Ann. Combinatorics, 10 (2006), no. 4, 395-413. MR 2007m:05010. Zbl 1116.05071. [From Thomas Zaslavsky (zaslav(AT)math.binghamton.edu), Jan 29 2010]
%H A108578 M. Beck and T. Zaslavsky, <a href="http://www.math.binghamton.edu/zaslav/Tpapers/index.html#SLS">Six little squares and how their numbers grow</a>, submitted. [From Thomas Zaslavsky (zaslav(AT)math.binghamton.edu), Jan 29 2010]
%F A108578 G.f.: (8*x^15*(1+2*x^3)) / ((1-x^3)*(1-x^6)*(1-x^9)). a(n) is given by a quasipolynomial of period 18.
%e A108578 a(5) = 8 because there are 8 3 X 3 magic squares with entries 1,...,9 and magic sum 15.
%o A108578 (PARI) a(n)=(1/9)*(2*n^2-32*n+[144,78,120,126,96,102][(n%18)/3+1])
%Y A108578 Equals 8 times the second differences of A055328.
%Y A108578 Cf. A108576, A108577, A108579.
%K A108578 nonn,new
%O A108578 1,1
%A A108578 Thomas Zaslavsky (zaslav(AT)math.binghamton.edu) and Ralf Stephan (ralf(AT)ark.in-berlin.de), Jun 11 2005
%E A108578 Edited by N, J. A. Sloane, Feb 05 2010

%I A108576
%S A108576 0,0,0,0,0,0,0,0,0,8,16,40,64,96,128,184,240,320,400,504,608,744,880,
%T A108576 1056,1232,1440,1648,1904,2160,2464,2768,3120,3472,3880,4288,4760,5232,
%U A108576 5760,6288,6888,7488,8160,8832,9576,10320,11144,11968,12880,13792,14784,15776
%N A108576 Number of 3 X 3 magic squares (with distinct positive entries) having all entries < n.
%H A108576 M. Beck and T. Zaslavsky, <a href="http://arXiv.org/abs/math.CO/0506315">An enumerative geometry for magic and magilatin labellings</a>, Ann. Combinatorics, 10 (2006), no. 4, 395-413. MR 2007m:05010. Zbl 1116.05071. [From Thomas Zaslavsky (zaslav(AT)math.binghamton.edu), Jan 29 2010]
%H A108576 M. Beck and T. Zaslavsky, <a href="http://www.math.binghamton.edu/zaslav/Tpapers/index.html#SLS">Six little squares and how their numbers grow</a>, submitted. [From Thomas Zaslavsky (zaslav(AT)math.binghamton.edu), Jan 29 2010]
%F A108576 G.f.: (8*x^10*(2*x^2+1)) / ((1-x^6)*(1-x^4)*(1-x)^2) a(n) is given by a quasipolynomial of period 12.
%e A108576 a(10) = 8 because there are 8 3 X 3 magic squares with distinct entries < 10 (they are the standard magic squares).
%o A108576 (PARI) a(n)=1/6*(n^3-16*n^2+(76-3*(n%2))*n-[96,58,96,102,112,90,96,70,96,90,112,102][(n%12)+1])
%Y A108576 Cf. A108577, A108578, A108579.
%K A108576 nonn,new
%O A108576 1,10
%A A108576 Thomas Zaslavsky (zaslav(AT)math.binghamton.edu) and Ralf Stephan (ralf(AT)ark.in-berlin.de), Jun 11 2005
%E A108576 Edited by N, J. A. Sloane, Feb 05 2010

%I A172390
%S A172390 1,8,24,0,168,0,2112,0,32040,0,536256,0,9542976,0,177126912,0,
%T A172390 3390361128,0,66436117440,0,1326185205696,0,26872637815296,0,
%U A172390 551301904867392,0,11428295231789568,0,239010764560888320,0
%V A172390 1,8,24,0,-168,0,2112,0,-32040,0,536256,0,-9542976,0,177126912,0,
%W A172390 -3390361128,0,66436117440,0,-1326185205696,0,26872637815296,0,
%X A172390 -551301904867392,0,11428295231789568,0,-239010764560888320,0
%N A172390 G.f. satisfies: A(x) = G(x/A(x))^2 and G(x)^2 = A(x*G(x)^2) where G(x) = Sum_{n>=0} C(2n,n)^2*x^n.
%F A172390 G.f.: A(x) = x/Series_Reversion(x*G(x)^2)) where G(x) = Sum_{n>=0} C(2n,n)^2*x^n = 1/agm(1, (1-16*x)^(1/2)) = g.f. of A002894 and G(x)^2 is the g.f. of A036917.
%F A172390 Self-convolution of A158101, which is a bisection of A158100; A158100 has g.f. F(x) that satisfies: F(x) = 1/AGM(1, 1 - 8*x/F(x) ).
%e A172390 G.f.: A(x) = 1 + 8*x + 24*x^2 - 168*x^4 + 2112*x^6 - 32040*x^8 +...
%e A172390 A(x) = G(x/A(x))^2 where G(x) = 1/AGM(1, (1-16x)^(1/2)) is the power series:
%e A172390 G(x) = 1 + 2^2*x + 6^2*x^2 + 20^2*x^3 + 70^2*x^4 + 252^2*x^5 +...+ C(2n,n)^2*x^n +...
%e A172390 The square root of g.f. A(x) begins:
%e A172390 A(x)^(1/2) = 1 + 4*x + 4*x^2 - 16*x^3 - 28*x^4 + 176*x^5 + 336*x^6 +...+ A158101(n)*x^n +...
%o A172390 (PARI) {a(n)=local(G=sum(m=0,n,binomial(2*m,m)^2*x^m)+x*O(x^n));polcoeff(x/serreverse(x*G^2),n)}
%Y A172390 Cf. A036917, A002894, A158101, A158100.
%K A172390 sign,new
%O A172390 0,2
%A A172390 Paul D. Hanna (pauldhanna(AT)juno.com), Feb 04 2010

%I A172484
%S A172484 4,10,18,27,39,57,77,99,123,149,177,207,240,274,310,348,387,427,469,513,
%T A172484 558,604,652,702,753,805,859,914,970,1027,1085,1145,1207,1270,1335,1401,
%U A172484 1469,1538,1608,1680,1754,1829,1905,1982,2060,2140,2222,2306,2391,2477
%N A172484 Partial sums of extravagant numbers, also called prodigal numbers, or wasteful numbers.
%C A172484 Every natural number, written in base 10, is either economical A046759 (also called frugal), or equidigital A046758, or extravagant (or prodigal or wasteful). An extravagant number is one for which the factorization requires more digits that the original number such as 30 = 2 * 3 * 5. The subsequence of economical partial sums of extravagant numbers begins: xxx, 18, 39, 57, 77, 99, 207, 240, 274, 310. The subsequence of equidigital partial sum of economical numbers begins: 10, 27, 123, 149, 177, 427, 469 (such as 1207 = 17 * 71). The subsequence of prime partial sums of economical numbers begins: xxx, 149, 859, 2477, 2833.
%F A172484 SUM[i=1..n] A046760(i) = Partial sum of {Write n as a product of primes raised to powers, let D(n) = number of digits in product, l(n) = number of digits in n; sequence gives n such that D(n)>l(n)}.
%e A172484 a(1) = A046760(1) = 4. a(2) = 4 + 6 = 10. a(67) = 4 + 6 + 8 + 9 + 12 + 18 + 20 + 22 + 24 + 26 + 28 + 30 + 33 + 34 + 36 + 38 + 39 + 40 + 42 + 44 + 45 + 46 + 48 + 50 + 51 + 52 + 54 + 55 + 56 + 57 + 58 + 60 + 62 + 63 + 65 + 66 + 68 + 69 + 70 + 72 + 74 + 75 + 76 + 77 + 78 + 80 + 82 + 84 + 85 + 86 + 87 + 88 + 90 + 91 + 92 + 93 + 94 + 95 + 96 + 98 + 99 + 100 + 102 + 104 + 108 + 110 + 114 = 4138 = 2 * 2069 which is thus an economical number, with 4 digits but 5 in its prime factorization.
%Y A172484 Cf. A046758, A046759, A046760.
%K A172484 base,easy,nonn,new
%O A172484 1,1
%A A172484 Jonathan Vos Post (jvospost3(AT)gmail.com), Feb 04 2010
%E A172484 27 inserted by R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Feb 06 2010

%I A172483
%S A172483 1,1,2,5,4,4,2,6,4,7,7,5,9,12,13,14,14,9,12,10,11,13,20,16,15,16,15,23,
%T A172483 19,22,26,27,28,26,22,20,27,25,27,28,26,35,29,29,29,30,45,30,36,22,30,
%U A172483 39,39,40,44,44,43,34,38,36,48,54,43,38,43,49,45,47,53,38,51,51,62,56
%N A172483 Are conescutive cousin primes infinite?
%C A172483 If you graph the order of the consecutive cousin primes along the x-axis (i.e., first pair of cousin primes, second, third,...) and the number of cousin primes in the sequence given above along the y-axis, a clear pattern emerges. As you go farther along the x-axis, greater are the number of consecutive cousin primes, on average, within the interval obtained. If one can prove that there's at least one consecutive cousin prime within each interval, this would imply that cousin primes are infinite. I suspect the number of consecutive primes within each interval will never be zero.
%D A172483 C. C. Clawson, Mathematical Mysteries: The Beauty and Magic of Numbers, Perseus Books, 1999.
%D A172483 M. D. Sautoy, The Music of the Primes: Searching to Solve the Greatest Mystery in Mathematics, HarperCollins Publishers Inc., 2004.
%e A172483 Take any pair of consecutive cousin primes. Let us say the very first one (7,11). Square the first term, you get 49, then take the product of the two primes, you get 7x11=77. Between these two numbers, namely (49,77) there is 1 consecutive cousin prime (67,71). Hence the very first term of the sequence is 1.
%o A172483 (Other) A SAS program written by Rick Aster was used as my starting point. The program can be found at this link: www.globalstatements.com/shortcuts/88a.html
%Y A172483 A171727
%K A172483 nonn,new
%O A172483 1,3
%A A172483 Jaspal Singh Cheema (Jaspal(AT)rogers.com), Feb 04 2010

%I A170931
%S A170931 2,4,24,112,544,2624,12672,61184,295424,1426432,6887424,33255424,
%T A170931 160571392,775307264,3743514624,18075287552,87275208704,421401985024,
%U A170931 2034708774912,9824443039744,47436607258624,229044201193472
%N A170931 Extended Lucas L(n,i)=n*(L(n,i-1)+L(n,i-2))=a^i+b^i where d=sqrt(n*(n+4));a=(n+d)/2;b=(n-d)/2
%C A170931 n=1 A000032 (classic Lucas) n=2 A080040 n=3 A085480 n=4 above n=5+ all new
%F A170931 a(n)=2*A084128(n) = 4*a(n-1)+4*a(n-2). G.f.: 2*(1-2*x)/(1-4*x-4*x^2). [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Feb 05 2010]
%e A170931 L(n,0)=2 L(n,1)=n
%Y A170931 see above
%K A170931 nonn,new
%O A170931 0,1
%A A170931 Claudio Peruzzi (claudio.peruzzi(AT)gmail.com), Feb 04 2010
%E A170931 More terms from R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Feb 05 2010

%I A170930
%S A170930 0,21,63,252,945,3591,13608,51597,195615,741636,2811753,10660167,
%T A170930 40415760,153227781,580930623,2202475212,8350217505,31658078151,
%U A170930 120024886968,455048895357,1725221346975,6540810726996,24798096221913
%N A170930 G(n,1) with n index G(n,i)=n*(G(n,i-1)+G(n,i-2))=(a^i-b^i)*d where d=sqrt(n*(n+4)); a=(n+d)/2; b=(n-d)/2
%C A170930 n=1 A022088 n=2 12*A002605 n=3 above n=4 ... new
%F A170930 a(n) = 3*a(n-1)+3*a(n-2) = 21*A030195(n). G.f.: 21*x/(1-3*x-3*x^2). [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Feb 05 2010]
%e A170930 G(n,0)=0 G(n,1)=n*(n+4)
%Y A170930 see above
%K A170930 nonn,new
%O A170930 0,2
%A A170930 Claudio Peruzzi (claudio.peruzzi(AT)gmail.com), Feb 04 2010
%E A170930 More terms from R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Feb 05 2010

%I A172482
%S A172482 3,16,47,104,195,328,511,752,1059,1440,1903
%N A172482 3*a(n)=(1+n)*(9+11*n+4*n^2).
%C A172482 Second bisection of first central column in Janet table A172002;see A131941(n+1). Companion to A172128=1,8,29,. a(n) mod 10=period 10:repeat 3,6,7,4,5,8,1,2,9,0 the ten digits.
%F A172482 a(n)=4a(n)-6a(n-2)+4a(n-3)-a(n-4) with 3 other recurrences.
%K A172482 nonn,uned,new
%O A172482 0,1
%A A172482 Paul Curtz (bpcrtz(AT)free.fr), Feb 04 2010

%I A175099
%S A175099 0,0,0,0,1,0,0,2,3,0,0,2,2,0,0,4,7,0,0,2,2,0,0,4,6,0,0,4,4,0,0,8,15,0,0,2,
%T A175099 2,0,0,4,6,0,0,4,4,0,0,8,14,0,0,4,4,0,0,8,12,0,0,8,8,0,0,16,31,0,0,2,2,
%U A175099 0,0,4,6,0,0,4,4,0,0,8,14,0,0,4,4,0,0,8,12,0,0,8,8,0,0,16,30,0,0,4,4,0
%N A175099 Number of closed rectangles added at the n-th stage of the leftist toothpicks A151566.
%H A175099 David Applegate, Omar E. Pol and N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/tooth.pdf">The Toothpick Sequence and Other Sequences from Cellular Automata</a>
%H A175099 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/toothlist.html">Catalog of Toothpick and Cellular Automata Sequences in the OEIS</a>
%F A175099 For a recurrence see A160018. - N. J. A. Sloane, Feb 07 2010
%Y A175099 Cf. A160125, A175098
%K A175099 nonn,new
%O A175099 0,8
%A A175099 R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Feb 04 2010

%I A172481
%S A172481 1,2,5,11,25,55,121,263,569,1223,2617
%N A172481 a(n+1)-2a(n)=0,1,1,3,5,11,=A001045 Jacobsthal.
%C A172481 Companion to A139790=1,2,3,5,7,9,7,-7,. a(n)+A139790=2,4,8,16,=A000079(n+1). A139790-a(n)=0,0,2,6,18,46,114,=A140960. a(n)-A001045=1,1,4,8,20,44,100,220,=A084219 unsigned.See A084247 signed.
%Y A172481 A151529.
%K A172481 nonn,uned,new
%O A172481 0,2
%A A172481 Paul Curtz (bpcrtz(AT)free.fr), Feb 04 2010

%I A175098
%S A175098 0,3,5,9,12,16,20,26,31,35,39,45,51,59,67,79,88,92,96,102,108,116,124,
%T A175098 136,146,154,162,174,186,202,218,242,259,263,267,273,279,287,295,307,
%U A175098 317,325,333,345,357,373,389,413,431,439,447,459,471,487,503,527,547
%N A175098 Number of lattice points covered at the n-th stage of the leftist toothpicks A151566
%H A175098 David Applegate, Omar E. Pol and N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/tooth.pdf">The Toothpick Sequence and Other Sequences from Cellular Automata</a>
%H A175098 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/toothlist.html">Catalog of Toothpick and Cellular Automata Sequences in the OEIS</a>
%Y A175098 Cf. A147614.
%K A175098 nonn,new
%O A175098 0,2
%A A175098 R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Feb 04 2010

%I A172480
%S A172480 5,7,13,17,29,31,37,41,43,53,61,67,73,89,97,101,109,113,137,149,157,173,
%T A172480 181,193,197,229,233,241,257,269,277,281,293,307,313,317,337,349,353,
%U A172480 367,373,389,397,401,409,421,433,449,457,461,487,509,521,541,557,569
%N A172480 Odd primes p such that there are as much primitive roots (mod p) in the interval [0,p/2] as in the interval [p/2,p].
%C A172480 The sequence contains all the primes of the form 4m+1 (A002144).
%C A172480 The sequence als contains some primes of the form 4m+3.
%t A172480 << NumberTheory`NumberTheoryFunctions` m = 2; s = {}; While[m < 10000, m++; p = Prime[m]; If[Mod[p, 4] == 1, s = {s, p}, q = (p - 1)/2; g = PrimitiveRoot[p]; se = Select[Range[p - 1], GCD[ #, p - 1] == 1 &]; e = Length[se]; j = 0; t = 0; While[j < e, j++; h = PowerMod[g, se[[j]], p]; If[h <= q, t = t + 1,] ]; If[e == 2t, s = {s, p},] ] ]; s = Flatten[s]
%Y A172480 Cf. A002144, A118818
%K A172480 hard,nonn,new
%O A172480 1,1
%A A172480 Emmanuel Vantieghem (manuvti(AT)hotmail.com), Feb 04 2010

%I A172479
%S A172479 1,1,1,1,1,1,1,2,2,1,1,2,4,2,1,1,3,6,6,3,1,1,4,12,12,12,4,1,1,5,20,30,
%T A172479 30,20,5,1,1,6,30,60,90,60,30,6,1,1,8,48,120,240,240,120,48,8,1,1,10,80,
%U A172479 240,600,800,600,240,80,10,1
%N A172479 A triangle based on Q partitions as products: c(n)=Product[PartitionsQ[m], {m, 1, n}];t(n,m)= c[n]/(c[m]*c[n - m])
%C A172479 Row sums are:
%C A172479 {1, 2, 3, 6, 10, 20, 46, 112, 284, 834, 2662,...}.
%F A172479 c(n)=Product[PartitionsQ[m], {m, 1, n}];
%F A172479 t(n,m)= c[n]/(c[m]*c[n - m])
%e A172479 {1},
%e A172479 {1, 1},
%e A172479 {1, 1, 1},
%e A172479 {1, 2, 2, 1},
%e A172479 {1, 2, 4, 2, 1},
%e A172479 {1, 3, 6, 6, 3, 1},
%e A172479 {1, 4, 12, 12, 12, 4, 1},
%e A172479 {1, 5, 20, 30, 30, 20, 5, 1},
%e A172479 {1, 6, 30, 60, 90, 60, 30, 6, 1},
%e A172479 {1, 8, 48, 120, 240, 240, 120, 48, 8, 1},
%e A172479 {1, 10, 80, 240, 600, 800, 600, 240, 80, 10, 1}
%t A172479 c[n_] := Product[PartitionsQ[m], {m, 1, n}];
%t A172479 t[n_, m_] := c[n]/(c[m]*c[n - m]);
%t A172479 Table[Table[t[n, m], {m, 0, n}], {n, 0, 10}];
%t A172479 Flatten[%]
%K A172479 nonn,tabl,uned,new
%O A172479 0,8
%A A172479 Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Feb 04 2010

%I A172478
%S A172478 1,4,72,13872,11762160,234312972480,41182101508222080
%N A172478 The number of ways to dissect an N by N square into polyominoes of size N and then fill it with numbers to make it a latin square, with the extra requirement that each digit occurs within each polyomino exactly once.
%C A172478 a(n) is the number of completed N by N jigsaw sudoku puzzles.
%e A172478 A 2 by 2 square can be covered by two dominoes by either positioning them vertically or horizontally. Both of these coverings allow for two 2 by 2 latin squares without violating the extra constraint.
%Y A172478 It is the number of pairs of an N by N latin square, A002860, with a dissection of the N by N square into polyominoes of size N, A172477, such that every number occurs in every polyomino exactly once.
%K A172478 nonn,new
%O A172478 1,2
%A A172478 Johan de Ruiter (johan.de.ruiter(AT)gmail.com), Feb 04 2010

%I A172477
%S A172477 1,2,10,117,4006,451206,158753814,187497290034
%N A172477 The number of ways to dissect an N by N square into polyominoes of size N.
%e A172477 A 2 by 2 square can be covered by two dominoes by either positioning them vertically or horizontally.
%Y A172477 Intersects with A167251, A167254, A167255, A167258.
%K A172477 nonn,new
%O A172477 1,2
%A A172477 Johan de Ruiter (johan.de.ruiter(AT)gmail.com), Feb 04 2010

%I A172476
%S A172476 0,0,0,1,1,2,2,2,3,3,4,4,4,5,5,6,6,6,7,7,8,8,8,9,9,10,10,11,11,11,12,12,
%T A172476 13,13,13,14,14,15,15,15,16,16,17,17,17,18,18,19,19,20,20,20,21,21,22,
%U A172476 22,22,23,23,24,24,24,25,25,26,26,26,27,27,28,28,28,29,29,30,30
%N A172476 a(n) = floor(n*sqrt(6)/6).
%K A172476 nonn,new
%O A172476 0,6
%A A172476 Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Feb 04 2010

%I A172475
%S A172475 0,0,1,2,3,4,5,6,6,7,8,9,10,11,12,12,13,14,15,16,17,18,19,19,20,21,22,
%T A172475 23,24,25,25,26,27,28,29,30,31,32,32,33,34,35,36,37,38,38,39,40,41,42,
%U A172475 43,44,45,45,46,47,48,49,50,51,51,52,53,54,55,56,57,58,58,59,60,61,62
%N A172475 a(n) = floor(n*sqrt(3)/2).
%F A172475 a(n) = A171970(n). [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Feb 05 2010]
%K A172475 nonn,new
%O A172475 0,4
%A A172475 Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Feb 04 2010

%I A172474
%S A172474 0,0,0,1,1,1,2,2,2,3,3,3,4,4,4,5,5,6,6,6,7,7,7,8,8,8,9,9,9,10,10,10,11,
%T A172474 11,12,12,12,13,13,13,14,14,14,15,15,15,16,16,16,17,17,18,18,18,19,19,
%U A172474 19,20,20,20,21,21,21,22,22,22,23,23,24,24,24,25,25,25,26,26
%N A172474 a(n) = floor(n*sqrt(2)/4).
%K A172474 nonn,new
%O A172474 0,7
%A A172474 Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Feb 04 2010

%I A172473
%S A172473 0,1,8,22,45,79,124,183,256,343,447,567,705,861,1037,1232,1448,1685,
%T A172473 1944,2225,2529,2858,3210,3587,3990,4419,4874,5357,5866,6404,6971,7566,
%U A172473 8192,8847,9532,10249,10996,11776,12588,13433,14310,15222,16167,17146
%N A172473 Floor(sqrt(2*n^5))
%K A172473 nonn,new
%O A172473 0,3
%A A172473 Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Feb 04 2010

%I A172472
%S A172472 0,1,4,7,11,15,20,26,32,38,44,51,58,66,74,82,90,99,108,117,126,136,145,
%T A172472 155,166,176,187,198,209,220,232,244,256,268,280,292,305,318,331,344,
%U A172472 357,371,384,398,412,426,441,455,470,485,500,515,530,545,561,576,592
%N A172472 Floor(sqrt(2*n^3))
%K A172472 nonn,new
%O A172472 0,3
%A A172472 Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Feb 04 2010

%I A172471
%S A172471 0,1,2,2,2,3,3,3,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,
%T A172471 8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,11,11,
%U A172471 11,11,11,11,11,11,11,11,11,12,12,12
%N A172471 Floor(sqrt(2*n))
%K A172471 nonn,new
%O A172471 0,3
%A A172471 Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Feb 04 2010

%I A172470
%S A172470 9,52,9,61,61,9,52,9,61,61,61,9,52,9,61,61,9,52,9,61,61,61,9,52,9,61,61,
%T A172470 9,52,9,52,9,61,61,9,52,9,61,61,61,9,52,9,61,61,9,52,9,61,61,61,9,52,9,
%U A172470 61,61,9,52,9,61,61,61,9,52,9,61,61,9,52,9,52,9,61,61,9,52,9,61,61,61,9
%N A172470 First differences of A172468
%C A172470 The numbers appear to be restricted to 9,52, or 61. Note that 61-52=9, (52+2)/(61-52)=(52+2)/9=6, (61+2)/(61-52)=(61+2)/9=7 and we have lcm(9,52,61)=28548=13^4-13 and (52+2)/9+(61+2)/9=6+7=13
%H A172470 Stephen Crowley, <a href="http://crowlogic.net/TheRiemannZetaHypothesisAndTheMellinTransformOfTheHarmonicSawtooth.pdf">The Harmonic Saw, Integral Transforms, Fractal Strings, and the Riemann Zeta Function</a>
%F A172470 A172468(n+1)-A172468(n)
%Y A172470 A172468, A172467, A167389
%K A172470 nonn,new
%O A172470 1,1
%A A172470 Stephen Crowley (crow(AT)crowlogic.net), Feb 04 2010

%I A172388
%S A172388 1,0,2,0,34,0,2942,0,1144834,0,1906714622,0,13264071114754,0,
%T A172388 380188784001777662,0,44530311225683389448194,0,
%U A172388 21199108233888497863938801662,0,40869840581497696551494454452682754
%V A172388 1,0,-2,0,34,0,-2942,0,1144834,0,-1906714622,0,13264071114754,0,
%W A172388 -380188784001777662,0,44530311225683389448194,0,
%X A172388 -21199108233888497863938801662,0,40869840581497696551494454452682754
%N A172388 a(n) = Sum_{k=0..n} (-1)^k*C(n,k)*2^(k*(n-k)).
%F A172388 O.g.f.: A(x) = Sum_{n>=0} x^n/(1+2^n*x)^(n+1).
%F A172388 E.g.f.: E(x) = Sum_{n>=0} exp(-2^n*x)*x^n/n!.
%e A172388 O.g.f.: A(x) = 1 - 2*x^2 + 34*x^4 - 2942*x^6 + 1144834*x^8 +...
%e A172388 A(x) = 1/(1+x) + x/(1+2*x)^2 + x^2/(1+2^2*x)^3 + x^3/(1+2^3*x)^4 +...+ x^n/(1+2^n*x)^(n+1) +...
%e A172388 E.g.f.: E(x) = 1 - 2*x^2/2! + 34*x^4/4! - 2942*x^6/6! + 1144834*x^8/8! +...
%e A172388 E(x) = exp(-x) + exp(-2*x)*x + exp(-2^2*x)*x^2/2! + exp(-2^3*x)*x^3/3! +...+ exp(-2^n*x)*x^n/n! +...
%o A172388 (PARI) {a(n)=sum(k=0,n,(-1)^k*binomial(n,k)*2^(k*(n-k)))}
%o A172388 (PARI) {a(n)=polcoeff(sum(k=0, n, x^k/(1+2^k*x +x*O(x^n))^(k+1)), n)}
%o A172388 (PARI) {a(n)=n!*polcoeff(sum(k=0, n, exp(-2^k*x +x*O(x^n))*x^k/k!), n)}
%Y A172388 Cf. variants: A172389, A047863.
%K A172388 sign,new
%O A172388 0,3
%A A172388 Paul D. Hanna (pauldhanna(AT)juno.com), Feb 03 2010

%I A172469
%S A172469 7,43,101,107,149,151,157,193,199,251,257,293,307,349,401,443,449,457,
%T A172469 499,557,593,599,601,607,643,701,743,751,757,857,907,1049,1051,1093,
%U A172469 1151,1193,1201,1249,1301,1307,1399,1451,1493,1499,1543,1549,1601,1607
%N A172469 Primes congruent to +/-1 or +/-7 modulo 25.
%C A172469 Equivalently, primes p such that the smallest extension of F_p containing the 5th roots of unity also contains the 25th roots of unity.
%C A172469 In this respect, the sequence is the n=5 instance of a family of sequences. For n=3, see A129805, and for n=2, see A002144.
%C A172469 Equivalently, the primes p for which, if p^t = 1 mod 5, then p^t = 1 mod 25.
%F A172469 A141927 U A141932 U A141946 U A141941. [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Feb 05 2010]
%K A172469 easy,nonn,new
%O A172469 1,1
%A A172469 Katherine Stange (stange(AT)pims.math.ca), Feb 03 2010
%E A172469 More terms from R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Feb 05 2010

%I A172389
%S A172389 1,1,2,7,44,481,9272,310087,18164624,1843946881,326808099872,
%T A172389 100310221406407,53656068398769344,49686835289802328801,
%U A172389 80090696216400251499392,223445962168511596412895367
%N A172389 a(n) = Sum_{k=0..n} C(n,k)*3^(k*(n-k))/2^n.
%F A172389 O.g.f.: A(x) = Sum_{n>=0} 2*x^n/(2 - 3^n*x)^(n+1).
%F A172389 E.g.f.: E(x) = Sum_{n>=0} exp(3^n*x/2)*(x/2)^n/n!.
%F A172389 a(n) = A135079(n)/2^n.
%e A172389 O.g.f.: A(x) = 1 + x + 2*x^2 + 7*x^3 + 44*x^4 + 481*x^5 + 9272*x^6 +...
%e A172389 A(x) = 2/(2-x) + 2*x/(2-3*x)^2 + 2*x^2/(2-3^2*x)^3 + 2*x^3/(2-3^3*x)^4 +...+ 2*x^n/(2-3^n*x)^(n+1) +...
%e A172389 E.g.f.: E(x) = 1 + x + 2*x^2/2! + 7*x^3/3! + 44*x^4/4! + 481*x^5/5! +...
%e A172389 E(x) = exp(x/2) + exp(3*x/2)*x/2 + exp(3^2*x/2)*(x/2)^2/2! + exp(3^3*x/2)*(x/2)^3/3! +...+ exp(3^n*x/2)*(x/2)^n/n! +...
%o A172389 (PARI) {a(n)=sum(k=0,n,binomial(n,k)*3^(k*(n-k)))/2^n}
%o A172389 (PARI) {a(n)=n!*polcoeff(sum(k=0, n, exp(3^k*x/2 +x*O(x^n))*(x/2)^k/k!), n)}
%o A172389 (PARI) {a(n)=polcoeff(sum(k=0, n, (x/2)^k/(1-3^k*x/2 +x*O(x^n))^(k+1)), n)}
%Y A172389 Cf. variants: A135079, A047863.
%K A172389 nonn,new
%O A172389 0,3
%A A172389 Paul D. Hanna (pauldhanna(AT)juno.com), Feb 03 2010

%I A172387
%S A172387 1,1,2,7,33,187,1208,8626,66782,553355,4862938,45025668,437029462,
%T A172387 4429568600,46738108168,512097036882,5814415004953,68293044651990,
%U A172387 828547526906649,10369747261970151,133732024450930096
%N A172387 G.f. satisfies: A(x) = x + x*Sum_{n>=1} G_{n}(x)^n, where G_{n}(x) is the n-th iteration of A(x) defined by G{n}(x) = G_{n-1}(A(x)) with G_0(x)=x.
%e A172387 G.f.: A(x) = x + x^2 + 2*x^3 + 7*x^4 + 33*x^5 + 187*x^6 +...
%e A172387 Related expansions:
%e A172387 A(A(x)) = x + 2*x^2 + 6*x^3 + 25*x^4 + 130*x^5 + 789*x^6 +...
%e A172387 A(A(x))^2 = x^2 + 4*x^3 + 16*x^4 + 74*x^5 + 396*x^6 +...
%e A172387 A(A(A(x))) = x + 3*x^2 + 12*x^3 + 60*x^4 + 353*x^5 + 2348*x^6 +...
%e A172387 A(A(A(x)))^3 = x^3 + 9*x^4 + 63*x^5 + 423*x^6 + 2895*x^7 +...
%e A172387 Coefficients in the n-th iteration of the g.f. begin:
%e A172387 n=1: [1, 1, 2, 7, 33, 187, 1208, 8626, 66782, 553355, ...];
%e A172387 n=2: [1, 2, 6, 25, 130, 789, 5376, 40140, 323198, 2775204, ...];
%e A172387 n=3: [1, 3, 12, 60, 353, 2348, 17208, 136574, 1159754, ...];
%e A172387 n=4: [1, 4, 20, 118, 788, 5800, 46236, 394382, 3568108, ...];
%e A172387 n=5: [1, 5, 30, 205, 1545, 12595, 109664, 1010914, 9803334, ...];
%e A172387 n=6: [1, 6, 42, 327, 2758, 24817, 235932, 2354038, 24532158, ...];
%e A172387 n=7: [1, 7, 56, 490, 4585, 45304, 469000, 5059950, 56677550, ...];
%e A172387 n=8: [1, 8, 72, 700, 7208, 77768, 873352, 10164212, 122210376, ...];
%e A172387 n=9: [1, 9, 90, 963, 10833, 126915, 1539720, 19271058, 248179134, ...];
%e A172387 n=10:[1, 10, 110, 1285, 15690, 198565, 2591528, 34766008, 478309118, ...]; ...
%e A172387 Coefficients in the n-th power of the n-th iteration of the g.f. begin:
%e A172387 n=1: [1, 1, 2, 7, 33, 187, 1208, 8626, 66782, 553355, ...];
%e A172387 n=2: [0, 1, 4, 16, 74, 396, 2398, 16093, 117752, 927818, ...];
%e A172387 n=3: [0, 0, 1, 9, 63, 423, 2895, 20634, 154323, 1213566, ...];
%e A172387 n=4: [0, 0, 0, 1, 16, 176, 1688, 15312, 136320, 1214472, ...];
%e A172387 n=5: [0, 0, 0, 0, 1, 25, 400, 5275, 62850, 707350, 7710070, ...];
%e A172387 n=6: [0, 0, 0, 0, 0, 1, 36, 792, 13842, 212028, 2989698, ...];
%e A172387 n=7: [0, 0, 0, 0, 0, 0, 1, 49, 1421, 31899, 614166, 10685675, ...];
%e A172387 n=8: [0, 0, 0, 0, 0, 0, 0, 1, 64, 2368, 66528, 1577280, ...];
%e A172387 n=9: [0, 0, 0, 0, 0, 0, 0, 0, 1, 81, 3726, 128223, 3676887, ...];
%e A172387 n=10:[0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 100, 5600, 231850, 7917900, ...]; ...
%e A172387 Column sums in the above table form this sequence shift left 1 place.
%o A172387 (PARI) {a(n)=local(a_n=0,G=x,F=x+sum(k=2,n-1,a(k)*x^k)); if(n<1,0,if(n==1,1, for(k=1,n-1,G=x; for(i=1,k,G=subst(F,x,G+x*O(x^n)));a_n=a_n+polcoeff(G^k,n-1));a_n))}
%Y A172387 Cf. A171780 (variant).
%K A172387 nonn,new
%O A172387 0,3
%A A172387 Paul D. Hanna (pauldhanna(AT)juno.com), Feb 03 2010

%I A172468
%S A172468 50,59,111,120,181,242,251,303,312,373,434,495,504,556,565,626,687,696,
%T A172468 748,757,818,879,940,949,1001,1010,1071,1132,1141,1193,1202,1254,1263,
%U A172468 1324,1385,1394,1446,1455,1516,1577,1638,1647,1699,1708,1769,1830,1839
%N A172468 numbers such that A172467(n)=1
%C A172468 It appears that the successive differences are limited to three numbers: 9, 52, and 61 where it is noted that 61-52=9, (52+2)/9=6, (61+2)/9=7 and we have lcm(9,52,61)=28548=13^4-13
%H A172468 Stephen Crowley, <a href="http://crowlogic.net/TheRiemannZetaHypothesisAndTheMellinTransformOfTheHarmonicSawtooth.pdf">The Harmonic Saw, Integral Transforms, and the Riemann Zeta Function</a>
%p A172468 [ListTools[SearchAll](1, [seq(round(evalf(floor((n+2)/ln(2))-2-(argument(exp(-(ln(2)+LambertW(n, -(1/2)*ln(2)))/ln(2)))*ln(2)+Im(LambertW(n, -(1/2)*ln(2))))/(2*Pi*ln(2)))), n = 1 .. 10000)])]
%Y A172468 A172467, A166986, A167389
%K A172468 hard,nonn,new
%O A172468 1,1
%A A172468 Stephen Crowley (crow(AT)crowlogic.net), Feb 03 2010

%I A172467
%S A172467 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
%T A172467 0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,
%U A172467 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0
%N A172467 A166986(n)/2-A167389(n)
%H A172467 Stephen Crowley, <a href="http://crowlogic.net/TheRiemannZetaHypothesisAndTheMellinTransformOfTheHarmonicSawtooth.pdf">The Harmonic Saw, Integral Transforms, and the Riemann Zeta Function</a>
%F A172467 (floor((n+2)/ln(2))-2)-(argument(exp(-(ln(2)+W(n,-(1/2)*ln(2)))/ln(2)))*ln(2)+Im(W(n,-(1/2)*ln(2))))/(2*Pi*ln(2))
%p A172467 [seq(round(evalf(floor((n+2)/ln(2))-2-(argument(exp(-(ln(2)+W(n, -(1/2)*ln(2)))/ln(2)))*ln(2)+Im(W(n, -(1/2)*ln(2))))/(2*Pi*ln(2)))), n = 1 .. 200)] the round() is for cosmetic purposes as this is an integer valued function
%K A172467 hard,nonn,new
%O A172467 1,1
%A A172467 Stephen Crowley (crow(AT)crowlogic.net), Feb 03 2010

%I A172466
%S A172466 1,45,65,87,117,362,1053,1257,1282,1539,1798,2962,2966,3478,5002,5242,
%T A172466 5932,9272,9374,9477,10550,10732,12975,13526,14427,20025,21782,21982,
%U A172466 21986,22436,23386,23728,25978,25994,27764,32146,35306,35414,36412
%N A172466 Numbers n such that sigma(sigma(phi(n))) = sigma(sigma(n))
%D A172466 M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 840.
%D A172466 T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 38
%H A172466 M. Abramowitz and I. A. Stegun, eds., <a href="http://www.nrbook.com/abramowitz_and_stegun/">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
%H A172466 C. K. Caldwell, The Prime Glossary, <a href="http://primes.utm.edu/glossary/page.php?sort=SigmaFunction">sigma function</a>
%H A172466 K. Matthews, <a href="http://www.numbertheory.org/php/factor.html">Factorizing n and calculating phi(n),omega(n),d(n),sigma(n) and mu(n)</a>
%e A172466 phi(45)= 24 sigma(phi(45))= sigma(24)= 60 sigma(sigma(phi(45)))= sigma(60)=168 sigma(45)= 78 sigma(sigma(45))= sigma(78)= 168
%p A172466 with(numtheory): for n from 1 to 1000000 do; if sigma(sigma(phi(n)))= sigma(sigma(n)) then print(n);fi ; od;
%Y A172466 Cf. A001157, A001158, A001160, A001065
%K A172466 nonn,new
%O A172466 1,2
%A A172466 Michel Lagneau (mn.lagneau2(AT)orange.fr), Feb 03 2010

%I A175097
%S A175097 3,5,31,61,163,193,227,383,401,521,631,653,883,1019,1151,1229,1433,1601,
%T A175097 1669,1801,1873,2437,2729,3191,3671,4013,4049,4127,4447,5507,5651,5701,
%U A175097 5813,6079,6199,6353,6569,6823,6857,7507,7529,7873,7907,8291,8419,8461
%N A175097 Primes in A139013.
%Y A175097 Cf. A062703, A074924, A139013.
%K A175097 nonn,new
%O A175097 1,1
%A A175097 Zak Seidov (zakseidov(AT)yahoo.com), Feb 03 2010

%I A172465
%S A172465 42,101,6720,9212,226570,276404,288086,299668,339098,392328,412276,
%T A172465 423395,530917,535759,559427,564209,666181
%N A172465 Numbers n such that phi(phi(n)) + sigma(sigma(n)) = p^8, where p is an integer.
%D A172465 W. L. Glaisher, Number-Divisor Tables. British Assoc. Math. Tables, Vol. 8, Camb. Univ. Press, 1940, p. 64.
%D A172465 S. W. Golomb, Equality among number-theoretic functions, Abstract 882-11-16, Abstracts Amer. Math. Soc., 14 (1993), 415-416.
%D A172465 R. K. Guy, Unsolved Problems in Number Theory, B42.
%H A172465 M. Abramowitz and I. A. Stegun, eds., <a href="http://www.nrbook.com/abramowitz_and_stegun/">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
%H A172465 K. Ford, <a href="http://www.ams.org/era/1998-04-05/S1079-6762-98-00043-2/home.html">The distribution of totients</a>, Electron. Res. Announc. Amer. Math. Soc. 4 (1998), 27-34.
%H A172465 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CarmichaelsTotientFunctionConjecture.html">Carmichael's Totient Function conjecture</a>
%e A172465 phi(phi(9)) + sigma(sigma(9))= 1 phi(phi(42)) + sigma(sigma(42))= 2^8 = 256 phi(phi(101)) + sigma(sigma(101))= 2^8 = 256 phi(phi(6720)) + sigma(sigma(6720))= 4^8 = 65536
%p A172465 with(numtheory):for n from 1 to 2000000 do;if floor(( phi(phi(n)) + sigma(sigma(n)))^.125) = (phi(phi(n)) + sigma(sigma(n)))^.125 then print (n);fi ; od;
%Y A172465 Cf. "A000010", "A002180", "A032446", "A058277".
%K A172465 nonn,new
%O A172465 1,1
%A A172465 Michel Lagneau (mn.lagneau2(AT)orange.fr), Feb 03 2010

%I A172464
%S A172464 9,42,101,339,407,420,471,915,1409,2572,2847,3706,4069,6631,6720,7229,
%T A172464 9212,14051,16641,31453,33067,33146,35701,37425,37675,37911,48016,48272,
%U A172464 53101,55956,56906,68895,73474,75023,83525,84676,86928,94525,101428
%N A172464 Numbers n such that phi(phi(n)) + sigma(sigma(n)) = p^4, where p is an integer.
%D A172464 W. L. Glaisher, Number-Divisor Tables. British Assoc. Math. Tables, Vol. 8, Camb. Univ. Press, 1940, p. 64.
%D A172464 S. W. Golomb, Equality among number-theoretic functions, Abstract 882-11-16, Abstracts Amer. Math. Soc., 14 (1993), 415-416.
%D A172464 R. K. Guy, Unsolved Problems in Number Theory, B42.
%H A172464 K. Ford, <a href="http://www.ams.org/era/1998-04-05/S1079-6762-98-00043-2/home.html">The distribution of totients</a>, Electron. Res. Announc. Amer. Math. Soc. 4 (1998), 27-34.
%H A172464 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/TotientValenceFunction.html">Totient Valence Function</a>
%H A172464 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CarmichaelsTotientFunctionConjecture.html">Carmichael's Totient Function conjecture</a>
%e A172464 phi(phi(9)) + sigma(sigma(9))= 1 phi(phi(42)) + sigma(sigma(42))= 4^4 = 256 phi(phi(101)) + sigma(sigma(101))= 4^4 = 256 phi(phi(339)) + sigma(sigma(339))= 6^4 = 1296
%p A172464 with(numtheory): for n from 1 to 2000000 do;if floor(( phi(phi(n)) + sigma(sigma(n)))^.25) =( phi(phi(n)) + sigma(sigma(n)))^.25 then print (n);fi ; od;
%Y A172464 "A000010" Cf. "A000010", "A002180", "A032446", "A058277".
%K A172464 nonn,new
%O A172464 1,1
%A A172464 Michel Lagneau (mn.lagneau2(AT)orange.fr), Feb 03 2010

%I A172463
%S A172463 13,30,61,98,169,242,321,418,525,638,787,944,1111,1290,1489,1800,2137,
%T A172463 2484,2843,3232,3232,3933,4642,5375,6114,6857,7608,8369,9138,10045,
%U A172463 10982,11923,12876,13843,14814,15797,16788,17797,18818,19849,20882
%N A172463 Partial sums of emirps, primes whose reversal is a different prime (A006567).
%C A172463 The subsequence of prime partial sums of emirps begins: 13, xxx, 787, 1489, 2137, 2843, 3232, 6857, 8369, 11923, 15797, 21943, 24103. The subsubsequence of emirp partial sums of emirps begins: 13, 32911 = emirp(736). Note that 787 is a prime when reversed, but not a different prime. The first square in the sequence is 169, and R(169) = 961 is also a square.
%F A172463 a(n) = SUM[i=1..n] {p such that p is prime and R(p) is prime} = SUM[i=1..n] {p such that p is in A000040 and A004086(p) is prime}.
%e A172463 a(x) = 13 + 17 + 31 + 37 + 71 + 73 + 79 + 97 + 107 + 113 + 149 + 157 + 167 + 179 + 199 + 311 + 337 + 347 + 359 + 389 + 701 + 709 + 733 + 739 + 743 + 751 + 761 + 769 + 907 + 937 + 941 = 11923, which is prime, and note that R(11923) = 32911 is also prime.
%Y A172463 Cf. A000040, A003684, A007628, A046732, A048051, A048052, A048053, A048054, A048895.
%K A172463 base,easy,nonn,new
%O A172463 1,1
%A A172463 Jonathan Vos Post (jvospost3(AT)gmail.com), Feb 03 2010

%I A172462
%S A172462 59,60,61,72,93,102,103,108,109,123,144,149,150,151,161,162,163,171,207,
%T A172462 213,236,257,258,264,265,267,268,271,291,312,313,318,333,334
%N A172462 Numbers n such that 2*n-+{1,3} are not primes
%e A172462 a(1)=59 because 2*59-1=117=nonprime, 2*59+1=119=nonprime, 2*59-3=115=nonprime and 2*59+3=121=nonprime.
%Y A172462 Cf. A000027, A104278.
%K A172462 nonn,new
%O A172462 1,1
%A A172462 Juri-Stepan Gerasimov (2stepan(AT)rambler.ru), Feb 03 2010

%I A172461
%S A172461 2,3,7,19,31,607,1279,2281
%N A172461 Primes p such that 2p-1 and 2^p-1 are both primes.
%e A172461 a(1)=2=prime because 2^2-1=3=prime and 2*2-1=3=prime; a(2)=3=prime because 2^3-1=7=prime and 2*3-1=5=prime.
%Y A172461 Cf. A000043, A000668, A005382
%K A172461 nonn,new
%O A172461 1,1
%A A172461 Juri-Stepan Gerasimov (2stepan(AT)rambler.ru), Feb 03 2010, Feb 06 2010

%I A172459
%S A172459 5,13,17,37,61,79,89,97,107,127,139,157,199,211,229,307,331,337,367,379,439
%N A172459 Primes p such that either 2p-1 or 2^p-1 is prime.
%e A172459 a(1)=5=prime because 2^5-1=31=prime and 2*5-1=9=nonprime; a(2)=13=prime because 2^13-1=8191=prime and 2*13-1=25=nonprime.
%Y A172459 Cf. A000043, A000668, A005382.
%K A172459 nonn,new
%O A172459 1,1
%A A172459 Juri-Stepan Gerasimov (2stepan(AT)rambler.ru), Feb 03 2010, Feb 06 2010
%E A172459 Corrected by Juri-Stepan Gerasimov (2stepan(AT)rambler.ru), Feb 06 2010

%I A172460
%S A172460 125,253,496,752,1095,1607,2232,2961,3985,5014,6229,7479,8759,10090,
%T A172460 11459,12917,14453,16134,17835,19550,21342,23191,25066,27114,29301,
%U A172460 31498,33707,36108,38668,41477,44602,48083,51667,55312,59033,63129
%N A172460 Partial sums of economical numbers A046759.
%C A172460 The subsequence of prime partial sum of economical numbers begins: 1607, 6229, 12917, 76367. The subsequence of economical partial sum of economical numbers begins: 125; what is the first nontrivial economical partial sum of economical numbers?
%F A172460 a(n) = SUM[i=1..n] {n written as a product of primes raised to powers, where D(n) = number of digits in product, l(n) = number of digits in n; sequence gives n such that D(n)<l(n)}.
%e A172460 a(41) = 125 + 128 + 243 + 256 + 343 + 512 + 625 + 729 + 1024 + 1029 + 1215 + 1250 + 1280 + 1331 + 1369 + 1458 + 1536 + 1681 + 1701 + 1715 + 1792 + 1849 + 1875 + 2048 + 2187 + 2197 + 2209 + 2401 + 2560 + 2809 + 3125 + 3481 + 3584 + 3645 + 3721 + 4096 + 4374 + 4375 + 4489 + 4802 + 4913.
%Y A172460 Cf. A000961, A046758, A046759, A046760.
%K A172460 base,easy,nonn,new
%O A172460 1,1
%A A172460 Jonathan Vos Post (jvospost3(AT)gmail.com), Feb 03 2010

%I A172458
%S A172458 1,4,8,10,12,14,16,18,20,22,24,26,27,33,34,35,39,40,42,44,45,48,49,50,
%T A172458 52,55,56,57,63,64,65,66,68,70,74,76,78,81,82,84,86,87,91,95,98,100,105,
%U A172458 106,111,112,115,116,117,119,121,125,126,128,129,132,134,136,138,140
%N A172458 Nonprimes n such that either 2*n-+1 is prime.
%e A172458 a(1)=1=nonprime because 2*1-1=1=nonprime and 2*1+1=3=prime; a(2)=4=nonprime because 2*4-1=7=prime and 2*4+1=9=nonprime.
%Y A172458 Cf. A000040, A141468.
%K A172458 nonn,new
%O A172458 1,2
%A A172458 Juri-Stepan Gerasimov (2stepan(AT)rambler.ru), Feb 03 2010

%I A172457
%S A172457 5,5,1,1,1,1,1,1,1,25,25
%V A172457 -5,5,-1,1,-1,1,-1,1,-1,25,-25
%N A172457 Numerators of differences of (A172298/A172282)=1,-1/4,1/36,0,1/900,0,1/1764,.
%C A172457 Differences:-5/4,5/18,-1/36,1/900,-1/900,1/1764,-1/1764,1/900,-1/900,25/4356,-25/4356,. From Bernoulli A027641*A164555. Pure twin numbers from fourth fraction.
%K A172457 nonn,uned,new
%O A172457 0,1
%A A172457 Paul Curtz (bpcrtz(AT)free.fr), Feb 03 2010

%I A172456
%S A172456 17,1277,1607,3527,4637,71327,97367,113147,191447,290657,312197,416387,
%T A172456 418337,421697,450797,566537,795647,886967,922067,1090877,1179317,
%U A172456 1300127,1464257,1632467,1749257,1866857,1901357,2073347,2322107
%N A172456 Initial members of primes sextuple (p, p+2, p+6, p +12, p+14, p+20)
%C A172456 The last digit of this primes numbers is 7
%D A172456 G. E. Andrews, MacMahon's prime numbers of measurement, Amer. Math. Monthly, 82 (1975), 922-923. R. K. Guy, Unsolved Problems in Number Theory, E30.
%D A172456 P. A. MacMahon, The prime numbers of measurement on a scale, Proc. Camb. Phil. Soc. 21 (1923), 651-654; reprinted in Coll. Papers I, pp. 797-800.
%D A172456 Problem E1910, Amer. Math. Monthly, 75 (1968), 80-81.
%H A172456 T. Forbes, <a href="http://mathworld.wolfram.com/PrimeTriplet.html">Prime k-tuplets</a>
%H A172456 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PrimeTriplet.html">Link to a section of The World of Mathematics</a>.
%e A172456 The two first sequences are : (17,19,23,29,31,37), (1277,1279,1283,1289,1291,1297)
%p A172456 for n from 1 by 2 to 400000 do; if isprime(n) and isprime(n+2) and isprime(n+6) and isprime(n+12) and isprime(n + 14) and isprime(n+20) then print(n) else fi;od;
%Y A172456 Initial members of prime quadruples (p, p+2, p+6, p +12) : A172454 Initial members of prime quintuple (p, p+2, p+6, p +12, p+14) : A078946. Cf. A073648, A098412.
%K A172456 nonn,new
%O A172456 1,1
%A A172456 Michel Lagneau (mn.lagneau2(AT)orange.fr), Feb 03 2010

%I A172454
%S A172454 5,11,17,41,101,227,347,641,1091,1277,1427,1481,1487,1607,2687,3527,
%T A172454 3917,4001,4127,4637,4787,4931,8231,9461,10331,11777,12107,13901,14627,
%U A172454 16061,19421,20747,21011,21557,22271,23741,25577,26681,26711,27737
%N A172454 Initial members of primes quadruple (p, p+2, p+6, p +12)
%D A172454 G. E. Andrews, MacMahon's prime numbers of measurement, Amer. Math. Monthly, 82 (1975), 922-923.
%D A172454 R. K. Guy, Unsolved Problems in Number Theory, E30.
%D A172454 P. A. MacMahon, The prime numbers of measurement on a scale, Proc. Camb. Phil. Soc. 21 (1923), 651-654; reprinted in Coll. Papers I, pp. 797-800. Problem E1910, Amer. Math. Monthly, 75 (1968), 80-81.
%H A172454 Eric Weisstein's World of Mathematics,<a href="http://mathworld.wolfram.com/PrimeTriplet.html"> Link to a section of The World of Mathematics</a>.
%e A172454 The two first sequences are : (5,7,11,17), (11,13,17,23)
%p A172454 for n from 1 by 2 to 110000 do; if isprime(n) and isprime(n+2) and isprime(n+6) and isprime(n+12) then print(n) else fi;od;
%Y A172454 Cf. A073648, A098412.
%K A172454 nonn,new
%O A172454 1,1
%A A172454 Michel Lagneau (mn.lagneau2(AT)orange.fr), Feb 03 2010

%I A172453
%S A172453 1,1,1,1,1,1,1,2,2,1,1,2,4,2,1,1,3,6,6,3,1,1,4,12,12,12,4,1,1,4,16,24,
%T A172453 24,16,4,1,1,4,16,32,48,32,16,4,1,1,5,20,40,80,80,40,20,5,1,1,6,30,60,
%U A172453 120,160,120,60,30,6,1
%N A172453 A combinatorial triangle based on A004001 product sequences:c(n)=If[n == 1, 1, Product[A004001[m], {m, 1, n}]];t(n,m)=c(n)/(c(m)*c(n-m))
%C A172453 Row sums are:
%C A172453 {1, 2, 3, 6, 10, 20, 46, 90, 154, 292, 594,...}.
%C A172453 The Modulo two pattern appears to be symmetrically chaotic:
%C A172453 ListDensityPlot[Table[Table[If[n >= m, Mod[t[ n, m], 2], 0], {m, 0, 32}], {n, 0, 32}], Mesh -> False, Axes -> False]
%F A172453 c(n)=If[n == 1, 1, Product[A004001[m], {m, 1, n}]]
%F A172453 t(n,m)=c(n)/(c(m)*c(n-m))
%e A172453 {1},
%e A172453 {1, 1},
%e A172453 {1, 1, 1},
%e A172453 {1, 2, 2, 1},
%e A172453 {1, 2, 4, 2, 1},
%e A172453 {1, 3, 6, 6, 3, 1},
%e A172453 {1, 4, 12, 12, 12, 4, 1},
%e A172453 {1, 4, 16, 24, 24, 16, 4, 1},
%e A172453 {1, 4, 16, 32, 48, 32, 16, 4, 1},
%e A172453 {1, 5, 20, 40, 80, 80, 40, 20, 5, 1},
%e A172453 {1, 6, 30, 60, 120, 160, 120, 60, 30, 6, 1}}
%t A172453 (*A004001*)
%t A172453 f[0] = 0; f[1] = 1; f[2] = 1;
%t A172453 f[n_] := f[n] = f[f[n - 1]] + f[n - f[n - 1]];
%t A172453 c[n_] := If[n == 1, 1, Product[f[m], {m, 1, n}]];
%t A172453 t[n_, m_] := c[n]/(c[m]*c[n - m]);
%t A172453 Table[Table[t[n, m], {m, 0, n}], {n, 0, 10}];
%t A172453 Flatten[%]
%Y A172453 A004001
%K A172453 nonn,tabl,uned,new
%O A172453 0,8
%A A172453 Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Feb 03 2010

%I A172452
%S A172452 1,1,1,2,4,12,48,192,768,3840,23040,161280,1128960,9031680,72253440,
%T A172452 578027520,4624220160,41617981440,416179814400,4577977958400,
%U A172452 54935735500800
%N A172452 A product sequence based on A004001:a(n)=If[n == 1, 1, Product[A004001[m], {m, 1, n}]]
%t A172452 (*A004001*)
%t A172452 f[0] = 0; f[1] = 1; f[2] = 1;
%t A172452 f[n_] := f[n] = f[f[n - 1]] + f[n - f[n - 1]];
%t A172452 c[n_] := If[n == 1, 1, Product[f[m], {m, 1, n}]];
%t A172452 Table[c[n], {n, 0, 20}]
%K A172452 nonn,uned,new
%O A172452 0,4
%A A172452 Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Feb 03 2010

%I A172451
%S A172451 1,2,4,6,22,333,355,103993,104348,1042060,1146408,4272943,541935
%N A172451 a(1) = 1, and for each k > =2, a(k) is the smallest number n such that 1/sin n > 1/sin a(k), so that 1/sin a(1)> 1/sin a(2)> ... > 1/sin a(k) > ...
%D A172451 J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 83, p. 29, Ellipses, Paris 2008. Also Entry 137, p. 47.
%H A172451 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Pi.html">Link to a section of The World of Mathematics</a>.
%e A172451 1/sin 1 = 1.1883951 1/sin 2 = 1.0997501 1/sin 4 = - 1.3213487
%p A172451 a:= evalf(1/ sin(1)); for n from 2 to 10000000 do; if a > evalf(1/sin(n)) then a:= evalf(1/sin(n)); print(n); else fi ; od;
%Y A172451 extension of "A172445"" Cf. "A172446", "A046959", "A046965". Adjacent sequences: "A046964"
%K A172451 nonn,new
%O A172451 1,2
%A A172451 Michel Lagneau (mn.lagneau2(AT)orange.fr), Feb 03 2010

%I A172449
%S A172449 0,0,0,0,0,0,0,92,1066,7828,44148,195270,707698,2211868,6120136,
%T A172449 15324708,35312064,75937606,153942964,296590536,546621416,968910732,
%U A172449 1659114170,2754780934,4449361442,7009572728,10796663102,16292133888
%N A172449 Number of ways to place 8 nonattacking queens on an 8 X n board
%H A172449 V. Kotesovec, <a href="http://web.telecom.cz/vaclav.kotesovec/math.htm">Number of ways of placing non-attacking queens and kings on boards of various sizes</a>
%F A172449 Explicit formula (Vaclav Kotesovec, 3.2.2010): a(n) = n^8 - 84n^7 + 3378n^6 - 85078n^5 + 1467563n^4 - 17723656n^3 + 145910074n^2 - 745654756n + 1802501048, n>=31
%Y A172449 A061989, A061990, A061991, A061992, A061993
%K A172449 nonn,new
%O A172449 1,8
%A A172449 Vaclav Kotesovec (kotesovec(AT)chello.cz), Feb 03 2010

%I A172448
%S A172448 1,2,8,33,344,1054,1764,2474,3184,3894,4604,5314,6024,6734,7444,8154,
%T A172448 8864,9574,10284,10994,11704,12414,13124,13834,14544,15254,15964,16674,
%U A172448 17384,18094,18804,19514,20224,20934,21644,22354,23064,23774,24484
%N A172448 a(1) = 1, and for each k > =2, a(k) is the smallest number n such that 1/cos n > 1/cos a(k), so that 1 /cos a(1) > 1/cos a(2) > ... > 1 / cos a(k) > ...
%D A172448 J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 83, p. 29, Ellipses, Paris 2008. Also Entry 137, p. 47.
%H A172448 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Pi.html">Link to a section of The World of Mathematics</a>.
%e A172448 1/cos 1 = 1.8508157 1/cos 2 = -2.402997962 1/cos 8 = -6.8728506
%p A172448 a:= evalf(1/ cos(1)); for n from 2 to 10000000 do; if a > evalf(1/cos(n)) then a:= evalf(1/cos(n)); print(n); else fi ; od;
%Y A172448 extension of "A172446" Cf. "A172445" , "A046959", "A046965". Adjacent sequences: "A046964"
%K A172448 nonn,new
%O A172448 1,2
%A A172448 Michel Lagneau (mn.lagneau2(AT)orange.fr), Feb 03 2010

%I A172447
%S A172447 1,4,17,70,283,1136,4549,18202,72815,291268
%N A172447 a(n+1)-4a(n)=0,1,2,3,4,5,=A001477.
%C A172447 Second bisection of submitted A172416=1,1,3,4,10,17,. Companion to A164044. a(n) mod 10=period 10:repeat 1,4,7,0,3,6,9,2,5,8 the ten digits;note 1+8=4+5=7+2=0+9=3+6=9. See A131579(n-3) and reversal A144468(n+3).
%F A172447 a(n)=6a(n-1)-9a(n-2)+4a(n-3) or 6*a(n-1)-9*a(n-2)+4*a(n-3).
%K A172447 nonn,uned,new
%O A172447 0,2
%A A172447 Paul Curtz (bpcrtz(AT)free.fr), Feb 03 2010

%I A172446
%S A172446 1,2,4,8,17,27,33,77,121,165,209,212,256,300,344,366,1054,1764,2474,
%T A172446 3184,3894,4604,5314,6024,6734,7444,8154,8864,9574,10284,10994,11704,
%U A172446 12414,13124,13384,14544,15254,15964,16674,17384,18094,18804,19514
%N A172446 a(1) = 1, and for each k > =2, a(k) is the smallest number n such that n /cos n > a(k) / cos a(k), so that a(1) /cos a(1) > a(2) /cos a(2) > ... > a(k) / cos a(k) > ...
%D A172446 J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 83, p. 29, Ellipses, Paris 2008. Also Entry 137, p. 47.
%H A172446 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Pi.html">Link to a section of The World of Mathematics</a>.
%e A172446 1/cos1 = 1.850815 2/cos2 = - 4.805995 4/cos 4 = - 6.119542
%p A172446 a:= evalf(1/cos(1)); for n from 2 to 10000000 do; if a > evalf(n/cos(n)) then a:= evalf(n/cos(n)); print(n); else fi ; od;
%Y A172446 See "A7172445" Cf. "A046959", "A046965". Adjacent sequences: "A046964"
%K A172446 nonn,new
%O A172446 1,2
%A A172446 Michel Lagneau (mn.lagneau2(AT)orange.fr), Feb 03 2010

%I A172445
%S A172445 1,4,6,12,16,22,66,110,154,198,201,245,289,333,355,1065,1775,2485,3195,
%T A172445 3905,4615,5325,6035,6745,7455,8165,8875,9585,10295,11005,11715,12425,
%U A172445 13135,13845,14555,15265,15975,16685,17395,18105,18815,19525,20235
%N A172445 a(1) = 1, and for each k > =2, a(k) is the smallest number n such that n /sin n > a(k) / sin a(k), so that a(1) / sin a(1) > a(2) / sin a(2) > ... > a(k) / sin a(k) > ...
%D A172445 J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 83, p. 29, Ellipses, Paris 2008. Also Entry 137, p. 47.
%H A172445 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Pi.html">Link to a section of The World of Mathematics</a>.
%e A172445 1/sin 1 = 1,1883951 4/sin 4 = - 5,285394 6/sin 6 = - 21,473397
%p A172445 a:= evalf(1/sin(1)); for n from 2 to 10000000 do; if a > evalf(n/sin(n)) then a:= evalf(n/sin(n)); print(n); else fi ; od;
%Y A172445 Cf. "A046959", "A046965". Adjacent sequences: "A046964"
%K A172445 nonn,new
%O A172445 1,2
%A A172445 Michel Lagneau (mn.lagneau2(AT)orange.fr), Feb 03 2010

%I A172440
%S A172440 1,1,3,11,49,134,1915,30437,1176925,47572678,2240962254,
%T A172440 119077789557,7053073003902,460586576005843,32870527083358387,
%U A172440 2544978866143616029,212452025172991768237,19021387591827001945347
%V A172440 1,1,-3,11,-49,134,-1915,-30437,-1176925,-47572678,-2240962254,
%W A172440 -119077789557,-7053073003902,-460586576005843,-32870527083358387,
%X A172440 -2544978866143616029,-212452025172991768237,-19021387591827001945347
%N A172440 G.f. satisfies: [x^n] A(x)^((n+1)^2) = (n+1)^2 for n>=1; that is, the coefficient of x^n in the (n+1)^2 power of g.f. A(x) equals (n+1)^2.
%e A172440 G.f.: A(x) = 1 + x - 3*x^2 + 11*x^3 - 49*x^4 + 134*x^5 - 1915*x^6 +...
%e A172440 Coefficients in the squared powers of A(x) begin:
%e A172440 A(x)^1: [(1), 1, -3, 11, -49, 134, -1915, -30437, ...];
%e A172440 A(x)^4: [1, (4), -6, 12, -45, -220, -4952, -148944, ...];
%e A172440 A(x)^9: [1, 9, (9), -33, 45, -1044, -13353, -387675, ...];
%e A172440 A(x)^16: [1, 16, 72, (16), -284, -1408, -36152, -857136, ...];
%e A172440 A(x)^25: [1, 25, 225, 775, (25), -6520, -78725, -1861575, ...];
%e A172440 A(x)^36: [1, 36, 522, 3756, 12411, (36), -229128, -4096368, ...];
%e A172440 A(x)^49: [1, 49, 1029, 11907, 80115, 283514, (49), -10015593, ...];
%e A172440 A(x)^64: [1, 64, 1824, 30272, 319760, 2177792, 8628896, (64), ...]; ...
%e A172440 where the coefficients [x^n] A(x)^((n+1)^2) form the squares.
%o A172440 (PARI) {a(n)=local(A=[1,1]);for(m=3,n+1,A=concat(A,0);A[ #A]=(m^2-Vec(Ser(A)^(m^2))[m])/m^2);A[n+1]}
%K A172440 sign,new
%O A172440 0,3
%A A172440 Paul D. Hanna (pauldhanna(AT)juno.com), Feb 02 2010

%I A172439
%S A172439 6,2,2,5,5,5,7,25,52,54,55,76,244,555,533,626,673,2563,5574,4272,6365,
%T A172439 26646,23322,57653,46567,35655,252565,266427,523722,524556,755646,
%U A172439 2546566,5237566,5554537,5365773,6553465,24656555,54253723,56677266
%N A172439 Fibonacci sequence rewritten using A006942.
%C A172439 For Fibonacci numbers containing two or more digits just concatenate the digits.
%C A172439 Digit 0 ==> 6;1 ==> 2;2 ==> 5;3 ==> 5;4 ==> 4;5 ==> 5;6 ==> 6;7 ==> 3;8 ==> 7;9 ==> 6
%e A172439 4181 is a Fibonacci number and using A006942 this is 4272.
%p A172439 A172439 := proc(n) if n = 0 then 6; else F := convert(combinat[fibonacci](n),base,10) ; dgs := [] ; for i from 1 to nops(F) do dgs := [op(dgs),op(1+op(i,F),[ 6, 2, 5, 5, 4, 5, 6, 3, 7, 6])] ; end do ; add( op(i,dgs)*10^(i-1),i=1..nops(dgs)) ; end if; end proc: seq(A172439(n),n=0..40) ; [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Feb 08 2010]
%Y A172439 Cf. A000045
%K A172439 nonn,new
%O A172439 0,1
%A A172439 Parthasarathy Nambi (PachaNambi(AT)yahoo.com), Feb 02 2010
%E A172439 More terms from R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Feb 08 2010

%I A172438
%S A172438 1,3,5,11,19,27,29,59,61,71,79,101,125,131,139,181,199,242,243,271,333,
%T A172438 349,379,387,409,423,449,461,477,521,569,571,603,631,641,661,739,747,
%U A172438 751,772,788,821,881,929,991,1017,1031,1039,1051,1058,1069,1075,1083
%N A172438 Numbers n such that tau(n^2+1) - tau(n^2) = 1 where the function tau(n) is the number of positive divisors of n.
%D A172438 M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 840.
%D A172438 G. E. Andrews, Some debts I owe, Seminaire Lotharingien Combinatoire, Paper B42a, Issue 42, 2000; see (7.1). T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 38.
%D A172438 P. A. MacMahon, Divisors of numbers and their continuations in the theory of partitions, Proc. London Math. Soc., 19 (1919), 75-113. D. S. Mitrinovic et al., Handbook of Number Theory, Kluwer, Chap. II. (For inequalities, etc.)
%H A172438 M. Abramowitz and I. A. Stegun, eds., <a href="http://www.nrbook.com/abramowitz_and_stegun/">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
%H A172438 S. Ramanujan, <a href="http://www.imsc.res.in/~rao/ramanujan/CamUnivCpapers/Cpaper8/page1.htm">On The Number Of Divisors Of A Number</a>
%e A172438 n=1, tau(2) - tau(1) = 2 - 1 = 1 n=3, tau(10) - tau(9) = 4 - 3 = 1 n = 5, tau(26)- tau(25) = 4 - 3 = 1 n = 387, tau(149770)- tau(149769) = 15 - 15 = 1
%p A172438 with(numtheory): for n from 1 to 100000 do; if tau(n^2+1)-tau(n^2)= 1 then print(n); else fi ; od;
%Y A172438 "A055927", "A000005" See "A002183", "A002182" for records. See "A000203" for the sum-of-divisors function sigma(n). Cf. "A001227", "A005237", "A005238", "A006601", "A006558", "A019273", "A039665", "A049051". Cf. "A001826", "A001842", "A051731", "A066446", "A129510", "A115361", "A129372", "A115361", "A127093", "A143319".
%K A172438 nonn,new
%O A172438 1,2
%A A172438 Michel Lagneau (mn.lagneau2(AT)orange.fr), Feb 02 2010

%I A172436
%S A172436 15,55,159,411,411,411,3647,15243,15243,15243,113343,1133759,1133759,
%T A172436 1133759,29149139
%N A172436 Smallest n such that the Moebius function take successively, from n, the k values 1,0,1,0,...,1 or 0
%C A172436 It's easy to prove that a(k) for k > =17 don't exist, because in all suite of 17 consecutive numbers that the first is free of squares, there are necessarily among this numbers two numbers r , s where 9 divide r and s, and then Mobius(r) = mobius(s) = 0 with r - s odd.
%D A172436 M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, p. 826.
%D A172436 T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 24. L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 161, #16. Deleglise, Marc and Rivat, Joel, Computing the summation of the Mobius function. Experiment. Math. 5 (1996), no. 4, 291-295.
%D A172436 G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, 1979, th. 262 and 287.
%H A172436 Ed Pegg Jr., <a href="http://www.maa.org/editorial/mathgames/mathgames_11_03_03.html">The Mobius function (and squarefree numbers)</a>
%H A172436 G. Villemin's Almanac of Numbers, <a href="http://villemin.gerard.free.fr/TABLES/aaaFArit/MobiusMe.htm">Nombres de Moebius et de Mertens</a>
%H A172436 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/MoebiusFunction.html">Link to a section of The World of Mathematics</a>.
%e A172436 a(2) = 15, a(3) = 55, a(4) = 159, a(5) = a(6) = a(7) = 411, a(8) = 3647,a(9) = a(10) = a(11) = 15243, a(12) = 113343, a(3) = a(14) = a(15) = 1133759 and a(16) = 29149139.
%p A172436 step 1: for n from 1 to 15000 do; if mobius(n)= 1 and mobius(n+1) = 0 then print(n); else fi ; od; step 2 : for n from 1 to 15000 do; if mobius(n)= 1 and mobius(n+1) = 0 and mobius(n+2) = 1 then print(n); else fi ; od; step 3 : for n from 1 to 15000 do; if mobius(n)= 1 and mobius(n+1) = 0 and mobius(n+2) = 1 and mobius(n+3) = 0 then print(n); else fi ; od; etc.
%Y A172436 Moebius (or Mobius) function mu(n): "A008683", "A007423" , "A002321" , "A002996"
%K A172436 nonn,new
%O A172436 1,1
%A A172436 Michel Lagneau (mn.lagneau2(AT)orange.fr), Feb 02 2010

%I A172435
%S A172435 2,5,10,17,28,41,58,95,174,287,484,683,1020,2213,5992,17931,37868,
%T A172435 231807,431740,1111111111111542851,11112222222222222653962
%N A172435 Partial sums of circular primes A016114.
%C A172435 Circular primes are a generalization of palindromatic primes (A002385): numbers which remain prime under cyclic shifts of digits. 484 is the first square partial sum of circular primes. The subsequence of prime partial sums of circular primes begins: 2, 5, 17, 41, 683, 2213. The subsubsequence of circular prime partial sums of circular primes begins 2, 5, 17, and what is the next? What are the analogues in other bases?
%e A172435 a(21) = 2 + 3 + 5 + 7 + 11 + 13 + 17 + 37 + 79 + 113 + 197 + 199 + 337 + 1193 + 3779 + 11939 + 19937 + 193939 + 199933 + 1111111111111111111 + 11111111111111111111111.
%Y A172435 Cf. A000040, A002385, A004023, A003459, A016114, A045978, A068652.
%K A172435 base,easy,more,nonn,new
%O A172435 1,1
%A A172435 Jonathan Vos Post (jvospost3(AT)gmail.com), Feb 02 2010

%I A172434
%S A172434 0,1,2,3,5,13
%N A172434 Nonegative numbers n such that neither 2^n-3^trivial prime is prime or nonprime.
%e A172434 a(1)=0 because 2^0-3^2=-8=nonprime and 2^0-3^3=-26=nonprime; a(2)=1 because 2^1-3^2=-7=nonprime and 2^1-3^3=-25=nonprime; a(3)=2 because 2^2-3^2=-5=nonprime and 2^2-3^3=-23=nonprime; a(4)=3 because 2^3-3^2=-1=nonprime and 2^3-3^3=-19=nonprime; a(5)=5 because 2^5-3^2=23=prime and 2^5-3^3=5=prime; a(6)=13 because 2^13-3^2=8187=nonprime and 2^13-3^3=8165=nonprime.
%Y A172434 Cf. A001477, A000040.
%K A172434 nonn,new
%O A172434 1,3
%A A172434 Juri-Stepan Gerasimov (2stepan(AT)rambler.ru), Feb 02 2010

%I A172433
%S A172433 2,6,9,16,17,27,26,36,38,48,43,67,59,67,72,88,75,102,86,111,115,123,99,
%T A172433 150,137,142,139,169,141,192
%N A172433 Let u(n) = Sum [n/[k]] and v(n) = Sum [n/k] where the summation index k ranges from 1 to infinity, although both sums are actually finites. Here [a] denotes the integer part of a. Now our sequence is u(n) - v(n).
%C A172433 One can pick out the values of the sequence at primes, obtaining the new sequence 6,9,17,26,43,59,75,86,99,141 which seems to be monotone, as opposed to the original sequence.
%K A172433 nonn,new
%O A172433 1,1
%A A172433 Ali A. Tanara (aatanara(AT)gmail.com), Feb 02 2010
%E A172433 The original submission had several non-ascii characters, and I had to guess at the definition. - N. J. A. Sloane, Feb 04 2010

%I A172432
%S A172432 0,1,2,3,4,5,6,7,8,9,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,
%T A172432 44,46,48,50,52,54,56,58,60,62,64,66,68,70,72,74,76,78,80,82,84,86,88,
%U A172432 90,92,94,96,98,102,105,108,111,114,117,120,123,126,129,132,135,138,141
%N A172432 A(n) = n devided by number of digits without reminder
%C A172432 56 is in sequence because 56 mod 2 = 0. 57 is not in sequence because 57 mod 2 = 0,5. 2 is number of digits.
%F A172432 {0} U A098952. [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Feb 05 2010]
%o A172432 (Other) (Liberty Basic)DO:x$ = STR$(a):l = LEN(x$):m = a MOD l:IF m = 0 THEN PRINT a:a = a + 1:LOOP UNTIL Inkey$ <> "":END
%K A172432 base,nonn,new
%O A172432 0,3
%A A172432 Boris Hostnik (megpplus(AT)siol.net), Feb 02 2010

%I A172431
%S A172431 1,1,2,1,4,3,1,6,10,4,1,8,21,20,5,1,10,36,56,35,6,1,12,55,120,126,56,7,
%T A172431 1,14,78,220,330,252,84,8,1,16,105,364,715,792,462,120,9,1,18,136,560,
%U A172431 1365,2002,1716,792,165,10
%N A172431 Even row Pascal-square read by anti-diagonals.
%C A172431 Apart from signs identical to A053123.Mirror of A078812.
%F A172431 As a decimal sequence: a(n)= 12*a(n-1)- a(n-2) with a(1)=1.
%e A172431 Array begins:
%e A172431 1,2,3,4,5,...
%e A172431 1,4,10,20,...
%e A172431 1,6,21,56,...
%e A172431 Example:
%e A172431 Starting with 1, every entry is twice the one to the left minus the second one to the left, plus the one above.
%e A172431 For n = 9 the a(9)= 10 solution is 2*4 - 1 + 3
%Y A172431 A078812, A053123, A007318, A001906
%K A172431 nonn,tabl,new
%O A172431 1,3
%A A172431 M. Dols (markdols99(AT)yahoo.com), Feb 02 2010

%I A172429
%S A172429 1,1,1,1,15,1,1,6,6,1,1,255,102,255,1,1,120,2040,2040,120,1,1,4095,
%T A172429 32760,1392300,32760,4095,1,1,5040,1375920,27518400,27518400,1375920,
%U A172429 5040,1,1,65535,22019760,15028486200,7072228800,15028486200,22019760
%V A172429 1,1,1,1,-15,1,1,6,6,1,1,-255,102,-255,1,1,120,2040,2040,120,1,1,-4095,
%W A172429 32760,-1392300,32760,-4095,1,1,5040,1375920,27518400,27518400,1375920,
%X A172429 5040,1,1,-65535,22019760,-15028486200,7072228800,-15028486200,22019760
%N A172429 Alternating q-form-Pascal triangle sequence:f(n,q)=If[Mod[1 - q^n, 2] == 0, , If[Mod[n, 2] == 0, (1 - q^n), n! ], If[Mod[n, 2] == 1, n!, (1 - q^n)]];q=4;c(n,q)=Product[f(i, q), {i, 1, n}];t(n,m,q)=c(n, q)/(c(m, q)*c(n - m, q))
%C A172429 Row sums are:
%C A172429 {1, 2, -13, 14, -406, 4322, -1334968, 57798722, -22940835148, 45439521523202,
%C A172429 -176478793961380348,...}
%F A172429 f(n,q)=If[Mod[1 - q^n, 2] == 0, If[Mod[n, 2] == 0, (1 - q^n), n! ], If[Mod[n, 2] == 1, n!, (1 - q^n)]];
%F A172429 q=4
%F A172429 ;c(n,q)=Product[f(i, q), {i, 1, n}];
%F A172429 t(n,m,q)=c(n, q)/(c(m, q)*c(n - m, q))
%e A172429 {1},
%e A172429 {1, 1},
%e A172429 {1, -15, 1},
%e A172429 {1, 6, 6, 1},
%e A172429 {1, -255, 102, -255, 1},
%e A172429 {1, 120, 2040, 2040, 120, 1},
%e A172429 {1, -4095, 32760, -1392300, 32760, -4095, 1},
%e A172429 {1, 5040, 1375920, 27518400, 27518400, 1375920, 5040, 1},
%e A172429 {1, -65535, 22019760, -15028486200, 7072228800, -15028486200, 22019760, -65535, 1},
%e A172429 {1, 362880, 1585422720, 1331755084800, 21386419891200, 21386419891200, 1331755084800, 1585422720, 362880, 1},
%e A172429 {1, -1048575, 25367126400, -277072438104000, 5476255247232000, -186877210311792000, 5476255247232000, -277072438104000, 25367126400, -1048575, 1}
%t A172429 Clear[t, n, m, c, q];
%t A172429 f[n_, q_] = If[Mod[1 - q^n, 2] == 0, If[Mod[n, 2] == 0, (1 - q^n), n! ], If[Mod[n, 2] == 1, n!, (1 - q^n)]];
%t A172429 c[n_, q_] = Product[f[i, q], {i, 1, n}];
%t A172429 t[n_, m_, q_] = c[n, q]/(c[m, q]*c[n - m, q]);
%t A172429 Table[Table[Table[t[n, m, q], {m, 0, n}], {n, 0, 10}], {q, 2, 12}];
%t A172429 Table[Flatten[Table[Table[t[n, m, q], {m, 0, n}], {n, 0, 10}]], {q, 2, 12}]
%K A172429 sign,tabl,uned,new
%O A172429 0,5
%A A172429 Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Feb 02 2010

%I A172430
%S A172430 2,1,6,7,0,3,4,9,8,5,2,1,6,7,0,3,4,9,8,5,2,1,6,7,0,3,4,9,8,5,2,1,6,7,0,
%T A172430 3,4,9,8,5
%N A172430 a(n)=A172285(n+1)=2,1,6,7,20,33,74, mod 10. Period 10:repeat 2,1,6,7,0,3,4,9,8,5.
%C A172430 A172285 is linked to Jacobsthal numbers A001045 via A053088. a(3n)=2,7,4,5,6,3,8,1,0,9 (period 10) = submitted A172423(n+2). See A135033.
%F A172430 a(n)=(1/5)*{2*(n mod 10)+2*[(n+1) mod 10]+[(n+2) mod 10]-2*[(n+3) mod 10]-[(n+5) mod 10]+4*[(n+6) mod 10]-2*[(n+8) mod 10]+[(n+9) mod 10]}, with n>=0 [From Paolo P. Lava (ppl(AT)spl.at), Feb 05 2010]
%K A172430 nonn,uned,new
%O A172430 0,1
%A A172430 Paul Curtz (bpcrtz(AT)free.fr), Feb 02 2010

%I A172428
%S A172428 1,1,1,1,8,1,1,6,6,1,1,80,60,80,1,1,120,1200,1200,120,1,1,728,10920,
%T A172428 145600,10920,728,1,1,5040,458640,9172800,9172800,458640,5040,1,1,
%U A172428 6560,4132800,501446400,752169600,501446400,4132800,6560,1,1,362880
%V A172428 1,1,1,1,-8,1,1,6,6,1,1,-80,60,-80,1,1,120,1200,1200,120,1,1,-728,10920,
%W A172428 -145600,10920,-728,1,1,5040,458640,9172800,9172800,458640,5040,1,1,
%X A172428 -6560,4132800,-501446400,752169600,-501446400,4132800,-6560,1,1,362880
%N A172428 Alternating q-form-Pascal triangle sequence:f(n,q)=If[Mod[1 - q^n, 2] == 0, , If[Mod[n, 2] == 0, (1 - q^n), n! ], If[Mod[n, 2] == 1, n!, (1 - q^n)]];q=3;c(n,q)=Product[f(i, q), {i, 1, n}];t(n,m,q)=c(n, q)/(c(m, q)*c(n - m, q))
%C A172428 Row sums are:
%C A172428 {1, 2, -6, 14, -98, 2642, -125214, 19272962, -242470718, 5049621077762,
%C A172428 -756108270204494,...}
%F A172428 f(n,q)=If[Mod[1 - q^n, 2] == 0, If[Mod[n, 2] == 0, (1 - q^n), n! ], If[Mod[n, 2] == 1, n!, (1 - q^n)]];
%F A172428 q=3
%F A172428 ;c(n,q)=Product[f(i, q), {i, 1, n}];
%F A172428 t(n,m,q)=c(n, q)/(c(m, q)*c(n - m, q))
%e A172428 {1},
%e A172428 {1, 1},
%e A172428 {1, -8, 1},
%e A172428 {1, 6, 6, 1},
%e A172428 {1, -80, 60, -80, 1},
%e A172428 {1, 120, 1200, 1200, 120, 1},
%e A172428 {1, -728, 10920, -145600, 10920, -728, 1},
%e A172428 {1, 5040, 458640, 9172800, 9172800, 458640, 5040, 1},
%e A172428 {1, -6560, 4132800, -501446400, 752169600, -501446400, 4132800, -6560, 1},
%e A172428 {1, 362880, 297561600, 249951744000, 2274560870400, 2274560870400, 249951744000, 297561600, 362880, 1},
%e A172428 {1, -59048, 2678417280, -2928402892800, 184489382246400, -1119235585628160, 184489382246400, -2928402892800, 2678417280, -59048, 1}
%t A172428 Clear[t, n, m, c, q];
%t A172428 f[n_, q_] = If[Mod[1 - q^n, 2] == 0, If[Mod[n, 2] == 0, (1 - q^n), n! ], If[Mod[n, 2] == 1, n!, (1 - q^n)]];
%t A172428 c[n_, q_] = Product[f[i, q], {i, 1, n}];
%t A172428 t[n_, m_, q_] = c[n, q]/(c[m, q]*c[n - m, q]);
%t A172428 Table[Table[Table[t[n, m, q], {m, 0, n}], {n, 0, 10}], {q, 2, 12}];
%t A172428 Table[Flatten[Table[Table[t[n, m, q], {m, 0, n}], {n, 0, 10}]], {q, 2, 12}]
%K A172428 sign,tabl,uned,new
%O A172428 0,5
%A A172428 Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Feb 02 2010

%I A172427
%S A172427 1,1,1,1,3,1,1,6,6,1,1,15,30,15,1,1,120,600,600,120,1,1,63,2520,
%T A172427 6300,2520,63,1,1,5040,105840,2116800,2116800,105840,5040,1,1,255,
%U A172427 428400,4498200,35985600,4498200,428400,255,1,1,362880,30844800
%V A172427 1,1,1,1,-3,1,1,6,6,1,1,-15,30,-15,1,1,120,600,600,120,1,1,-63,2520,
%W A172427 -6300,2520,-63,1,1,5040,105840,2116800,2116800,105840,5040,1,1,-255,
%X A172427 428400,-4498200,35985600,-4498200,428400,-255,1,1,362880,30844800
%N A172427 Alternating q-form-Pascal triangle sequence:f(n,q)=If[Mod[1 - q^n, 2] == 0, , If[Mod[n, 2] == 0, (1 - q^n), n! ], If[Mod[n, 2] == 1, n!, (1 - q^n)]];q=2;c(n,q)=Product[f(i, q), {i, 1, n}];t(n,m,q)=c(n, q)/(c(m, q)*c(n - m, q))
%C A172427 Row sums are:
%C A172427 {1, 2, -1, 14, 2, 1442, -1384, 4455362, 27845492, 269522588162, 2596108836356,...}
%F A172427 f(n,q)=If[Mod[1 - q^n, 2] == 0, If[Mod[n, 2] == 0, (1 - q^n), n! ], If[Mod[n, 2] == 1, n!, (1 - q^n)]];
%F A172427 q=2
%F A172427 ;c(n,q)=Product[f(i, q), {i, 1, n}];
%F A172427 t(n,m,q)=c(n, q)/(c(m, q)*c(n - m, q))
%e A172427 {1},
%e A172427 {1, 1},
%e A172427 {1, -3, 1},
%e A172427 {1, 6, 6, 1},
%e A172427 {1, -15, 30, -15, 1},
%e A172427 {1, 120, 600, 600, 120, 1},
%e A172427 {1, -63, 2520, -6300, 2520, -63, 1},
%e A172427 {1, 5040, 105840, 2116800, 2116800, 105840, 5040, 1},
%e A172427 {1, -255, 428400, -4498200, 35985600, -4498200, 428400, -255, 1},
%e A172427 {1, 362880, 30844800, 25909632000, 108820454400, 108820454400, 25909632000, 30844800, 362880, 1},
%e A172427 {1, -1023, 123742080, -5259038400, 1767036902400, -927694373760, 1767036902400, -5259038400, 123742080, -1023, 1}
%t A172427 Clear[t, n, m, c, q];
%t A172427 f[n_, q_] = If[Mod[1 - q^n, 2] == 0, If[Mod[n, 2] == 0, (1 - q^n), n! ], If[Mod[n, 2] == 1, n!, (1 - q^n)]];
%t A172427 c[n_, q_] = Product[f[i, q], {i, 1, n}];
%t A172427 t[n_, m_, q_] = c[n, q]/(c[m, q]*c[n - m, q]);
%t A172427 Table[Table[Table[t[n, m, q], {m, 0, n}], {n, 0, 10}], {q, 2, 12}];
%t A172427 Table[Flatten[Table[Table[t[n, m, q], {m, 0, n}], {n, 0, 10}]], {q, 2, 12}]
%K A172427 sign,tabl,uned,new
%O A172427 0,5
%A A172427 Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Feb 02 2010

%I A172426
%S A172426 5,12,27,75,363,1587,2523,5043,8427,20667,23763,38307,51483,89787,96123,
%T A172426 109443,162867,171363,189003,236883,257547
%N A172426 Number of non-trivial solutions (x,y,z) for each prime number p of the Fermat equation x^p + y^p + z^p = 0 mod (n) where n is prime of the form n = 2p + 1, and x, y, z are integers such that x < = y.
%C A172426 Solution to a Diophantine equation in finite fields Z/n. Historical reminder : Sophie Germain's work led to Fermat's Last Theorem being broken into two cases : x^p + y^p= z^p has no integer solutions for which x,y, and z are relatively prime to p, i.e. in which none of x,y, and z are divisible by p, and then x^p + y^p = z^p has no integer solutions for which one of the three numbers is divisible by p.
%C A172426 This result was presented by Legendre in an 1823 paper to the French Academy of Sciences and included in a supplement to his second edition of Theorie des Nombres, with a footnote crediting the result to Sophie Germain. The Sophie Germain's Theorem introduce an auxiliary prime n satisfying the two conditions : x^p + y^p + z^p = 0 mod (n) implies that x = 0 mod n, or y = 0 mod n, or z = 0 mod n, and x^p = p mod n is impossible for any value of x. Then Case I of Fermat's Last Theorem is true for p. This sequence give solutions for each prime number p, and n = 2p + 1
%D A172426 Del Centina, Andrea. "Unpublished manuscripts of Sophie Germain and a revaluation of her work on Fermat's Last Theorem," Arch. Hist. Exact Sci., Vol 62 (2008), 349-392.
%D A172426 Legendre, A. M., "Recherches sur quelques objets d'analyse indeterminee et particulierement sur le theoreme de Fermat," Mem. Acad. Sci. Inst. France 6 (1823), 1-60.
%D A172426 Sampson, J.H. "Sophie Germain and the theory of numbers," Arch. Hist. Exact Sci. 41 (1990), 157-161.
%D A172426 Schoof, "Wiles' proof of the Taniyama-Weil conjecture for semi-stable elliptic curves over Q", Chap. 14 in 'Ou En Sont Les Mathematiques ?' Soc. Math. de France (SMF), Vuibert, Paris 2002.
%H A172426 C. K. Caldwell, The Prime Glossary,<a href="http://primes.utm.edu/glossary/page.php?sort=FermatsLastTheorem"> Fermat's Last Theorem</a>
%H A172426 Del Centina, Andrea. <a href="http://web.unife.it/progetti/geometria/Germain.html">Letters of Sophie Germain preserved in Florence</a>, Historia Mathematica, Vol. 32 (2005), 60-75.
%e A172426 We consider the case p = 1, n = 3. We have 5 solutions mod 3 : (0,1,2), (0,2,1),(1,1,1),(1,2,0),(2,2,2) with p = 2, n = 5 we have 12 solutions mod 5 : (0,1,2), (0,1,3),(0,2,1),(0,2,4),(0,3,1),(0,3,4), (0,4,2),(0,4,3),(1,2,0),(1,3,0),(2,4,0),(3,4,0) With p = 3, n = 7, we have 27 solutions mod 7 : (0,1,3),(0,1,5), (0,1,6),(0,2,3),(0,2,5),(0,2,6),(0,3,1),(0,3,2),(0,3,4), (0,4,3),(0,4,5),(0,4,6),(0,5,1),(0,5,2),(0,5,4),(0,6,1),(0,6,2),(0,6,4), (1,3,0),(1,5,0),(1,6,0),(2,3,0),(2,5,0),(2,6,0),(3,4,0),(4,5,0),(4,6,0)
%Y A172426 Fermat's last theorem: "A019590"
%K A172426 nonn,new
%O A172426 1,1
%A A172426 Michel Lagneau (mn.lagneau2(AT)orange.fr), Feb 02 2010

%I A172424
%S A172424 24,36,224,432,624,735,2232,3276,4224,6624,23328,32832,33264,34272,
%T A172424 34992,42336,42624,43632,73332,82944,83232,92232,93744,229392,234432,
%U A172424 244224,248832,272832,282624,344736,442368,622272,628224,772632,843264
%N A172424 Numbers n with digits different from 0 and 1 such that the sum of digits and the product of digits divides n.
%D A172424 Charles Ashbacher, Journal of Recreational Mathematics, Vol. 33 (2005), pp. 227.
%D A172424 J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 166, p. 53, Ellipses, Paris 2008.
%D A172424 J.M. De Koning & A. Mercier, Introduction a la theorie des nombres, Modulo, 2e edition, 1997
%D A172424 J.M. De Koning & A. Mercier, 1001 problemes en theorie classique des nombres, Ellipses, Paris,2004
%H A172424 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Digit.html">Link to a section of The World of Mathematics</a>.
%e A172424 4 + 2 = 6 and 2*4 = 8 divide 24 3 + 6 = 9 and 3*6 = 18 divide 36 2+2+4 = 8 and 2*2*4 = 32 divide 224 23328, 2 +3+3+2+8 = 18 and 2*3*3*2*8 = 288 divide 23328
%Y A172424 Sequence in context: "A108854", "A106007", "A002796", "A171492", "A034838", "A071249"
%K A172424 nonn,base,new
%O A172424 1,1
%A A172424 Michel Lagneau (mn.lagneau2(AT)orange.fr), Feb 02 2010

%I A172423
%S A172423 0,9,2,7,4,5,6,3,8,1,0,9,2,7,4,5,6,3,8,1,0,9,2,7,4,5,6,3,8,1,0,9,2,7,4,
%T A172423 5,6,3,8,1,0,9,2,7,4,5,6,3,8,1
%N A172423 Period 10:repeat 0,9,2,7,4,5,6,3,8,1.
%C A172423 Linked to 0,2,1,6,7,20,=A172285.Explained soon.
%F A172423 Mix (0,2,4,6,8=A135033(n-1)) , (9,7,5,3,1=A096230).
%F A172423 a(n)=(1/5)*{(n mod 10)+4*[(n+1) mod 10]-2*[(n+2) mod 10]+2*[(n+3) mod 10]+2*[(n+6) mod 10]-2*[(n+7) mod 10]+4*[(n+8) mod 10]-4*[(n+9) mod 10]}, with n>=0 [From Paolo P. Lava (ppl(AT)spl.at), Feb 05 2010]
%K A172423 nonn,uned,new
%O A172423 0,2
%A A172423 Paul Curtz (bpcrtz(AT)free.fr), Feb 02 2010

%I A172422
%S A172422 256,512,1024,2048,4096,6561,8192,16384,19683,32768,59049,65536,131072,
%T A172422 177147,262144,294912
%N A172422 Numbers c which have measure of smoothness J bigger than 7. Where J=Log[c]/Log[N(c)], where N(c) is product of distinct prime divisors of c
%C A172422 This sequence is subset of A049094 and A172418 and A172419 and A172420 and A172421.
%t A172422 aa = {}; Do[kk = FactorInteger[c]; nn = 1; Do[nn = nn*kk[[n]][[1]], {n, 1, Length[kk]}]; If[Log[c]/Log[nn] > 7, AppendTo[aa, c]], {c, 2, 10000}]; aa (*Artur Jasinski*)
%Y A172422 A049094, A172418-A172422.
%K A172422 nonn,new
%O A172422 1,1
%A A172422 Artur Jasinski (grafix(AT)csl.pl), Feb 02 2010

%I A172421
%S A172421 128,256,512,1024,2048,2187,4096,6561,8192,16384,19683,32768,49152,
%T A172421 52488,55296,59049,62208,65536,69984,73728,78125,78732,82944,93312,
%U A172421 98304,104976,110592,118098,124416,131072,139968,147456,157464,165888
%N A172421 Numbers c which have measure of smoothness J bigger than 6. Where J=Log[c]/Log[N(c)], where N(c) is product of distinct prime divisors of c
%C A172421 This sequence is subset of A049094 and A172418 and A172419 and A172420.
%t A172421 aa = {}; Do[kk = FactorInteger[c]; nn = 1; Do[nn = nn*kk[[n]][[1]], {n, 1, Length[kk]}]; If[Log[c]/Log[nn] > 6, AppendTo[aa, c]], {c, 2, 10000}]; aa (*Artur Jasinski*)
%Y A172421 A049094, A172418-A172422.
%K A172421 nonn,new
%O A172421 1,1
%A A172421 Artur Jasinski (grafix(AT)csl.pl), Feb 02 2010

%I A172420
%S A172420 64,128,256,512,729,1024,2048,2187,4096,6561,8192,8748,9216,10368,11664,
%T A172420 12288,13122,13824,15552,15625,16384,17496,18432,19683,20736,23328,
%U A172420 24576,26244,27648,31104,32768,34992,36864,39366,41472,46656,49152
%N A172420 Numbers c which have measure of smoothness J bigger than 5. Where J=Log[c]/Log[N(c)], where N(c) is product of distinct prime divisors of c
%C A172420 This sequence is subset of A049094 and A172418 and A172419.
%t A172420 aa = {}; Do[kk = FactorInteger[c]; nn = 1; Do[nn = nn*kk[[n]][[1]], {n, 1, Length[kk]}]; If[Log[c]/Log[nn] > 5, AppendTo[aa, c]], {c, 2, 10000}]; aa (*Artur Jasinski*)
%Y A172420 A049094, A172418-A172422.
%K A172420 nonn,new
%O A172420 1,1
%A A172420 Artur Jasinski (grafix(AT)csl.pl), Feb 02 2010

%I A172419
%S A172419 32,64,128,243,256,512,729,1024,1458,1536,1728,1944,2048,2187,2304,2592,
%T A172419 2916,3072,3125,3456,3888,4096,4374,4608,5184,5832,6144,6561,6912,7776,
%U A172419 8192,8748,9216,10240,10368,11664,12288,12500,12800,13122,13824,15552
%N A172419 Numbers c which have measure of smoothness J bigger than 4. Where J=Log[c]/Log[N(c)], where N(c) is product of distinct prime divisors of c
%C A172419 This sequence is subset of A049094 and A172418.
%p A172419 aa = {}; Do[kk = FactorInteger[c]; nn = 1; Do[nn = nn*kk[[n]][[1]], {n, 1, Length[kk]}]; If[Log[c]/Log[nn] > 4, AppendTo[aa, c]], {c, 2, 10000}]; aa (*Artur Jasinski*)
%Y A172419 A049094, A172418-A172422.
%K A172419 nonn,new
%O A172419 1,1
%A A172419 Artur Jasinski (grafix(AT)csl.pl), Feb 02 2010

%I A172418
%S A172418 16,32,64,81,128,243,256,288,324,384,432,486,512,576,625,648,729,768,
%T A172418 864,972,1024,1152,1250,1280,1296,1458,1536,1600,1728,1944,2000,2048,
%U A172418 2187,2304,2401,2500,2560,2592,2916,3072,3125,3136,3200,3456,3584,3645
%N A172418 Numbers c which have measure of smoothness J bigger than 3. Where J=Log[c]/Log[N(c)], where N(c) is product of distinct prime divisors of c
%C A172418 This sequence is subset of A049094.
%p A172418 aa = {}; Do[kk = FactorInteger[c]; nn = 1; Do[nn = nn*kk[[n]][[1]], {n, 1, Length[kk]}]; If[Log[c]/Log[nn] > 3, AppendTo[aa, c]], {c, 2, 10000}]; aa (*Artur Jasinski*)
%Y A172418 A049094, A172418-A172422.
%K A172418 nonn,new
%O A172418 1,1
%A A172418 Artur Jasinski (grafix(AT)csl.pl), Feb 02 2010

%I A172417
%S A172417 1,2,2,5,5,5,14,14,14,14,42,42,42,42,42,132,132,132,132,132,132
%N A172417 n*Catalan number(n+1) triangle
%C A172417 Row sums = A001791
%e A172417 Triangle begins:
%e A172417 .....1
%e A172417 ....2,2
%e A172417 ...5,5,5
%e A172417 14,14,14,14
%Y A172417 Cf. A001791, A000108
%K A172417 nonn,tabl,new
%O A172417 1,2
%A A172417 M. Dols (markdols99(AT)yahoo.com), Feb 02 2010

%I A172416
%S A172416 1,1,3,4,10,17,37,70,144,283,571,1136,2278,4549,9105
%N A172416 a(n+1)-2a(n)=-1,1,-2,2,-3,3,=A008619 signed.
%C A172416 Differences=0,2,1,6,7,20,=A172285;then Jacobsthal. Main diagonal:1,2,5,12,28,=A045623. First bisection:1,3,10,37,=A164044 ;second is its companion 1,4,17,70.
%F A172416 a(n)=a(n-1)+3a(n-2)-a(n-3)-2a(n-4). See A133993.
%K A172416 nonn,uned,new
%O A172416 0,3
%A A172416 Paul Curtz (bpcrtz(AT)free.fr), Feb 02 2010

%I A172415
%S A172415 12,24,35,56,4752,7744,16500,91728,917280
%N A172415 56 is in the sequence because 56 = (5 + 2) * (6 + 2) (5 and 6 are digits in 56 and 2 is number of digits in 56)
%o A172415 (Other) DO: rezultat = 1: stevilostr$ = STR$(stevilo) :stevilostevk = LEN(stevilostr$) :FOR k = 1 TO stevilostevk: stevka$ = MID$(stevilostr$, k, 1): stevka = VAL(stevka$): rezultat = rezultat * (stevka + stevilostevk): NEXT k: IF stevilo = rezultat THEN PRINT stevilo, rezultat: stevilo = stevilo +1: LOOP UNTIL inkey$ <> ""
%K A172415 fini,full,nonn,new
%O A172415 0,1
%A A172415 Boris Hostnik (megpplus(AT)siol.net), Feb 02 2010

%I A172414
%S A172414 1,1,1,1,2,2,2,2,2,5,5,5,5,5,5,14,14,14,14,14,14,14,14,14,42,42,42,42,
%T A172414 42,42,42,42,42,42,42
%N A172414 (2n+1)*Catalan triangle
%C A172414 Row sums = A001700
%e A172414 Triangle begins:
%e A172414 ......1
%e A172414 ....1,1,1
%e A172414 ..2,2,2,2,2
%e A172414 5,5,5,5,5,5,5,
%Y A172414 Cf. A001700, A000108
%K A172414 nonn,tabf,new
%O A172414 1,5
%A A172414 M. Dols (markdols99(AT)yahoo.com), Feb 02 2010

%I A172413
%S A172413 0,1,2,3,4,5,6,7,8,9,18
%N A172413 18 is in the sequence because 18 = 1 * 2 + 8 * 2 (1 and 8 are digits in 18 and 2 is number of digits in 18)
%o A172413 (Other) (Liberty Basic) DO stevilostr$ = STR$(stevilo) stevilostevk = LEN(stevilostr$) FOR k = 1 TO stevilostevk stevka$ = MID$(stevilostr$, k, 1) stevka = VAL(stevka$) rezultat = rezultat + (stevka * (stevilostevk)) NEXT k IF stevilo = rezultat THEN PRINT stevilo, rezultat rezultat = 0 stevilo = stevilo +1 LOOP UNTIL inkey$ <> ""
%K A172413 fini,full,nonn,base,new
%O A172413 0,3
%A A172413 Boris Hostnik (megpplus(AT)siol.net), Feb 02 2010

%I A172411
%S A172411 1,2,5,10
%N A172411 Numbers n such that 2^n+3^trivial prime are both primes.
%C A172411 Or numbers n such that 2^n+3^{2,3} are both primes.
%C A172411 No further terms between 10 and 5000. [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Feb 07 2010]
%F A172411 A057196 INTERSECT A157007 [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Feb 07 2010]
%e A172411 a(1)=1 because 2^1+3^2=11=prime and 2^1+3^3=29=prime; a(2)=2 because 2^2+3^2=13=prime and 2^2+3^3=31=prime: a(3)=5 because 2^5+3^2=41=prime and 2^5+3^3=59=prime; a(4)=10 because 2^10+3^2=1033 and 2^10+3^3=1051=prime.
%Y A172411 Cf. A000027, A000040.
%K A172411 nonn,more,new
%O A172411 1,2
%A A172411 Juri-Stepan Gerasimov (2stepan(AT)rambler.ru), Feb 02 2010

%I A172410
%S A172410 1,2,5,7,10,16,17,22,26,31,35,37,40,50,52,61,65,77,82,92,96,107,112,115,
%T A172410 127
%N A172410 Numbers n such that 2*n+3^trivial prime are both primes.
%C A172410 Or numbers n such that 2*n+3^{2,3} are both primes.
%e A172410 a(1)=1 because 2*1+3^2=11=prime and 2*1+3^3=29=prime; a(2)=2 because 2*2+3^2=13=prime and 2*2+3^3=31=prime.
%Y A172410 Cf. A000027, A000040.
%K A172410 nonn,new
%O A172410 1,2
%A A172410 Juri-Stepan Gerasimov (2stepan(AT)rambler.ru), Feb 02 2010

%I A172408
%S A172408 127,1852,2818,3146,3615,3764,4419,5889,7994,8058,8330,10171,10561
%N A172408 Arises in a refined modular approach to the Diophantine equation x^2+y^(2n)=z^3.
%C A172408 Dahmen, plus or minus alpha, Column 1 of Table 1, p. 10, elements of S'_k,p with corresponding values of a_p(E[nonascii character here]_alpha)^2 (mod L).
%H A172408 Sander R. Dahmen, <a href="http://arxiv.org/abs/1002.0020">A refined modular approach to the Diophantine equation x^2+y^(2n)=z^3</a>, Jan 29, 2010.
%K A172408 nonn,new
%O A172408 1,1
%A A172408 Jonathan Vos Post (jvospost3(AT)gmail.com), Feb 02 2010

%I A172403
%S A172403 1,1,5,51,1059,44620,3795202,649054326,222639357434,152968659433948,
%T A172403 210361428050679489,578800452225641673965,3185715127946958245708501,
%U A172403 35071788327149162320178667272,772254422082165524711277630023576
%N A172403 G.f.: 1/(1-x) = Sum_{n>=0} a(n)*x^n/(1+x)^(2^(n+2)-4).
%F A172403 Column 2 of triangle A172400.
%e A172403 1/(1-x) = 1 + x/(1+x)^4 + 5*x^2/(1+x)^12 + 51*x^3/(1+x)^28 + 1059*x^4/(1+x)^60 +...
%o A172403 (PARI) {a(n)=if(n==0,1,polcoeff(-(1-x)*sum(m=0,n-1,a(m)*x^m/(1+x +x*O(x^n))^(2^(m+2)-4)),n))}
%Y A172403 Cf. A172400, A172401, A172402.
%K A172403 nonn,new
%O A172403 0,3
%A A172403 Paul D. Hanna (pauldhanna(AT)juno.com), Feb 01 2010

%I A172402
%S A172402 1,1,3,16,166,3464,146167,12437880,2127406440,729774938584,
%T A172402 501412103054958,689540993399039000,1897244652767973627366,
%U A172402 10442429807446111573892528,114961543001288361817819197405
%N A172402 G.f.: 1/(1-x) = Sum_{n>=0} a(n)*x^n/(1+x)^(2^(n+1)-2).
%F A172402 Column 1 of triangle A172400.
%e A172402 1/(1-x) = 1 + x/(1+x)^2 + 3*x^2/(1+x)^6 + 16*x^3/(1+x)^14 + 166*x^4/(1+x)^30 +...
%o A172402 (PARI) {a(n)=if(n==0,1,polcoeff(-(1-x)*sum(m=0,n-1,a(m)*x^m/(1+x +x*O(x^n))^(2^(m+1)-2)),n))}
%Y A172402 Cf. A172400, A172401, A172403.
%K A172402 nonn,new
%O A172402 0,3
%A A172402 Paul D. Hanna (pauldhanna(AT)juno.com), Feb 01 2010

%I A172401
%S A172401 1,1,2,6,32,332,6928,292334,24875760,4254812880,1459549877168,
%T A172401 1002824206109916,1379081986798078000,3794489305535947254732,
%U A172401 20884859614892223147785056,229923086002576723635638394810
%N A172401 G.f.: 1/(1-x) = Sum_{n>=0} a(n)*x^n/(1+x)^(2^n-1).
%F A172401 Column 0 of triangle A172400.
%e A172401 1/(1-x) = 1 + x/(1+x) + 2*x^2/(1+x)^3 + 6*x^3/(1+x)^7 + 32*x^4/(1+x)^15 +...
%o A172401 (PARI) {a(n)=if(n==0,1,polcoeff(-(1-x)*sum(m=0,n-1,a(m)*x^m/(1+x +x*O(x^n))^(2^m-1)),n))}
%Y A172401 Cf. A172400, A172402, A172403.
%K A172401 nonn,new
%O A172401 0,3
%A A172401 Paul D. Hanna (pauldhanna(AT)juno.com), Feb 01 2010

%I A172407
%S A172407 1,3,7,9,13,19,21,27,31,33,37,43,49,51,57,61,63,69,73,79,87,91,93,97,99,
%T A172407 103,117,121,127,129,139,141,147,153,157,163,169,171,181,183,187,189,
%U A172407 201,213,217,219,223,229,231,241,247,253,259,261,267,271,273,283,297
%N A172407 Positive numbers n such that n+10 is a prime.
%e A172407 a(1)=1 because 1+10=11=prime.
%Y A172407 Cf. A000027, A000040, A006093, A040976, A086801, A172367.
%K A172407 nonn,new
%O A172407 1,2
%A A172407 Juri-Stepan Gerasimov (2stepan(AT)rambler.ru), Feb 01 2010

%I A172400
%S A172400 1,1,1,2,1,1,6,3,1,1,32,16,5,1,1,332,166,51,9,1,1,6928,3464,1059,181,17,
%T A172400 1,1,292334,146167,44620,7557,681,33,1,1,24875760,12437880,3795202,
%U A172400 641035,57097,2641,65,1,1,4254812880,2127406440,649054326,109540639
%N A172400 G.f.: 1/(1-x) = (1-x*y) * Sum_{k>=0} Sum_{n>=k} T(n,k)*x^n*y^k/(1+x)^(2^n-2^k).
%F A172400 Unsigned column 0 of matrix inverse forms A001192, which is the number of full sets of size n.
%e A172400 Triangle begins:
%e A172400 1;
%e A172400 1, 1;
%e A172400 2, 1, 1;
%e A172400 6, 3, 1, 1;
%e A172400 32, 16, 5, 1, 1;
%e A172400 332, 166, 51, 9, 1, 1;
%e A172400 6928, 3464, 1059, 181, 17, 1, 1;
%e A172400 292334, 146167, 44620, 7557, 681, 33, 1, 1;
%e A172400 24875760, 12437880, 3795202, 641035, 57097, 2641, 65, 1, 1;
%e A172400 4254812880, 2127406440, 649054326, 109540639, 9723237, 443921, 10401, 129, 1, 1; ...
%e A172400 Matrix inverse of this triangle begins:
%e A172400 1;
%e A172400 -1,1;
%e A172400 -1,-1,1;
%e A172400 -2,-2,-1,1;
%e A172400 -9,-9,-4,-1,1;
%e A172400 -88,-88,-38,-8,-1,1;
%e A172400 -1802,-1802,-772,-156,-16,-1,1;
%e A172400 -75598,-75598,-32313,-6456,-632,-32,-1,1; ...
%e A172400 in which unsigned column 0 = A001192, number of full sets of size n.
%o A172400 (PARI) {T(n,k)=if(n==k,1,polcoeff(-(1-x)*sum(m=0,n-k-1,T(m+k,k)*x^m/(1+x +x*O(x^n))^(2^(m+k)-2^k)),n-k))}
%o A172400 (PARI) {T(n,k)=local(M,N); M=matrix(n+1,n+1,r,c,if(r>=c,polcoeff(1/(1-x+O(x^(r-c+1)))^1*(1+x)^(2^(r-1)-2^(c-1)),r-c))); N=matrix(n+1,n+1,r,c,if(r>=c,polcoeff(1/(1-x+O(x^(r-c+1)))^2*(1+x)^(2^(r-1)-2^(c-1)),r-c))); (M^-1*N)[n+1,k+1]}
%Y A172400 Cf. A001192, columns: A172401, A172402, A172403.
%K A172400 nonn,tabl,new
%O A172400 0,4
%A A172400 Paul D. Hanna (pauldhanna(AT)juno.com), Feb 01 2010

%I A172386
%S A172386 2,983,1327373,12695039657
%N A172386 The prime which gives n primes as quickly as possible by concatenation of numbers incremented in sequence.
%C A172386 This is analogous to A172257, with increments in place of decrements.
%e A172386 23 comes from 2, explaining the first term. Until 983, no prime gives
%e A172386 primes with concatenation of 5 and 7 numbers.
%Y A172386 A172257
%K A172386 base,nonn,new
%O A172386 1,1
%A A172386 James G. Merickel (merk7(AT)verizon.net), Feb 01 2010

%I A172385
%S A172385 1,1,1,0,2,4,1,14,34,2,189,439,263,3796,6997,14437,96643,106774,
%T A172385 671097,2800836,57519,31088662,82674287,155322877,1455331563,
%U A172385 1936970102,14267868745
%V A172385 1,1,1,0,-2,-4,-1,14,34,2,-189,-439,263,3796,6997,-14437,-96643,-106774,
%W A172385 671097,2800836,57519,-31088662,-82674287,155322877,1455331563,
%X A172385 1936970102,-14267868745
%N A172385 a(n)=if(n=0,1,sum(C(n-k-1)*(-1)^k*a(n-1-2k),k,0,floor((n-1)/2)).
%F A172385 G.f.: A(x)=1+(x/(1+x^2))*A(x/(1+x^2)).
%e A172385 Eigensequence for the number triangle
%e A172385 1,
%e A172385 1, 0,
%e A172385 0, 1, 0,
%e A172385 -1, 0, 1, 0,
%e A172385 0, -2, 0, 1, 0,
%e A172385 1, 0, -3, 0, 1, 0,
%e A172385 0, 3, 0, -4, 0, 1, 0,
%e A172385 -1, 0, 6, 0, -5, 0, 1, 0,
%e A172385 0, -4, 0, 10, 0, -6, 0, 1, 0,
%e A172385 1, 0, -10, 0, 15, 0, -7, 0, 1, 0,
%e A172385 0, 5, 0, -20, 0, 21, 0, -8, 0, 1, 0
%e A172385 (augmented version of Riordan array (1/(1+x^2),x/(1+x^2)).
%Y A172385 Cf. A172383.
%K A172385 easy,sign,new
%O A172385 0,5
%A A172385 Paul Barry (pbarry(AT)wit.ie), Feb 01 2010

%I A172384
%S A172384 1061,2152,3753,5654,15715,25806,41807,60808,167669,277560,446161,
%T A172384 645062,1751943,2861824,4467905,6273966,8083057,9969068,11858079,
%U A172384 13767160,24574041,35383922,46445733,57537544,69147225,80845916
%N A172384 Partial sums of A048895.
%C A172384 None of these partial sums of "bemirps: primes that yield a different prime when turned upside down with reversals of both being two more different primes" is itself prime. So, what is the first (nontrivial) prime partial sum of bemirps? Of emirps? Of "norep emirps": primes with distinct digits which remain prime when reversed? Of emirpimes? I suspect that G. L. Honaker, Jr. would be delighted to have any of these.
%e A172384 a(26) = 1061 + 1091 + 1601 + 1901 + 10061 + 10091 + 16001 + 19001 + 106861 + 109891 + 168601 + 198901 + 1106881 + 1109881 + 1606081 + 1806061 + 1809091 + 1886011 + 1889011 + 1909081 + 10806881 + 10809881 + 11061811 + 11091811 + 11609681 + 11698691.
%Y A172384 Cf. A000040, A003684, A006567, A007628, A046732, A048051, A048052, A048053, A048054, A048895.
%K A172384 base,easy,nonn,new
%O A172384 1,1
%A A172384 Jonathan Vos Post (jvospost3(AT)gmail.com), Feb 01 2010

%I A172383
%S A172383 1,1,1,2,4,8,19,46,118,322,903,2653,8053,25194,81387,269667,917529,
%T A172383 3197480,11393821,41497060,154186653,584151512,2254240317,8852998343,
%U A172383 35361762709,143540660088
%N A172383 a(n)=if(n=0,1,sum(C(n-k-1)*a(n-1-2k),k,0,floor((n-1)/2)).
%F A172383 G.f.: A(x)=1+(x/(1-x^2))*A(x/(1-x^2)).
%e A172383 Eigensequence for number triangle
%e A172383 1,
%e A172383 1, 0,
%e A172383 0, 1, 0,
%e A172383 1, 0, 1, 0,
%e A172383 0, 2, 0, 1, 0,
%e A172383 1, 0, 3, 0, 1, 0,
%e A172383 0, 3, 0, 4, 0, 1, 0,
%e A172383 1, 0, 6, 0, 5, 0, 1, 0,
%e A172383 0, 4, 0, 10, 0, 6, 0, 1, 0,
%e A172383 1, 0, 10, 0, 15, 0, 7, 0, 1, 0,
%e A172383 0, 5, 0, 20, 0, 21, 0, 8, 0, 1, 0
%e A172383 (augmented version of Riordan array (1/(1-x^2),x/(1-x^2)).
%K A172383 easy,nonn,new
%O A172383 0,4
%A A172383 Paul Barry (pbarry(AT)wit.ie), Feb 01 2010

%I A172382
%S A172382 1,0,1,1,3,8,25,84,301,1139,4528,18849,81968,371622,1753164,8589608,
%T A172382 43624741,229255133,1244544545,6968765313,40195073827,238527741350,
%U A172382 1454722738679,9108812009826,58503226812174,385086101416594
%N A172382 Diagonal sums of number triangle A172380.
%C A172382 a(n+2) gives the diagonal sums of A172381.
%K A172382 nonn,new
%O A172382 0,5
%A A172382 Paul Barry (pbarry(AT)wit.ie), Feb 01 2010

%I A172381
%S A172381 1,1,1,2,3,1,5,10,6,1,14,36,31,10,1,42,137,156,75,15,1,132,544,787,510,
%T A172381 155,21,1,429,2235,4017,3331,1380,287,28,1,1430,9445,20809,21405,11411,
%U A172381 3255,490,36,1,4862,40876,109486,136921,90665,33390,6916,786,45,1,16796
%N A172381 Triangle whose inverse has production matrix with general term (-1)^(n-k+1)*C(k+1, n-k+1).
%C A172381 Product of A033184 and A172380. Diagonal sums are A172382(n+2).
%e A172381 Triangle begins
%e A172381 1,
%e A172381 1, 1,
%e A172381 2, 3, 1,
%e A172381 5, 10, 6, 1,
%e A172381 14, 36, 31, 10, 1,
%e A172381 42, 137, 156, 75, 15, 1,
%e A172381 132, 544, 787, 510, 155, 21, 1,
%e A172381 429, 2235, 4017, 3331, 1380, 287, 28, 1,
%e A172381 1430, 9445, 20809, 21405, 11411, 3255, 490, 36, 1,
%e A172381 4862, 40876, 109486, 136921, 90665, 33390, 6916, 786, 45, 1,
%e A172381 16796, 180544, 584955, 877252, 704720, 322728, 86443, 13536, 1200, 55, 1
%e A172381 Production array of inverse is
%e A172381 -1, 1,
%e A172381 0, -2, 1,
%e A172381 0, 1, -3, 1,
%e A172381 0, 0, 3, -4, 1,
%e A172381 0, 0, -1, 6, -5, 1,
%e A172381 0, 0, 0, -4, 10, -6, 1,
%e A172381 0, 0, 0, 1, -10, 15, -7, 1,
%e A172381 0, 0, 0, 0, 5, -20, 21, -8, 1,
%e A172381 0, 0, 0, 0, -1, 15, -35, 28, -9, 1,
%e A172381 0, 0, 0, 0, 0, -6, 35, -56, 36, -10, 1,
%e A172381 0, 0, 0, 0, 0, 1, -21, 70, -84, 45, -11, 1
%K A172381 nonn,tabl,new
%O A172381 0,4
%A A172381 Paul Barry (pbarry(AT)wit.ie), Feb 01 2010

%I A172380
%S A172380 1,0,1,0,1,1,0,2,3,1,0,5,10,6,1,0,14,36,31,10,1,0,42,137,156,75,15,1,0,
%T A172380 132,544,787,510,155,21,1,0,429,2235,4017,3331,1380,287,28,1,0,1430,
%U A172380 9445,20809,21405,11411,3255,490,36,1,0,4862,40876,109486,136921,90665
%N A172380 Eigentriangle of Catalan triangle A033184.
%C A172380 Row sums are A091768. Production matrix of inverse is matrix with general term (-1)^(n-k+1)C(k,n-k+1).
%C A172380 Diagonal sums are A172382. Product of A033184 and A172380 is the matrix A172381.
%e A172380 Triangle begins
%e A172380 1,
%e A172380 0, 1,
%e A172380 0, 1, 1,
%e A172380 0, 2, 3, 1,
%e A172380 0, 5, 10, 6, 1,
%e A172380 0, 14, 36, 31, 10, 1,
%e A172380 0, 42, 137, 156, 75, 15, 1,
%e A172380 0, 132, 544, 787, 510, 155, 21, 1,
%e A172380 0, 429, 2235, 4017, 3331, 1380, 287, 28, 1,
%e A172380 0, 1430, 9445, 20809, 21405, 11411, 3255, 490, 36, 1
%e A172380 Production matrix of inverse is
%e A172380 0, 1,
%e A172380 0, -1, 1,
%e A172380 0, 0, -2, 1,
%e A172380 0, 0, 1, -3, 1,
%e A172380 0, 0, 0, 3, -4, 1,
%e A172380 0, 0, 0, -1, 6, -5, 1,
%e A172380 0, 0, 0, 0, -4, 10, -6, 1,
%e A172380 0, 0, 0, 0, 1, -10, 15, -7, 1,
%e A172380 0, 0, 0, 0, 0, 5, -20, 21, -8, 1,
%e A172380 0, 0, 0, 0, 0, -1, 15, -35, 28, -9, 1,
%e A172380 0, 0, 0, 0, 0, 0, -6, 35, -56, 36, -10, 1
%K A172380 nonn,tabl,new
%O A172380 0,8
%A A172380 Paul Barry (pbarry(AT)wit.ie), Feb 01 2010

%I A172367
%S A172367 1,3,7,9,13,15,19,25,27,33,37,39,43,49,55,57,63,67,69,75,79,85,93,97,99,
%T A172367 103,105,109,123,127,133,135,145,147,153,159,163,169,175,177,187,189,
%U A172367 193,195,207,219,223,225,229,235,237,247,253,259,265,267,273,277,279
%N A172367 Odd numbers n such that n+4 is a prime.
%F A172367 a(n)=(n+2)th prime-4.
%e A172367 a(1)=5-4=1, a(2)=7-4=3.
%Y A172367 Cf. A000040, A006093, A040976, A086801.
%K A172367 nonn,new
%O A172367 1,2
%A A172367 Juri-Stepan Gerasimov (2stepan(AT)rambler.ru), Feb 01 2010

%I A172379
%S A172379 1357,1358,1368,1468,2468,2469,2479,2579,3579
%N A172379 The unique d=3 cycle arising in higher dimensional cluster combinatorics and representation theory
%C A172379 Oppermann, p.43, Computer experiments have not detected any similar phenomena when d = 2. Higher Auslander algebras were introduced by Iyama generalizing classical concepts from representation theory of finite dimensional algebras. Recently these higher analogues of classical representation theory have been increasingly studied. Cyclic polytopes are classical objects of study in convex geometry. In particular, their triangulations have been studied with a view towards generalizing the rich combinatorial structure of triangulations of polygons. In this paper, we demonstrate a connection between these two seemingly unrelated subjects.
%C A172379 We study triangulations of even-dimensional cyclic polytopes and tilting modules for higher Auslander algebras of linearly oriented type A which are summands of the cluster tilting module. We show that such tilting modules correspond bijectively to triangulations. Moreover mutations of tilting modules correspond to bistellar flips of triangulations. For any d-representation finite algebra we introduce a certain d-dimensional cluster category and study its cluster tilting objects. For higher Auslander algebras of linearly oriented type A we obtain a similar correspondence between cluster tilting objects and triangulations of a certain cyclic polytope. Finally we study certain functions on generalized laminations in cyclic polytopes, and show that they satisfy analogues of tropical cluster exchange relations. Moreover we observe that the terms of these exchange relations are closely related to the terms occuring in the mutation of cluster tilting objects.
%H A172379 Steffen Oppermann, Hugh Thomas, <a href="http://arxiv.org/abs/1001.5437">Higher dimensional cluster combinatorics and representation theory</a>, Jan 30, 2010.
%K A172379 fini,full,nonn,new
%O A172379 1,1
%A A172379 Jonathan Vos Post (jvospost3(AT)gmail.com), Feb 01 2010

%I A172365
%S A172365 3,5,7,11,17,31,47,71,139,67,101,199,71,107,211,127,193,379,223,317,631,
%T A172365 307,461,919
%N A172365 List of primes p1, p2 and p3 such that 3*p1-2=2*p2-3=p3.
%e A172365 a(1)=3=p1, a(2)=5=p2, a(3)=7=p3 because 3*3-2=2*5-3=7; a(4)=11=p1, a(5)=17=p2, a(6)=31=p3 because 3*11-2=2*17-3=31.
%Y A172365 Cf. A000040, A172287, A172888.
%K A172365 nonn,new
%O A172365 1,1
%A A172365 Juri-Stepan Gerasimov (2stepan(AT)rambler.ru), Feb 01 2010

%I A172235
%S A172235 1,3,5,7,15,23,25,39,53,67,75,89,103,111,131,157,177,197,211,225,245,
%T A172235 265,285,311,313,321,353,379,405,449,453,543,587,625,675,701,727,765,
%U A172235 791,823,855,875,901,921
%N A172235 nth partial sum of primes of the form 3*k+1/2+-1/2 minus nth partial sum of primes of the form 3*m-1.
%F A172235 a(n)=A172189(n)-A172188(n).
%e A172235 a(1)=(3*1+1/2-1/2)-(3*1-1)=1; a(2)=(3+3*2+1/2+1/2)-(2+3*2-1)=3.
%Y A172235 Cf. A172188, A172189.
%K A172235 nonn,new
%O A172235 1,2
%A A172235 Juri-Stepan Gerasimov (2stepan(AT)rambler.ru), Feb 01 2010

%I A175096
%S A175096 1,2,1,2,1,2,1,2,1,2,2,2,2,2,1,2,1,4,2,4,1,4,2,2,2,4,1,2,2,2,1,2,1,4,2,
%T A175096 2,2,8,2,4,2,2,3,8,3,4,2,2,2,8,1,8,3,2,2,2,2,4,2,2,2,2,1,2,1,4,2,4,2,8,
%U A175096 2,4,1,6,6,4,6,8,2,4,2,6,6,6,1,6,3,8,6,6,3,8,3,4,2,2,2,8,1,4,6,4,2,8,6
%N A175096 Write n in binary (without leading 0's). a(n) = the number of distinct numerical values made by permutating the runs of 0's and the runs of 1's, such that the runs (of nonzero length) of 1's alternate with the runs (of nonzero length) of 0's. The permutated binary numbers (those not equal to n) may start with leading 0's.
%C A175096 Each "run" of binary digit b (0 or 1) is bounded by digits equal to 1-b, or is bounded by the edge of the binary string (which is n written in binary).
%C A175096 For all odd n, the values of all permutations of binary n are themselves odd, since there are an odd number of runs (the first and last runs being of 1's).
%e A175096 20 in binary is 10100. So we have a run of one 1, followed by a run of one 0, followed by a run of one 1, followed finally by a run of two 0's. The permutations of the runs of 0's and the run's of 1's form these distinct binary numbers: 00101 (5 in decimal), 01001 (9 in decimal), 10010 (18 in decimal), and 10100 (20 in decimal). So, a(20) = 4 since there are 4 such permutations.
%K A175096 base,nonn,new
%O A175096 1,2
%A A175096 Leroy Quet (q1qq2qqq3qqqq(AT)yahoo.com), Feb 01 2010
%E A175096 Extended by Ray Chandler (rayjchandler(AT)sbcglobal.net), Feb 07 2010

%I A175095
%S A175095 0,2,3,5,6,9,14,15,17,21,27,30,33,35,45,51,62,63,65,75,93,99,119,126,
%T A175095 129,135,155,189,195,231,254,255,257,279,315,381,387,455,495,510,513,
%U A175095 527,567,635,765,771,903,975,1022,1023,1025,1071,1143,1275,1533,1539
%N A175095 Those distinct nonnegative integers that are of the form (2^j -1)*(2^k +1), j>=0, k>=0.
%Y A175095 Cf. A000051, A000225
%K A175095 easy,nonn,new
%O A175095 1,2
%A A175095 Leroy Quet (q1qq2qqq3qqqq(AT)yahoo.com), Feb 01 2010
%E A175095 Extended by Ray Chandler (rayjchandler(AT)sbcglobal.net) and R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Feb 06 2010

%I A172378
%S A172378 1,1,1,1,10,1,1,110,110,1,1,1199,13189,1199,1,1,13080,1568292,1568292,
%T A172378 13080,1,1,142680,186625440,2034217296,186625440,142680,1,1,1556401,
%U A172378 22206729468,2640582013104,2640582013104,22206729468,1556401,1,1
%N A172378 Second beta integer combination triangle of a Narayana type: a=3:f(n, a) = a*f(n - 1, a) + f(n - 2, a);c(n,a)=If[n == 0, 1, Product[f(i, a), {i, 1, n}]];w(n,m,q)=c(n - 1, q)*c(n, q)/(c(m - 1, q)*c(n - m, q)*c(m - 1, q)*c(n - m + 1, q)*f(m, q))
%C A172378 Row sums are:
%C A172378 {1, 2, 12, 222, 15589, 3162746, 2407753538, 5325580597948, 44250590408026536,
%C A172378 1067713385571585752220,...}
%F A172378 a=3:
%F A172378 f(n, a) = a*f(n - 1, a) + f(n - 2, a);
%F A172378 c(n,a)=If[n == 0, 1, Product[f(i, a), {i, 1, n}]];
%F A172378 w(n,m,q)=c(n - 1, q)*c(n, q)/(c(m - 1, q)*c(n - m, q)*c(m - 1, q)*c(n - m + 1, q)*f(m, q))
%e A172378 {1},
%e A172378 {1, 1},
%e A172378 {1, 10, 1},
%e A172378 {1, 110, 110, 1},
%e A172378 {1, 1199, 13189, 1199, 1},
%e A172378 {1, 13080, 1568292, 1568292, 13080, 1},
%e A172378 {1, 142680, 186625440, 2034217296, 186625440, 142680, 1},
%e A172378 {1, 1556401, 22206729468, 2640582013104, 2640582013104, 22206729468, 1556401, 1},
%e A172378 {1, 16977730, 2642415594973, 3427453246279524, 37390399050322080, 3427453246279524, 2642415594973, 16977730, 1},
%e A172378 {1, 185198630, 314425233650990, 4448834073451222609, 529407544286922803880, 529407544286922803880, 4448834073451222609, 314425233650990, 185198630, 1}
%t A172378 Clear[t, n, m, c, q, w, f, a] f[0, a_] := 0; f[1, a_] := 1;
%t A172378 f[n_, a_] := f[n, a] = a*f[n - 1, a] + f[n - 2, a];
%t A172378 c[n_, a_] := If[n == 0, 1, Product[f[i, a], {i, 1, n}]];
%t A172378 w[n_, m_, q_] := c[n - 1, q]*c[n, q]/(c[m - 1, q]*c[n - m, q]*c[m - 1, q]*c[n - m + 1, q]*f[m, q]);
%t A172378 Table[Table[Table[w[n, m, q], {m, 1, n}], {n, 1, 10}], {q, 1, 12}];
%t A172378 Table[Flatten[Table[Table[w[n, m, q], {m, 1, n}], {n, 1, 10}]], {q, 1, 12}]
%K A172378 nonn,uned,new
%O A172378 1,5
%A A172378 Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Feb 01 2010

%I A172377
%S A172377 1,1,1,1,5,1,1,30,30,1,1,174,1044,174,1,1,1015,35322,35322,1015,1,1,
%T A172377 5915,1200745,6964321,1200745,5915,1,1,34476,40785108,1379896154,
%U A172377 1379896154,40785108,34476,1,1,200940,1385521488,273178653384
%N A172377 Second beta integer combination triangle of a Narayana type: a=2:f(n, a) = a*f(n - 1, a) + f(n - 2, a);c(n,a)=If[n == 0, 1, Product[f(i, a), {i, 1, n}]];w(n,m,q)=c(n - 1, q)*c(n, q)/(c(m - 1, q)*c(n - m, q)*c(m - 1, q)*c(n - m + 1, q)*f(m, q))
%C A172377 Row sums are:
%C A172377 {1, 2, 4, 14, 77, 682, 9570, 218220, 8079864, 483294396,...}
%F A172377 a=2:
%F A172377 f(n, a) = a*f(n - 1, a) + f(n - 2, a);
%F A172377 c(n,a)=If[n == 0, 1, Product[f(i, a), {i, 1, n}]];
%F A172377 w(n,m,q)=c(n - 1, q)*c(n, q)/(c(m - 1, q)*c(n - m, q)*c(m - 1, q)*c(n - m + 1, q)*f(m, q))
%e A172377 {1},
%e A172377 {1, 1},
%e A172377 {1, 5, 1},
%e A172377 {1, 30, 30, 1},
%e A172377 {1, 174, 1044, 174, 1},
%e A172377 {1, 1015, 35322, 35322, 1015, 1},
%e A172377 {1, 5915, 1200745, 6964321, 1200745, 5915, 1},
%e A172377 {1, 34476, 40785108, 1379896154, 1379896154, 40785108, 34476, 1},
%e A172377 {1, 200940, 1385521488, 273178653384, 1593542144740, 273178653384, 1385521488, 200940, 1},
%e A172377 {1, 1171165, 47066779020, 54089142449784, 1838719986152140, 1838719986152140, 54089142449784, 47066779020, 1171165, 1}
%t A172377 Clear[t, n, m, c, q, w, f, a] f[0, a_] := 0; f[1, a_] := 1;
%t A172377 f[n_, a_] := f[n, a] = a*f[n - 1, a] + f[n - 2, a];
%t A172377 c[n_, a_] := If[n == 0, 1, Product[f[i, a], {i, 1, n}]];
%t A172377 w[n_, m_, q_] := c[n - 1, q]*c[n, q]/(c[m - 1, q]*c[n - m, q]*c[m - 1, q]*c[n - m + 1, q]*f[m, q]);
%t A172377 Table[Table[Table[w[n, m, q], {m, 1, n}], {n, 1, 10}], {q, 1, 12}];
%t A172377 Table[Flatten[Table[Table[w[n, m, q], {m, 1, n}], {n, 1, 10}]], {q, 1, 12}]
%K A172377 nonn,uned,new
%O A172377 1,5
%A A172377 Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Feb 01 2010

%I A172376
%S A172376 1,1,1,1,12,1,1,180,180,1,1,2565,38475,2565,1,1,36936,7895070,7895070,
%T A172376 36936,1,1,530712,1633531536,23277824388,1633531536,530712,1,1,7628985,
%U A172376 337399490610,69234375473172,69234375473172,337399490610,7628985,1,1
%N A172376 A beta integer combination triangle of a Narayana type: a=3:f(n, a) = a*f(n - 1, a) + a*f(n - 2, a);c(n,a)=If[n == 0, 1, Product[f(i, a), {i, 1, n}]];w(n,m,q)=c(n - 1, q)*c(n, q)/(c(m - 1, q)*c(n - m, q)*c(m - 1, q)*c(n - m + 1, q)*f(m, q))
%C A172376 Row sums are:
%C A172376 {1, 2, 14, 362, 43607, 15864014, 26545948886, 139143565185536,
%C A172376 3371062783875324716, 253833990333055824621332,...}
%F A172376 a=3:
%F A172376 f(n, a) = a*f(n - 1, a) + a*f(n - 2, a);
%F A172376 c(n,a)=If[n == 0, 1, Product[f(i, a), {i, 1, n}]];
%F A172376 w(n,m,q)=c(n - 1, q)*c(n, q)/(c(m - 1, q)*c(n - m, q)*c(m - 1, q)*c(n - m + 1, q)*f(m, q))
%e A172376 {1},
%e A172376 {1, 1},
%e A172376 {1, 12, 1},
%e A172376 {1, 180, 180, 1},
%e A172376 {1, 2565, 38475, 2565, 1},
%e A172376 {1, 36936, 7895070, 7895070, 36936, 1},
%e A172376 {1, 530712, 1633531536, 23277824388, 1633531536, 530712, 1},
%e A172376 {1, 7628985, 337399490610, 69234375473172, 69234375473172, 337399490610, 7628985, 1},
%e A172376 {1, 109656180, 69713779364775, 205544107079102610, 2959835141939077584, 205544107079102610, 69713779364775, 109656180, 1},
%e A172376 {1, 1576188396, 14403233205473940, 610455833755903367505, 126306524929537227280824, 126306524929537227280824, 610455833755903367505, 14403233205473940, 1576188396, 1}
%t A172376 Clear[t, n, m, c, q, w, f, a] f[0, a_] := 0; f[1, a_] := 1;
%t A172376 f[n_, a_] := f[n, a] = a*f[n - 1, a] + a*f[n - 2, a];
%t A172376 c[n_, a_] := If[n == 0, 1, Product[f[i, a], {i, 1, n}]];
%t A172376 w[n_, m_, q_] := c[n - 1, q]*c[n, q]/(c[m - 1, q]*c[n - m, q]*c[m - 1, q]*c[n - m + 1, q]*f[m, q]);
%t A172376 Table[Table[Table[w[n, m, q], {m, 1, n}], {n, 1, 10}], {q, 1, 12}];
%t A172376 Table[Flatten[Table[Table[w[n, m, q], {m, 1, n}], {n, 1, 10}]], {q, 1, 12}]
%K A172376 nonn,uned,new
%O A172376 1,5
%A A172376 Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Feb 01 2010

%I A172375
%S A172375 1,1,1,1,6,1,1,48,48,1,1,352,2816,352,1,1,2640,154880,154880,2640,1,1,
%T A172375 19680,8659200,63500800,8659200,19680,1,1,146944,481976320,26508697600,
%U A172375 26508697600,481976320,146944,1,1,1096704,26859012096,11012194959360
%N A172375 A beta integer combination triangle of a Narayana type: a=2:f(n, a) = a*f(n - 1, a) + a*f(n - 2, a);c(n,a)=If[n == 0, 1, Product[f(i, a), {i, 1, n}]];w(n,m,q)=c(n - 1, q)*c(n, q)/(c(m - 1, q)*c(n - m, q)*c(m - 1, q)*c(n - m + 1, q)*f(m, q))
%C A172375 Row sums are:
%C A172375 {1, 2, 8, 98, 3522, 315042, 80858562, 53981641730, 104669572331522,
%C A172375 521363491264516610,...}
%F A172375 a=2:
%F A172375 f(n, a) = a*f(n - 1, a) + a*f(n - 2, a);
%F A172375 c(n,a)=If[n == 0, 1, Product[f(i, a), {i, 1, n}]];
%F A172375 w(n,m,q)=c(n - 1, q)*c(n, q)/(c(m - 1, q)*c(n - m, q)*c(m - 1, q)*c(n - m + 1, q)*f(m, q))
%e A172375 {1},
%e A172375 {1, 1},
%e A172375 {1, 6, 1},
%e A172375 {1, 48, 48, 1},
%e A172375 {1, 352, 2816, 352, 1},
%e A172375 {1, 2640, 154880, 154880, 2640, 1},
%e A172375 {1, 19680, 8659200, 63500800, 8659200, 19680, 1},
%e A172375 {1, 146944, 481976320, 26508697600, 26508697600, 481976320, 146944, 1},
%e A172375 {1, 1096704, 26859012096, 11012194959360, 82591462195200, 11012194959360, 26859012096, 1096704, 1},
%e A172375 {1, 8186112, 1496290295808, 4580643358900224, 256099605974876160, 256099605974876160, 4580643358900224, 1496290295808, 8186112, 1}
%t A172375 Clear[t, n, m, c, q, w, f, a] f[0, a_] := 0; f[1, a_] := 1;
%t A172375 f[n_, a_] := f[n, a] = a*f[n - 1, a] + a*f[n - 2, a];
%t A172375 c[n_, a_] := If[n == 0, 1, Product[f[i, a], {i, 1, n}]];
%t A172375 w[n_, m_, q_] := c[n - 1, q]*c[n, q]/(c[m - 1, q]*c[n - m, q]*c[m - 1, q]*c[n - m + 1, q]*f[m, q]);
%t A172375 Table[Table[Table[w[n, m, q], {m, 1, n}], {n, 1, 10}], {q, 1, 12}];
%t A172375 Table[Flatten[Table[Table[w[n, m, q], {m, 1, n}], {n, 1, 10}]], {q, 1, 12}]
%K A172375 nonn,uned,new
%O A172375 1,5
%A A172375 Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Feb 01 2010

%I A172374
%S A172374 2,278,1826,4498070,2645182700
%N A172374 The first number generating n primes through the concatenation of numbers increasing in minimal increments in the quickest way.
%C A172374 Analogous to A152396 with increments replacing decrements.
%e A172374 278279 and 278279280281 are prime and no number <278 gives primes when
%e A172374 treated similarly. Concatenation of 2, 4, 8, 10, and 14 numbers give
%e A172374 primes beginning with 2645182700.
%Y A172374 A152396
%K A172374 base,nonn,new
%O A172374 1,1
%A A172374 James G. Merickel (merk7(AT)verizon.net), Feb 01 2010

%I A172373
%S A172373 1,1,1,1,2,1,1,6,6,1,1,15,45,15,1,1,40,300,300,40,1,1,104,2080,5200,
%T A172373 2080,104,1,1,273,14196,94640,94640,14196,273,1,1,714,97461,1689324,
%U A172373 4504864,1689324,97461,714,1,1,1870,667590,30375345,210602392,210602392
%N A172373 A beta integer combination triangle of a Narayana type: a=1:f(n, a) = a*f(n - 1, a) + a*f(n - 2, a);c(n,a)=If[n == 0, 1, Product[f(i, a), {i, 1, n}]];w(n,m,q)=c(n - 1, q)*c(n, q)/(c(m - 1, q)*c(n - m, q)*c(m - 1, q)*c(n - m + 1, q)*f(m, q))
%C A172373 Row sums are:
%C A172373 {1, 2, 4, 14, 77, 682, 9570, 218220, 8079864, 483294396,...}
%F A172373 a=1:
%F A172373 f(n, a) = a*f(n - 1, a) + a*f(n - 2, a);
%F A172373 c(n,a)=If[n == 0, 1, Product[f(i, a), {i, 1, n}]];
%F A172373 w(n,m,q)=c(n - 1, q)*c(n, q)/(c(m - 1, q)*c(n - m, q)*c(m - 1, q)*c(n - m + 1, q)*f(m, q))
%e A172373 {1},
%e A172373 {1, 1},
%e A172373 {1, 2, 1},
%e A172373 {1, 6, 6, 1},
%e A172373 {1, 15, 45, 15, 1},
%e A172373 {1, 40, 300, 300, 40, 1},
%e A172373 {1, 104, 2080, 5200, 2080, 104, 1},
%e A172373 {1, 273, 14196, 94640, 94640, 14196, 273, 1},
%e A172373 {1, 714, 97461, 1689324, 4504864, 1689324, 97461, 714, 1},
%e A172373 {1, 1870, 667590, 30375345, 210602392, 210602392, 30375345, 667590, 1870, 1}
%t A172373 Clear[t, n, m, c, q, w, f, a] f[0, a_] := 0; f[1, a_] := 1;
%t A172373 f[n_, a_] := f[n, a] = a*f[n - 1, a] + a*f[n - 2, a];
%t A172373 c[n_, a_] := If[n == 0, 1, Product[f[i, a], {i, 1, n}]];
%t A172373 w[n_, m_, q_] := c[n - 1, q]*c[n, q]/(c[m - 1, q]*c[n - m, q]*c[m - 1, q]*c[n - m + 1, q]*f[m, q]);
%t A172373 Table[Table[Table[w[n, m, q], {m, 1, n}], {n, 1, 10}], {q, 1, 12}];
%t A172373 Table[Flatten[Table[Table[w[n, m, q], {m, 1, n}], {n, 1, 10}]], {q, 1, 12}]
%K A172373 nonn,uned,new
%O A172373 1,5
%A A172373 Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Feb 01 2010

%I A172372
%S A172372 1,3,6,2,6,16,18,22,28,15,3,5,21,46,13,58,60,33,35,8,13,41,44,6,96,4,34,
%T A172372 53,108,112,42,130,8,46,148,75,78,81,166,43,43,178,180,95,192,98,99,30,
%U A172372 222,113,228,232,7,30,50,256,262,268,5,69,28,141,146,153,155,312,79,110
%N A172372 Repunits numbers x(n) = (10^n - 1)/9. s(n) is the number of digits "1" such that every odd prime number n, n different from 5, divide the smallest x(n).
%C A172372 If p is a odd prime different of 5, then p divides an infinite number of terms of this sequence {1, 11, 111, 1111, ... }. The proof is elementary : with n = 3, if we consider the numbers 111, 111 111, 111 111 111, ..., this all numbers including 3,6,9,... times the number "1", we obtain an infinite of numbers of this form. Let the prime number n > =7. A number N build with only "1" can be writting N = (10^p - 1)/9. Now according to the Fermat theorem, 10^(n-1) = 1 (mod n), then 10^(m(n-1)) = 1 (mod n) for m = 1,2,3,... . Because n different from 3, n divide (10^(m(n-1)) / 9, for m = 0,1,2,3,... . Hence the result. For each odd prime n different of 5, we consider the smallest repunit number x(n) = 11...11 where p divide x(n), and s(n) = number of digits of x(n). Exemple : for n = 7 then 7 divide 111111 and s(7) = 6 ; for n = 11 then 11 divide 11 and s(11) = 2, etc.
%D A172372 David Wells, The Factors of the Repunits 11 through R_40, The Penguin Dictionary of Curious and Interesting Numbers, p. 219 Penguin 1986.
%D A172372 D. Wells, The Penguin Dictionary of Curious and Interesting Numbers, Penguin Books 1997. David Wells, Curious and Interesting Numbers (Revised), Penguin Books, page 114.
%H A172372 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Repunit.html">Link to a section of The World of Mathematics</a>.
%H A172372 S. S. Wagstaff, Jr.,<a href="http://homes.cerias.purdue.edu/~ssw/cun/index.html">The Cunningham Project</a>
%F A172372 a(n) = (10^n-1)/9
%e A172372 1 divide 1 and s(1) = 1 3 divide 111 and s(2) = 3 7 divide 111 111 and s(3) = 6 11 divide 11 and s(4) = 2 13 divide 111 111 and s(5) = 6 17 divide 1111 1111 1111 1111 and s(6) = 16, etc.
%Y A172372 A002275 "A002275", Repunits: (10^n - 1)/9. "A095250" a(n) = 11111111... (n times) = (10^n-1)/9 reduced mod n
%K A172372 nonn,new
%O A172372 1,2
%A A172372 Michel Lagneau (mn.lagneau2(AT)orange.fr), Feb 01 2010

%I A172371
%S A172371 0,0,1,0,1,1,0,1,1,1,0,1,1,2,2,0,1,1,3,3,2,0,1,1,4,4,5,3,0,1,1,5,5,10,8,
%T A172371 4,0,1,1,6,6,17,15,13,5,0,1,1,7,7,26,24,34,21,7
%N A172371 Cubic recursion anti-diagonal triangle sequence: f(n,a)=a*f(n-2,a)+f(n-3,a)
%C A172371 Row sums are:
%C A172371 {0, 1, 2, 3, 6, 10, 18, 34, 64, 128,...}
%F A172371 f(n,a)=a*f(n-2,a)+f(n-3,a);
%F A172371 Output(n,a)=antidiagonal(f(n,a))
%e A172371 {0},
%e A172371 {0, 1},
%e A172371 {0, 1, 1},
%e A172371 {0, 1, 1, 1},
%e A172371 {0, 1, 1, 2, 2},
%e A172371 {0, 1, 1, 3, 3, 2},
%e A172371 {0, 1, 1, 4, 4, 5, 3},
%e A172371 {0, 1, 1, 5, 5, 10, 8, 4},
%e A172371 {0, 1, 1, 6, 6, 17, 15, 13, 5},
%e A172371 {0, 1, 1, 7, 7, 26, 24, 34, 21, 7}
%t A172371 f[0, a_] := 0; f[1, a_] := 1; f[2, a_] := 1;
%t A172371 f[n_, a_] := f[n, a] = a*f[n - 2, a] + f[n - 3, a];
%t A172371 m1 = Table[f[n, a], {n, 0, 10}, {a, 1, 11}];
%t A172371 Table[Table[m1[[m, n - m + 1]], {m, 1, n}], {n, 1, 10}];
%t A172371 Flatten[%]
%K A172371 nonn,tabl,uned,new
%O A172371 0,14
%A A172371 Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Feb 01 2010

%I A172370
%S A172370 3,5,8,7,3,15,9,16,21,24,11,5,1,2,35,13,24,33,40,45,48,15,7,39,3,55,15,
%T A172370 63,17,32,5,56,65,8,77,80,19,9,51,4,3,21,91,6,99,21,40,57,72,85,96,105,
%U A172370 112,117,120,23,11,7,5,95,1,119,1,5,35,143,25,48,69,88,105,120,133,144
%N A172370 Triangle A120072 read by reversal rows.
%C A172370 Successive numerators of extended Rydberg-Ritz spectra of hydrogen atom are 1) A067998=0,-1,A005563 Lyman; 2) A144477=0,-3,-1,-3,A061037 Balmer; 3) A171709=0,-5,-8,-1,-8,-5,A061039 Paschen; 4) submitted A171825=0,-7,-3,-15,-1,-15,-3,-7,A061041 Brackett; 5) 0,-9,-16,-21,-24,-1,-24,-21,-16,-9,A061043 Pfund; 6) 0,-11,-5,-1,-2,-35,-1,-35,-2,-1,-5,-11,A061045 Humphreys; 7) 0,-13,-24,..,-24,-13,A061047 Hansen-Strong; .. . See A172157 and A165795 (good Balmer values omitted in array) .Opposite rows of a(n) are ,just after 0, in preceding terms of Rydberg-Ritz. Sequence of a(n) with 0's will be submitted. Without 0,preceding terms are palindromic. Thanks to Richard Mathar.
%K A172370 nonn,uned,new
%O A172370 0,1
%A A172370 Paul Curtz (bpcrtz(AT)free.fr), Feb 01 2010

%I A172369
%S A172369 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,1,1,7,28,28,28,7,1,1,10,70,280,
%T A172369 280,70,10,1,1,13,130,910,3640,910,130,13,1,1,25,325,3250,22750,22750,
%U A172369 3250,325,25,1,1,46,1150,14950,149500,261625,149500,14950,1150,46,1
%N A172369 Seventh (second quartic) type of beta integer triangle sequence: a=3;f(n,a)=f(n-1,a)+a*f(n-4,a);c(n,a)=If[n == 0, 1, Product[f(i, a), {i, 1, n}]];t(n,m,a)=c(n, a)/(c(m, a)*c(n - m, a))
%C A172369 Row sums are:
%C A172369 {1, 2, 3, 4, 5, 18, 100, 722, 5748, 52702, 592919,...}.
%F A172369 a=3;
%F A172369 f(n,a)=f(n-1,a)+a*f(n-4,a);
%F A172369 c(n,a)=If[n == 0, 1, Product[f(i, a), {i, 1, n}]];
%F A172369 t(n,m,a)=c(n, a)/(c(m, a)*c(n - m, a))
%e A172369 {1},
%e A172369 {1, 1},
%e A172369 {1, 1, 1},
%e A172369 {1, 1, 1, 1},
%e A172369 {1, 1, 1, 1, 1},
%e A172369 {1, 4, 4, 4, 4, 1},
%e A172369 {1, 7, 28, 28, 28, 7, 1},
%e A172369 {1, 10, 70, 280, 280, 70, 10, 1},
%e A172369 {1, 13, 130, 910, 3640, 910, 130, 13, 1},
%e A172369 {1, 25, 325, 3250, 22750, 22750, 3250, 325, 25, 1},
%e A172369 {1, 46, 1150, 14950, 149500, 261625, 149500, 14950, 1150, 46, 1}
%t A172369 Clear[f, c, a, t];
%t A172369 f[0, a_] := 0; f[1, a_] := 1; f[2, a_] := 1; f[3, a_] := 1;
%t A172369 f[n_, a_] := f[n, a] = a*f[n - 1, a] + f[n - 4, a];
%t A172369 c[n_, a_] := If[n == 0, 1, Product[f[i, a], {i, 1, n}]];
%t A172369 t[n_, m_, a_] := c[n, a]/(c[m, a]*c[n - m, a]);
%t A172369 Table[Table[Table[t[n, m, a], {m, 0, n}], {n, 0, 10}], {a, 1, 10}];
%t A172369 Table[Flatten[Table[Table[t[n, m, a], {m, 0, n}], {n, 0, 10}]], {a, 1, 10}]
%K A172369 nonn,tabl,uned,new
%O A172369 0,17
%A A172369 Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Feb 01 2010

%I A172368
%S A172368 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,1,1,5,15,15,15,5,1,1,7,35,105,
%T A172368 105,35,7,1,1,9,63,315,945,315,63,9,1,1,15,135,945,4725,4725,945,135,15,
%U A172368 1,1,25,375,3375,23625,39375,23625,3375,375,25,1
%N A172368 Seventh (second quartic) type of beta integer triangle sequence: a=2;f(n,a)=f(n-1,a)+a*f(n-4,a);c(n,a)=If[n == 0, 1, Product[f(i, a), {i, 1, n}]];t(n,m,a)=c(n, a)/(c(m, a)*c(n - m, a))
%C A172368 Row sums are:
%C A172368 {1, 2, 3, 4, 5, 14, 57, 296, 1721, 11642, 94177,...}.
%F A172368 a=2;
%F A172368 f(n,a)=f(n-1,a)+a*f(n-4,a);
%F A172368 c(n,a)=If[n == 0, 1, Product[f(i, a), {i, 1, n}]];
%F A172368 t(n,m,a)=c(n, a)/(c(m, a)*c(n - m, a))
%e A172368 {1},
%e A172368 {1, 1},
%e A172368 {1, 1, 1},
%e A172368 {1, 1, 1, 1},
%e A172368 {1, 1, 1, 1, 1},
%e A172368 {1, 3, 3, 3, 3, 1},
%e A172368 {1, 5, 15, 15, 15, 5, 1},
%e A172368 {1, 7, 35, 105, 105, 35, 7, 1},
%e A172368 {1, 9, 63, 315, 945, 315, 63, 9, 1},
%e A172368 {1, 15, 135, 945, 4725, 4725, 945, 135, 15, 1},
%e A172368 {1, 25, 375, 3375, 23625, 39375, 23625, 3375, 375, 25, 1}
%t A172368 Clear[f, c, a, t];
%t A172368 f[0, a_] := 0; f[1, a_] := 1; f[2, a_] := 1; f[3, a_] := 1;
%t A172368 f[n_, a_] := f[n, a] = a*f[n - 1, a] + f[n - 4, a];
%t A172368 c[n_, a_] := If[n == 0, 1, Product[f[i, a], {i, 1, n}]];
%t A172368 t[n_, m_, a_] := c[n, a]/(c[m, a]*c[n - m, a]);
%t A172368 Table[Table[Table[t[n, m, a], {m, 0, n}], {n, 0, 10}], {a, 1, 10}];
%t A172368 Table[Flatten[Table[Table[t[n, m, a], {m, 0, n}], {n, 0, 10}]], {a, 1, 10}]
%K A172368 nonn,tabl,uned,new
%O A172368 0,17
%A A172368 Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Feb 01 2010

%I A172364
%S A172364 1,1,1,1,1,1,1,1,1,1,1,3,3,3,1,1,10,30,30,10,1,1,31,310,930,310,31,1,1,
%T A172364 94,2914,29140,29140,2914,94,1,1,285,26790,830490,2768300,830490,26790,
%U A172364 285,1,1,865,246525,23173350,239457950,239457950,23173350,246525,865,1
%N A172364 Sixth (quartic) type of beta integer triangle sequence: a=3;f(n,a)=a*f(n-1,a)+f(n-4,a);c(n,a)=If[n == 0, 1, Product[f(i, a), {i, 1, n}]];t(n,m,a)=c(n, a)/(c(m, a)*c(n - m, a))
%C A172364 Row sums are:
%C A172364 {1, 2, 3, 4, 11, 82, 1614, 64298, 4483432, 525757382, 104749766604,...}.
%F A172364 a=3;
%F A172364 f(n,a)=a*f(n-1,a)+f(n-4,a);
%F A172364 c(n,a)=If[n == 0, 1, Product[f(i, a), {i, 1, n}]];
%F A172364 t(n,m,a)=c(n, a)/(c(m, a)*c(n - m, a))
%e A172364 {1},
%e A172364 {1, 1},
%e A172364 {1, 1, 1},
%e A172364 {1, 1, 1, 1},
%e A172364 {1, 3, 3, 3, 1},
%e A172364 {1, 10, 30, 30, 10, 1},
%e A172364 {1, 31, 310, 930, 310, 31, 1},
%e A172364 {1, 94, 2914, 29140, 29140, 2914, 94, 1},
%e A172364 {1, 285, 26790, 830490, 2768300, 830490, 26790, 285, 1},
%e A172364 {1, 865, 246525, 23173350, 239457950, 239457950, 23173350, 246525, 865, 1},
%e A172364 {1, 2626, 2271490, 647374650, 20284405700, 62881657670, 20284405700, 647374650, 2271490, 2626, 1}
%t A172364 Clear[f, c, a, t];
%t A172364 f[0, a_] := 0; f[1, a_] := 1; f[2, a_] := 1; f[3, a_] := 1;
%t A172364 f[n_, a_] := f[n, a] = a*f[n - 1, a] + f[n - 4, a];
%t A172364 c[n_, a_] := If[n == 0, 1, Product[f[i, a], {i, 1, n}]];
%t A172364 t[n_, m_, a_] := c[n, a]/(c[m, a]*c[n - m, a]);
%t A172364 Table[Table[Table[t[n, m, a], {m, 0, n}], {n, 0, 10}], {a, 1, 10}];
%t A172364 Table[Flatten[Table[Table[t[n, m, a], {m, 0, n}], {n, 0, 10}]], {a, 1, 10}]
%K A172364 nonn,tabl,uned,new
%O A172364 0,12
%A A172364 Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Feb 01 2010

%I A172363
%S A172363 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,1,1,3,6,6,6,3,1,1,4,12,24,24,
%T A172363 12,4,1,1,5,20,60,120,60,20,5,1,1,7,35,140,420,420,140,35,7,1,1,10,70,
%U A172363 350,1400,2100,1400,350,70,10,1
%N A172363 Sixth (quartic) type of beta integer triangle sequence: a=1;f(n,a)=a*f(n-1,a)+f(n-4,a);c(n,a)=If[n == 0, 1, Product[f(i, a), {i, 1, n}]];t(n,m,a)=c(n, a)/(c(m, a)*c(n - m, a))
%C A172363 Row sums are:
%C A172363 {1, 2, 3, 4, 5, 10, 26, 82, 292, 1206, 5762,...}.
%F A172363 a=1;
%F A172363 f(n,a)=a*f(n-1,a)+f(n-4,a);
%F A172363 c(n,a)=If[n == 0, 1, Product[f(i, a), {i, 1, n}]];
%F A172363 t(n,m,a)=c(n, a)/(c(m, a)*c(n - m, a))
%e A172363 {1},
%e A172363 {1, 1},
%e A172363 {1, 1, 1},
%e A172363 {1, 1, 1, 1},
%e A172363 {1, 1, 1, 1, 1},
%e A172363 {1, 2, 2, 2, 2, 1},
%e A172363 {1, 3, 6, 6, 6, 3, 1},
%e A172363 {1, 4, 12, 24, 24, 12, 4, 1},
%e A172363 {1, 5, 20, 60, 120, 60, 20, 5, 1},
%e A172363 {1, 7, 35, 140, 420, 420, 140, 35, 7, 1},
%e A172363 {1, 10, 70, 350, 1400, 2100, 1400, 350, 70, 10, 1}
%t A172363 Clear[f, c, a, t];
%t A172363 f[0, a_] := 0; f[1, a_] := 1; f[2, a_] := 1; f[3, a_] := 1;
%t A172363 f[n_, a_] := f[n, a] = a*f[n - 1, a] + f[n - 4, a];
%t A172363 c[n_, a_] := If[n == 0, 1, Product[f[i, a], {i, 1, n}]];
%t A172363 t[n_, m_, a_] := c[n, a]/(c[m, a]*c[n - m, a]);
%t A172363 Table[Table[Table[t[n, m, a], {m, 0, n}], {n, 0, 10}], {a, 1, 10}];
%t A172363 Table[Flatten[Table[Table[t[n, m, a], {m, 0, n}], {n, 0, 10}]], {a, 1, 10}]
%K A172363 nonn,tabl,uned,new
%O A172363 0,17
%A A172363 Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Feb 01 2010

%I A172362
%S A172362 1,33,594,7722,81081,729729,5837832,42532776,287096238,1818276174,
%T A172362 10909657044,62482581252,343654196886,1824010737318,9380626649064,
%U A172362 46903133245320,228652774570935,1089463220014455,5084161693400790
%N A172362 Binomial(n+10, 10)*3^n.
%C A172362 With a different offset, number of n-permutations (n>=10) of 4 objects: u, v, z, x with repetition allowed, containing exactly ten, (10) u's.
%t A172362 Table[Binomial[n + 10, 10]*3^n, {n, 0, 20}] Maple: seq(binomial(n+10, 10)*3^n, n=0..30);
%Y A172362 Cf. A027472, A036216, A036217, A036219, A036220, A036221, A036222, A036223,
%K A172362 nonn,new
%O A172362 0,2
%A A172362 Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Feb 01 2010

%I A172361
%S A172361 1,0,3,6,38,160,905,4830,28308,166992,1024758,6389460,40724244,
%T A172361 263385408,1728855843,11484066594,77130790880,523010474272,
%U A172361 3577392455780,24659960867256,171191809159176,1196062991373120
%N A172361 Number of walks within N^2 (the first quadrant of Z^2) starting and ending at (0,0), and consisting of n steps taken from {(-1, -1), (-1, 0), (-1, 1), (0, -1), (0, 1), (1, -1), (1, 0), (1, 1)}
%H A172361 M. Bouquet-Melou and M. Mishna, 2008. Walks with small steps in the quarter plane, <a href="http://arxiv.org/abs/0810.4387">ArXiv 0810.4387</a>.
%t A172361 aux[i_Integer, j_Integer, n_Integer] := Which[Min[i, j, n] < 0 || Max[i, j] > n, 0, n == 0,KroneckerDelta[i, j, n], True, aux[i, j, n] = aux[ -1 + i, -1 + j, -1 + n] + aux[ -1 + i, j, -1 + n] + aux[ -1 + i, 1 + j, -1 + n] + aux[i, -1 + j, -1 + n] + aux[i, 1 + j, -1 + n] + aux[1 + i, -1 + j, -1 + n] + aux[1 + i, j, -1 + n] + aux[1 + i, 1 + j, -1 + n]]; Table[aux[0, 0, n], {n, 0, 25}]
%K A172361 nonn,new
%O A172361 0,3
%A A172361 Manuel Kauers (manuel(AT)kauers.de), Feb 01 2010

%I A172360
%S A172360 1,1,1,1,1,1,1,1,1,1,1,6,6,6,1,1,6,36,36,6,1,1,11,66,396,66,11,1,1,36,
%T A172360 396,2376,2376,396,36,1,1,41,1476,16236,16236,16236,1476,41,1,1,91,3731,
%U A172360 134316,246246,246246,134316,3731,91,1,1,221,20111,824551,4947306
%N A172360 Fifth (third cubic) type of beta integer triangle sequence: a=5;f(n,a)=f(n-2,a)+a*f(n-3,a);c(n,a)=If[n == 0, 1, Product[f(i, a), {i, 1, n}]];t(n,m,a)=c(n, a)/(c(m, a)*c(n - m, a))
%C A172360 Row sums are:
%C A172360 {1, 2, 3, 4, 20, 86, 552, 5618, 51744, 768770, 20654441,...}.
%F A172360 a=5;
%F A172360 f(n,a)=a*f(n-2,a)+a*f(n-3,a);
%F A172360 c(n,a)=If[n == 0, 1, Product[f(i, a), {i, 1, n}]];
%F A172360 t(n,m,a)=c(n, a)/(c(m, a)*c(n - m, a))
%e A172360 {1},
%e A172360 {1, 1},
%e A172360 {1, 1, 1},
%e A172360 {1, 1, 1, 1},
%e A172360 {1, 6, 6, 6, 1},
%e A172360 {1, 6, 36, 36, 6, 1},
%e A172360 {1, 11, 66, 396, 66, 11, 1},
%e A172360 {1, 36, 396, 2376, 2376, 396, 36, 1},
%e A172360 {1, 41, 1476, 16236, 16236, 16236, 1476, 41, 1},
%e A172360 {1, 91, 3731, 134316, 246246, 246246, 134316, 3731, 91, 1},
%e A172360 {1, 221, 20111, 824551, 4947306, 9070061, 4947306, 824551, 20111, 221, 1}
%t A172360 Clear[f, c, a, t];
%t A172360 f[0, a_] := 0; f[1, a_] := 1; f[2, a_] := 1;
%t A172360 f[n_, a_] := f[n, a] = f[n - 2, a] + a*f[n - 3, a];
%t A172360 c[n_, a_] := If[n == 0, 1, Product[f[i, a], {i, 1, n}]];
%t A172360 t[n_, m_, a_] := c[n, a]/(c[m, a]*c[n - m, a]);
%t A172360 Table[Table[Table[t[n, m, a], {m, 0, n}], {n, 0, 10}], {a, 1, 10}];
%t A172360 Table[Flatten[Table[Table[t[n, m, a], {m, 0, n}], {n, 0, 10}]], {a, 1, 10}]
%Y A172360 cf. A010048
%K A172360 nonn,tabl,uned,new
%O A172360 0,12
%A A172360 Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Feb 01 2010

%I A172359
%S A172359 1,1,1,1,1,1,1,1,1,1,1,5,5,5,1,1,5,25,25,5,1,1,9,45,225,45,9,1,1,25,225,
%T A172359 1125,1125,225,25,1,1,29,725,6525,6525,6525,725,29,1,1,61,1769,44225,
%U A172359 79605,79605,44225,1769,61,1,1,129,7869,228201,1141005,2053809,1141005
%N A172359 Fifth (third cubic) type of beta integer triangle sequence: a=4;f(n,a)=f(n-2,a)+a*f(n-3,a);c(n,a)=If[n == 0, 1, Product[f(i, a), {i, 1, n}]];t(n,m,a)=c(n, a)/(c(m, a)*c(n - m, a))
%C A172359 Row sums are:
%C A172359 {1, 2, 3, 4, 17, 62, 335, 2752, 21085, 251322, 4808219,...}.
%F A172359 a=4;
%F A172359 f(n,a)=a*f(n-2,a)+a*f(n-3,a);
%F A172359 c(n,a)=If[n == 0, 1, Product[f(i, a), {i, 1, n}]];
%F A172359 t(n,m,a)=c(n, a)/(c(m, a)*c(n - m, a))
%e A172359 {1},
%e A172359 {1, 1},
%e A172359 {1, 1, 1},
%e A172359 {1, 1, 1, 1},
%e A172359 {1, 5, 5, 5, 1},
%e A172359 {1, 5, 25, 25, 5, 1},
%e A172359 {1, 9, 45, 225, 45, 9, 1},
%e A172359 {1, 25, 225, 1125, 1125, 225, 25, 1},
%e A172359 {1, 29, 725, 6525, 6525, 6525, 725, 29, 1},
%e A172359 {1, 61, 1769, 44225, 79605, 79605, 44225, 1769, 61, 1},
%e A172359 {1, 129, 7869, 228201, 1141005, 2053809, 1141005, 228201, 7869, 129, 1}
%t A172359 Clear[f, c, a, t];
%t A172359 f[0, a_] := 0; f[1, a_] := 1; f[2, a_] := 1;
%t A172359 f[n_, a_] := f[n, a] = f[n - 2, a] + a*f[n - 3, a];
%t A172359 c[n_, a_] := If[n == 0, 1, Product[f[i, a], {i, 1, n}]];
%t A172359 t[n_, m_, a_] := c[n, a]/(c[m, a]*c[n - m, a]);
%t A172359 Table[Table[Table[t[n, m, a], {m, 0, n}], {n, 0, 10}], {a, 1, 10}];
%t A172359 Table[Flatten[Table[Table[t[n, m, a], {m, 0, n}], {n, 0, 10}]], {a, 1, 10}]
%Y A172359 cf. A010048
%K A172359 nonn,tabl,uned,new
%O A172359 0,12
%A A172359 Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Feb 01 2010

%I A172358
%S A172358 1,1,1,1,1,1,1,1,1,1,1,3,3,3,1,1,3,9,9,3,1,1,5,15,45,15,5,1,1,9,45,135,
%T A172358 135,45,9,1,1,11,99,495,495,495,99,11,1,1,19,209,1881,3135,3135,1881,
%U A172358 209,19,1,1,29,551,6061,18183,30305,18183,6061,551,29,1
%N A172358 Fifth (third cubic) type of beta integer triangle sequence: a=2;f(n,a)=f(n-2,a)+a*f(n-3,a);c(n,a)=If[n == 0, 1, Product[f(i, a), {i, 1, n}]];t(n,m,a)=c(n, a)/(c(m, a)*c(n - m, a))
%C A172358 Row sums are:
%C A172358 {1, 2, 3, 4, 11, 26, 87, 380, 1707, 10490, 79955,...}.
%F A172358 a=2;
%F A172358 f(n,a)=a*f(n-1,a)+a*f(n-3,a);
%F A172358 c(n,a)=If[n == 0, 1, Product[f(i, a), {i, 1, n}]];
%F A172358 t(n,m,a)=c(n, a)/(c(m, a)*c(n - m, a))
%e A172358 {1},
%e A172358 {1, 1},
%e A172358 {1, 1, 1},
%e A172358 {1, 1, 1, 1},
%e A172358 {1, 3, 3, 3, 1},
%e A172358 {1, 3, 9, 9, 3, 1},
%e A172358 {1, 5, 15, 45, 15, 5, 1},
%e A172358 {1, 9, 45, 135, 135, 45, 9, 1},
%e A172358 {1, 11, 99, 495, 495, 495, 99, 11, 1},
%e A172358 {1, 19, 209, 1881, 3135, 3135, 1881, 209, 19, 1},
%e A172358 {1, 29, 551, 6061, 18183, 30305, 18183, 6061, 551, 29, 1}
%t A172358 Clear[f, c, a, t];
%t A172358 f[0, a_] := 0; f[1, a_] := 1; f[2, a_] := 1;
%t A172358 f[n_, a_] := f[n, a] = f[n - 2, a] + a*f[n - 3, a];
%t A172358 c[n_, a_] := If[n == 0, 1, Product[f[i, a], {i, 1, n}]];
%t A172358 t[n_, m_, a_] := c[n, a]/(c[m, a]*c[n - m, a]);
%t A172358 Table[Table[Table[t[n, m, a], {m, 0, n}], {n, 0, 10}], {a, 1, 10}];
%t A172358 Table[Flatten[Table[Table[t[n, m, a], {m, 0, n}], {n, 0, 10}]], {a, 1, 10}]
%Y A172358 cf. A010048
%K A172358 nonn,tabl,uned,new
%O A172358 0,12
%A A172358 Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Feb 01 2010

%I A172357
%S A172357 58,185,194,274,287,342,344,382,493,566,667,856,858,926,1012,1014,1157,
%T A172357 1165,1230,1232,1234,1267,1318,1385,1393,1418,1482,1484,1679,1681,1795,
%U A172357 1841,1915,1917,2060,2062,2064,2232,2340,2342,2567,2569,2627,2805,3013
%N A172357 n such the Liouville function L(n) take successively, from n, the values 1,-1,1,-1,1,-1
%C A172357 Short history of conjecture L(n) <= 0 for all n >= 2 by Deborah Tepper Haimo. George Polya conjectured 1919 that L(n) <= 0 for all n >= 2. The conjecture was generally deemed true for nearly 40 years, until 1958, when C. B. Haselgrove proved that L(n) > 0 for infinitely many n. In 1962, R. S. Lehman found that L(906180359) = 1 and in 1980, M. Tanaka discovered that the smallest counterexample of the Polya conjecture occurs when n = 906150257
%D A172357 H. Gupta, On a table of values of L(n), Proceedings of the Indian Academy of Sciences. Section A, 12 (1940), 407-409.
%D A172357 H. Gupta, A table of values of Liouville's function L(n), Research Bulletin of East Panjab University, No. 3 (Feb. 1950), 45-55. D. T. Haimo, Experimentation and Conjecture Are Not Enough, The American Mathematical Monthly Volume 102 Number 2, 1995, page 105.
%D A172357 R. S. Lehman, On Liouville's function, Math. Comp., 14 (1960), 311-320. M. Tanaka, A numerical investigation on cumulative sum of the Liouville function, Tokyo J. Math. 3 (1980), 187-189.
%H A172357 Peter Borwein, Ron Ferguson, and Michael J. Mossinghoff, <a href="http://www.ams.org/mcom/2008-77-263/S0025-5718-08-02036-X/home.html">Sign changes in sums of the Liouville function</a>. Math. Comp. 77 (2008), 1681-1694. [From T. D. Noe (noe(AT)sspectra.com), Jul 17 2009]
%F A172357 L(n) = (-1)^omega(n) where omega(n) is the number of distinct primes factors of n
%p A172357 with(numtheory): for n from 1 to 4300 do;if (-1)^bigomega(n)=1 and (-1)^bigomega(n+1) = -1 and (-1)^bigomega(n+2) = 1 and (-1)^bigomega(n+3) = -1 and (-1)^bigomega(n+4) = 1 and (-1)^bigomega(n+5) = -1 then print(n); else fi ; od;
%Y A172357 Cf. "A051470", "A028488", "A002819"
%K A172357 nonn,new
%O A172357 1,1
%A A172357 Michel Lagneau (mn.lagneau2(AT)orange.fr), Feb 01 2010

%I A172356
%S A172356 1,1,1,1,1,1,1,1,1,1,1,2,2,2,1,1,3,6,6,3,1,1,4,12,24,12,4,1,1,6,24,72,
%T A172356 72,24,6,1,1,9,54,216,324,216,54,9,1,1,13,117,702,1404,1404,702,117,13,
%U A172356 1,1,19,247,2223,6669,8892,6669,2223,247,19,1
%N A172356 Fourth (second cubic) type of beta integer triangle sequence: a=1;f(n,a)=a*f(n-1,a)+f(n-3,a);c(n,a)=If[n == 0, 1, Product[f(i, a), {i, 1, n}]];t(n,m,a)=c(n, a)/(c(m, a)*c(n - m, a))
%C A172356 Row sums are:
%C A172356 {1, 2, 3, 4, 8, 20, 58, 206, 884, 4474, 27210,...}.
%F A172356 a=1;
%F A172356 f(n,a)=a*f(n-1,a)+a*f(n-3,a);
%F A172356 c(n,a)=If[n == 0, 1, Product[f(i, a), {i, 1, n}]];
%F A172356 t(n,m,a)=c(n, a)/(c(m, a)*c(n - m, a))
%e A172356 {1},
%e A172356 {1, 1},
%e A172356 {1, 1, 1},
%e A172356 {1, 1, 1, 1},
%e A172356 {1, 2, 2, 2, 1},
%e A172356 {1, 3, 6, 6, 3, 1},
%e A172356 {1, 4, 12, 24, 12, 4, 1},
%e A172356 {1, 6, 24, 72, 72, 24, 6, 1},
%e A172356 {1, 9, 54, 216, 324, 216, 54, 9, 1},
%e A172356 {1, 13, 117, 702, 1404, 1404, 702, 117, 13, 1},
%e A172356 {1, 19, 247, 2223, 6669, 8892, 6669, 2223, 247, 19, 1}
%t A172356 Clear[f, c, a, t];
%t A172356 f[0, a_] := 0; f[1, a_] := 1; f[2, a_] := 1;
%t A172356 f[n_, a_] := f[n, a] = a*f[n - 1, a] + f[n - 3, a];
%t A172356 c[n_, a_] := If[n == 0, 1, Product[f[i, a], {i, 1, n}]];
%t A172356 t[n_, m_, a_] := c[n, a]/(c[m, a]*c[n - m, a]);
%t A172356 Table[Table[Table[t[n, m, a], {m, 0, n}], {n, 0, 10}], {a, 1, 10}];
%t A172356 Table[Flatten[Table[Table[t[n, m, a], {m, 0, n}], {n, 0, 10}]], {a, 1, 10}]
%Y A172356 cf. A010048
%K A172356 nonn,tabl,uned,new
%O A172356 0,12
%A A172356 Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Feb 01 2010

%I A172355
%S A172355 1,1,1,1,1,1,1,5,5,1,1,6,30,6,1,1,26,156,156,26,1,1,35,910,1092,910,35,
%T A172355 1,1,136,4760,24752,24752,4760,136,1,1,201,27336,191352,829192,191352,
%U A172355 27336,201,1,1,715,143715,3909048,22802780,22802780,3909048,143715,715
%N A172355 Third (cubic) type of beta integer triangle sequence: a=5;f(n,a)=a*f(n-2,a)+f(n-3,a);c(n,a)=If[n == 0, 1, Product[f(i, a), {i, 1, n}]];t(n,m,a)=c(n, a)/(c(m, a)*c(n - m, a))
%C A172355 a=2 :A010048.
%C A172355 Row sums are:
%C A172355 {1, 2, 3, 12, 44, 366, 2984, 59298, 1266972, 53712518, 2554657926,...}.
%C A172355 The majority of these triangles turn out rational except for a={1,2,5}.
%C A172355 Since these are new types of combination I name them a-form( in analogy to q-form)
%C A172355 or beta integer combinations.
%C A172355 Three types exist so far:
%C A172355 f(n,a)=a*f(n-1,a)+f(n-2,a);
%C A172355 f(n,a)=f(n-1,a)+a*f(n-2,a);
%C A172355 f(n,a)=a*f(n-2,a)+f(n-3,a).
%F A172355 a=5;
%F A172355 f(n,a)=a*f(n-2,a)+a*f(n-3,a);
%F A172355 c(n,a)=If[n == 0, 1, Product[f(i, a), {i, 1, n}]];
%F A172355 t(n,m,a)=c(n, a)/(c(m, a)*c(n - m, a))
%e A172355 {1},
%e A172355 {1, 1},
%e A172355 {1, 1, 1},
%e A172355 {1, 5, 5, 1},
%e A172355 {1, 6, 30, 6, 1},
%e A172355 {1, 26, 156, 156, 26, 1},
%e A172355 {1, 35, 910, 1092, 910, 35, 1},
%e A172355 {1, 136, 4760, 24752, 24752, 4760, 136, 1},
%e A172355 {1, 201, 27336, 191352, 829192, 191352, 27336, 201, 1},
%e A172355 {1, 715, 143715, 3909048, 22802780, 22802780, 3909048, 143715, 715, 1},
%e A172355 {1, 1141, 815815, 32795763, 743370628, 1000691230, 743370628, 32795763, 815815, 1141, 1}
%t A172355 Clear[f, c, a, t];
%t A172355 f[0, a_] := 0; f[1, a_] := 1; f[2, a_] := 1;
%t A172355 f[n_, a_] := f[n, a] = a*f[n - 2, a] + f[n - 3, a];
%t A172355 c[n_, a_] := If[n == 0, 1, Product[f[i, a], {i, 1, n}]];
%t A172355 t[n_, m_, a_] := c[n, a]/(c[m, a]*c[n - m, a]);
%t A172355 Table[Table[Table[t[n, m, a], {m, 0, n}], {n, 0, 10}], {a, 1, 10}];
%t A172355 Table[Flatten[Table[Table[t[n, m, a], {m, 0, n}], {n, 0, 10}]], {a, 1, 10}]
%Y A172355 A010048
%K A172355 nonn,tabl,uned,new
%O A172355 0,8
%A A172355 Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Feb 01 2010

%I A172354
%S A172354 195,1491,1547,2139,2715,2749,2751,2847,2967,3359,3615,3819,4011,4013,
%T A172354 4015,4047,4155,4547,5019,5449,5647,5741,5779,6351,6353,6355,6447,6547,
%U A172354 6563,6565,6567,6947,6959,6961,6963,7347,7503,7545,7683,8007,9339,10091
%N A172354 n such that the Moebius function take successively, from n, the values -1,0,-1,0,-1,0
%D A172354 M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, p. 826.
%D A172354 Deleglise, Marc and Rivat, Joel, Computing the summation of the Mobius function. Experiment. Math. 5 (1996), no. 4, 291-295.
%D A172354 G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, 1979, th. 262 and 287.
%H A172354 Ed Pegg Jr., <a href="http://www.maa.org/editorial/mathgames/mathgames_11_03_03.html">The Mobius function (and squarefree numbers)</a>
%H A172354 Primefan, <a href="http://primefan.tripod.com/Mertens2500.html"> Mobius and Mertens Values For n=1 to 2500 </a>
%H A172354 G. Villemin's Almanac of Numbers,<a href="http://villemin.gerard.free.fr/TABLES/aaaFArit/MobiusMe.htm"> Nombres de Moebius et de Mertens</a>
%p A172354 with(numtheory): for n from 1 to 15000 do;if mobius(n)= -1 and mobius(n+1) = 0 and mobius(n+2)= -1 and mobius(n+3)= 0 and mobius(n+4)= -1 and mobius(n+5) = 0 then print(n); else fi ; od;
%Y A172354 Moebius (or Mobius) function mu(n): "A008683" , "A007423" , "A002321" , "A002996"
%K A172354 nonn,new
%O A172354 1,1
%A A172354 Michel Lagneau (mn.lagneau2(AT)orange.fr), Feb 01 2010

%I A172353
%S A172353 1,1,1,1,1,1,1,1,1,1,1,2,2,2,1,1,2,4,4,2,1,1,3,6,12,6,3,1,1,4,12,24,24,
%T A172353 12,4,1,1,5,20,60,60,60,20,5,1,1,7,35,140,210,210,140,35,7,1,1,9,63,315,
%U A172353 630,945,630,315,63,9,1
%N A172353 Third (cubic) type of beta integer triangle sequence: a=1;f(n,a)=a*f(n-2,a)+f(n-3,a);c(n,a)=If[n == 0, 1, Product[f(i, a), {i, 1, n}]];t(n,m,a)=c(n, a)/(c(m, a)*c(n - m, a))
%C A172353 Row sums are:
%C A172353 {1, 2, 3, 4, 8, 14, 32, 82, 232, 786, 2981,...}
%F A172353 a=1;
%F A172353 f(n,a)=a*f(n-2,a)+a*f(n-3,a);
%F A172353 c(n,a)=If[n == 0, 1, Product[f(i, a), {i, 1, n}]];
%F A172353 t(n,m,a)=c(n, a)/(c(m, a)*c(n - m, a))
%e A172353 {1},
%e A172353 {1, 1},
%e A172353 {1, 1, 1},
%e A172353 {1, 1, 1, 1},
%e A172353 {1, 2, 2, 2, 1},
%e A172353 {1, 2, 4, 4, 2, 1},
%e A172353 {1, 3, 6, 12, 6, 3, 1},
%e A172353 {1, 4, 12, 24, 24, 12, 4, 1},
%e A172353 {1, 5, 20, 60, 60, 60, 20, 5, 1},
%e A172353 {1, 7, 35, 140, 210, 210, 140, 35, 7, 1},
%e A172353 {1, 9, 63, 315, 630, 945, 630, 315, 63, 9, 1}
%t A172353 Clear[f, c, a, t];
%t A172353 f[0, a_] := 0; f[1, a_] := 1; f[2, a_] := 1;
%t A172353 f[n_, a_] := f[n, a] = a*f[n - 2, a] + f[n - 3, a];
%t A172353 c[n_, a_] := If[n == 0, 1, Product[f[i, a], {i, 1, n}]];
%t A172353 t[n_, m_, a_] := c[n, a]/(c[m, a]*c[n - m, a]);
%t A172353 Table[Table[Table[t[n, m, a], {m, 0, n}], {n, 0, 10}], {a, 1, 10}];
%t A172353 Table[Flatten[Table[Table[t[n, m, a], {m, 0, n}], {n, 0, 10}]], {a, 1, 10}]
%Y A172353 Cf. A010048
%K A172353 nonn,tabl,uned,new
%O A172353 0,12
%A A172353 Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Feb 01 2010

%I A172352
%S A172352 1,1,1,1,1,1,1,8,8,1,1,15,120,15,1,1,71,1065,1065,71,1,1,176,12496,
%T A172352 23430,12496,176,1,1,673,118448,1051226,1051226,118448,673,1,1,1905,
%U A172352 1282065,28205430,133505702,28205430,1282065,1905,1,1,6616,12603480
%N A172352 Second type of beta integer triangle sequence: a=7;f(n,a)=f(n-1,a)+a*f(n-2,a);c(n,a)=If[n == 0, 1, Product[f(i, a), {i, 1, n}]];t(n,m,a)=c(n, a)/(c(m, a)*c(n - m, a))
%C A172352 a = 1 : A010048;
%C A172352 a = 2 : A015109;
%C A172352 Row sums are:
%C A172352 {1, 2, 3, 18, 152, 2274, 48776, 2340696, 192484504, 27026705688, 6379354108992,...}
%F A172352 a=7;
%F A172352 f(n,a)=f(n-1,a)+a*f(n-2,a);
%F A172352 c(n,a)=If[n == 0, 1, Product[f(i, a), {i, 1, n}]];
%F A172352 t(n,m,a)=c(n, a)/(c(m, a)*c(n - m, a))
%e A172352 {1},
%e A172352 {1, 1},
%e A172352 {1, 1, 1},
%e A172352 {1, 8, 8, 1},
%e A172352 {1, 15, 120, 15, 1},
%e A172352 {1, 71, 1065, 1065, 71, 1},
%e A172352 {1, 176, 12496, 23430, 12496, 176, 1},
%e A172352 {1, 673, 118448, 1051226, 1051226, 118448, 673, 1},
%e A172352 {1, 1905, 1282065, 28205430, 133505702, 28205430, 1282065, 1905, 1},
%e A172352 {1, 6616, 12603480, 1060267755, 12440474992, 12440474992, 1060267755, 12603480, 6616, 1},
%e A172352 {1, 19951, 131995816, 31431503685, 1410226798667, 3495773472752, 1410226798667, 31431503685, 131995816, 19951, 1}
%t A172352 Clear[f, c, a, t];
%t A172352 f[0, a_] := 0; f[1, a_] := 1;
%t A172352 f[n_, a_] := f[n, a] = f[n - 1, a] + a*f[n - 2, a];
%t A172352 c[n_, a_] := If[n == 0, 1, Product[f[i, a], {i, 1, n}]];
%t A172352 t[n_, m_, a_] := c[n, a]/(c[m, a]*c[n - m, a]);
%t A172352 Table[Table[Table[t[n, m, a], {m, 0, n}], {n, 0, 10}], {a, 1, 10}];
%t A172352 Table[Flatten[Table[Table[t[n, m, a], {m, 0, n}], {n, 0, 10}]], {a, 1, 10}]
%Y A172352 A010048, A015109
%K A172352 nonn,tabl,uned,new
%O A172352 0,8
%A A172352 Roger L. Bagula and Gary W. Adamson (rlbagulatftn(AT)yahoo.com), Feb 01 2010

%I A172351
%S A172351 1,1,1,1,1,1,1,7,7,1,1,13,91,13,1,1,55,715,715,55,1,1,133,7315,13585,
%T A172351 7315,133,1,1,463,61579,483835,483835,61579,463,1,1,1261,583843,
%U A172351 11093017,46931995,11093017,583843,1261,1,1,4039,5093179,336877411
%N A172351 Second type of beta integer triangle sequence: a=6;f(n,a)=f(n-1,a)+a*f(n-2,a);c(n,a)=If[n == 0, 1, Product[f(i, a), {i, 1, n}]];t(n,m,a)=c(n, a)/(c(m, a)*c(n - m, a))
%C A172351 a = 1 : A010048;
%C A172351 a = 2 : A015109;
%C A172351 Row sums are:
%C A172351 {1, 2, 3, 16, 119, 1542, 28483, 1091756, 70288239, 7576979362, 1345651717403,...}
%F A172351 a=6;
%F A172351 f(n,a)=f(n-1,a)+a*f(n-2,a);
%F A172351 c(n,a)=If[n == 0, 1, Product[f(i, a), {i, 1, n}]];
%F A172351 t(n,m,a)=c(n, a)/(c(m, a)*c(n - m, a))
%e A172351 {1},
%e A172351 {1, 1},
%e A172351 {1, 1, 1},
%e A172351 {1, 7, 7, 1},
%e A172351 {1, 13, 91, 13, 1},
%e A172351 {1, 55, 715, 715, 55, 1},
%e A172351 {1, 133, 7315, 13585, 7315, 133, 1},
%e A172351 {1, 463, 61579, 483835, 483835, 61579, 463, 1},
%e A172351 {1, 1261, 583843, 11093017, 46931995, 11093017, 583843, 1261, 1},
%e A172351 {1, 4039, 5093179, 336877411, 3446515051, 3446515051, 336877411, 5093179, 4039, 1},
%e A172351 {1, 11605, 46872595, 8443763185, 300727873435, 727214675761, 300727873435, 8443763185, 46872595, 11605, 1}
%t A172351 Clear[f, c, a, t];
%t A172351 f[0, a_] := 0; f[1, a_] := 1;
%t A172351 f[n_, a_] := f[n, a] = f[n - 1, a] + a*f[n - 2, a];
%t A172351 c[n_, a_] := If[n == 0, 1, Product[f[i, a], {i, 1, n}]];
%t A172351 t[n_, m_, a_] := c[n, a]/(c[m, a]*c[n - m, a]);
%t A172351 Table[Table[Table[t[n, m, a], {m, 0, n}], {n, 0, 10}], {a, 1, 10}];
%t A172351 Table[Flatten[Table[Table[t[n, m, a], {m, 0, n}], {n, 0, 10}]], {a, 1, 10}]
%Y A172351 A010048, A015109
%K A172351 nonn,tabl,uned,new
%O A172351 0,8
%A A172351 Roger L. Bagula and Gary W. Adamson (rlbagulatftn(AT)yahoo.com), Feb 01 2010

%I A172350
%S A172350 1,1,1,1,1,1,1,6,6,1,1,11,66,11,1,1,41,451,451,41,1,1,96,3936,7216,3936,
%T A172350 96,1,1,301,28896,197456,197456,28896,301,1,1,781,235081,3761296,
%U A172350 14019376,3761296,235081,781,1,1,2286,1785366,89565861,781665696
%N A172350 Second type of beta integer triangle sequence: a=5;f(n,a)=f(n-1,a)+a*f(n-2,a);c(n,a)=If[n == 0, 1, Product[f(i, a), {i, 1, n}]];t(n,m,a)=c(n, a)/(c(m, a)*c(n - m, a))
%C A172350 a = 1 : A010048;
%C A172350 a = 2 : A015109;
%C A172350 Row sums are:
%C A172350 {1, 2, 3, 14, 90, 986, 15282, 453308, 22013694, 1746038420, 222562828116,...}
%F A172350 a=5;
%F A172350 f(n,a)=f(n-1,a)+a*f(n-2,a);
%F A172350 c(n,a)=If[n == 0, 1, Product[f(i, a), {i, 1, n}]];
%F A172350 t(n,m,a)=c(n, a)/(c(m, a)*c(n - m, a))
%e A172350 {1},
%e A172350 {1, 1},
%e A172350 {1, 1, 1},
%e A172350 {1, 6, 6, 1},
%e A172350 {1, 11, 66, 11, 1},
%e A172350 {1, 41, 451, 451, 41, 1},
%e A172350 {1, 96, 3936, 7216, 3936, 96, 1},
%e A172350 {1, 301, 28896, 197456, 197456, 28896, 301, 1},
%e A172350 {1, 781, 235081, 3761296, 14019376, 3761296, 235081, 781, 1},
%e A172350 {1, 2286, 1785366, 89565861, 781665696, 781665696, 89565861, 1785366, 2286, 1},
%e A172350 {1, 6191, 14152626, 1842200151, 50409295041, 118031520096, 50409295041, 1842200151, 14152626, 6191, 1}
%t A172350 Clear[f, c, a, t];
%t A172350 f[0, a_] := 0; f[1, a_] := 1;
%t A172350 f[n_, a_] := f[n, a] = f[n - 1, a] + a*f[n - 2, a];
%t A172350 c[n_, a_] := If[n == 0, 1, Product[f[i, a], {i, 1, n}]];
%t A172350 t[n_, m_, a_] := c[n, a]/(c[m, a]*c[n - m, a]);
%t A172350 Table[Table[Table[t[n, m, a], {m, 0, n}], {n, 0, 10}], {a, 1, 10}];
%t A172350 Table[Flatten[Table[Table[t[n, m, a], {m, 0, n}], {n, 0, 10}]], {a, 1, 10}]
%Y A172350 A010048, A015109
%K A172350 nonn,tabl,uned,new
%O A172350 0,8
%A A172350 Roger L. Bagula and Gary W. Adamson (rlbagulatftn(AT)yahoo.com), Feb 01 2010

%I A172349
%S A172349 1,1,1,1,1,1,1,5,5,1,1,9,45,9,1,1,29,261,261,29,1,1,65,1885,3393,1885,
%T A172349 65,1,1,181,11765,68237,68237,11765,181,1,1,441,79821,1037673,3343613,
%U A172349 1037673,79821,441,1,1,1165,513765,18598293,134321005,134321005
%N A172349 Second type of beta integer triangle sequence: a=4;f(n,a)=f(n-1,a)+a*f(n-2,a);c(n,a)=If[n == 0, 1, Product[f(i, a), {i, 1, n}]];t(n,m,a)=c(n, a)/(c(m, a)*c(n - m, a))
%C A172349 a = 1 : A010048;
%C A172349 a = 2 : A015109;
%C A172349 Row sums are:
%C A172349 {1, 2, 3, 12, 65, 582, 7295, 160368, 5579485, 306868458, 26280601275,...}
%F A172349 a=4;
%F A172349 f(n,a)=f(n-1,a)+a*f(n-2,a);
%F A172349 c(n,a)=If[n == 0, 1, Product[f(i, a), {i, 1, n}]];
%F A172349 t(n,m,a)=c(n, a)/(c(m, a)*c(n - m, a))
%e A172349 {1},
%e A172349 {1, 1},
%e A172349 {1, 1, 1},
%e A172349 {1, 5, 5, 1},
%e A172349 {1, 9, 45, 9, 1},
%e A172349 {1, 29, 261, 261, 29, 1},
%e A172349 {1, 65, 1885, 3393, 1885, 65, 1},
%e A172349 {1, 181, 11765, 68237, 68237, 11765, 181, 1},
%e A172349 {1, 441, 79821, 1037673, 3343613, 1037673, 79821, 441, 1},
%e A172349 {1, 1165, 513765, 18598293, 134321005, 134321005, 18598293, 513765, 1165, 1},
%e A172349 {1, 2929, 3412285, 300963537, 6052711133, 13566421505, 6052711133, 300963537, 3412285, 2929, 1}
%t A172349 Clear[f, c, a, t];
%t A172349 f[0, a_] := 0; f[1, a_] := 1;
%t A172349 f[n_, a_] := f[n, a] = f[n - 1, a] + a*f[n - 2, a];
%t A172349 c[n_, a_] := If[n == 0, 1, Product[f[i, a], {i, 1, n}]];
%t A172349 t[n_, m_, a_] := c[n, a]/(c[m, a]*c[n - m, a]);
%t A172349 Table[Table[Table[t[n, m, a], {m, 0, n}], {n, 0, 10}], {a, 1, 10}];
%t A172349 Table[Flatten[Table[Table[t[n, m, a], {m, 0, n}], {n, 0, 10}]], {a, 1, 10}]
%Y A172349 A010048, A015109
%K A172349 nonn,tabl,uned,new
%O A172349 0,8
%A A172349 Roger L. Bagula and Gary W. Adamson (rlbagulatftn(AT)yahoo.com), Feb 01 2010

%I A172348
%S A172348 2,6,13,26,48,75,103,135,199,270,338,443,508,581,706,878,1001,1124,1305,
%T A172348 1413,1565,1764,1978,2299,2571,2724,2886,3052,3213,3710,4259,4581,4859,
%U A172348 5259,5668,5954,6409,6797,7184,7696,8029,8515,9062,9325,9608
%N A172348 Indices k = k(n) of semiprimes a(k) = prime(n) * prime(n+1) (A006094)
%C A172348 prime(n) * prime(n+1) is the product of two SUCCESSIVE primes
%D A172348 Edmund Landau, Handbuch der Lehre von der Verteilung der Primzahlen, Band I, B. G. Teubner, Leipzig u. Berlin, 1909
%D A172348 Derrick H. Lehmer, Guide to Tables in the Theory of Numbers Washington, D.C. 1941
%e A172348 6 = 2 x 3 = prime(1) x prime(2) = semiprime(2) => k(1) = 2
%e A172348 15 = 3 x 5 = prime(2) x prime(3) = semiprime(6) => k(2) = 6
%e A172348 35 = 5 x 7 = prime(3) x prime(4) = semiprime(13) => k(3) = 13
%Y A172348 A001358, A006094, A006881
%K A172348 base,nonn,uned,new
%O A172348 1,1
%A A172348 Ulrich Krug (leuchtfeuer37(AT)gmx.de), Feb 01 2010

%I A172347
%S A172347 1,1,1,1,1,1,1,4,4,1,1,7,28,7,1,1,19,133,133,19,1,1,40,760,1330,760,40,
%T A172347 1,1,97,3880,18430,18430,3880,97,1,1,217,21049,210490,571330,210490,
%U A172347 21049,217,1,1,508,110236,2673223,15275560,15275560,2673223,110236,508
%N A172347 Second type of beta integer triangle sequence: a=3;f(n,a)=f(n-1,a)+a*f(n-2,a);c(n,a)=If[n == 0, 1, Product[f(i, a), {i, 1, n}]];t(n,m,a)=c(n, a)/(c(m, a)*c(n - m, a))
%C A172347 a = 1 : A010048;
%C A172347 a = 2 : A015109;
%C A172347 Row sums are:
%C A172347 {1, 2, 3, 10, 44, 306, 2932, 44816, 1034844, 36119056, 1882089488,...}
%F A172347 a=3;
%F A172347 f(n,a)=f(n-1,a)+a*f(n-2,a);
%F A172347 c(n,a)=If[n == 0, 1, Product[f(i, a), {i, 1, n}]];
%F A172347 t(n,m,a)=c(n, a)/(c(m, a)*c(n - m, a))
%e A172347 {1},
%e A172347 {1, 1},
%e A172347 {1, 1, 1},
%e A172347 {1, 4, 4, 1},
%e A172347 {1, 7, 28, 7, 1},
%e A172347 {1, 19, 133, 133, 19, 1},
%e A172347 {1, 40, 760, 1330, 760, 40, 1},
%e A172347 {1, 97, 3880, 18430, 18430, 3880, 97, 1},
%e A172347 {1, 217, 21049, 210490, 571330, 210490, 21049, 217, 1},
%e A172347 {1, 508, 110236, 2673223, 15275560, 15275560, 2673223, 110236, 508, 1},
%e A172347 {1, 1159, 588772, 31940881, 442609351, 931809160, 442609351, 31940881, 588772, 1159, 1}
%t A172347 Clear[f, c, a, t];
%t A172347 f[0, a_] := 0; f[1, a_] := 1;
%t A172347 f[n_, a_] := f[n, a] = f[n - 1, a] + a*f[n - 2, a];
%t A172347 c[n_, a_] := If[n == 0, 1, Product[f[i, a], {i, 1, n}]];
%t A172347 t[n_, m_, a_] := c[n, a]/(c[m, a]*c[n - m, a]);
%t A172347 Table[Table[Table[t[n, m, a], {m, 0, n}], {n, 0, 10}], {a, 1, 10}];
%t A172347 Table[Flatten[Table[Table[t[n, m, a], {m, 0, n}], {n, 0, 10}]], {a, 1, 10}]
%Y A172347 A010048, A015109
%K A172347 nonn,tabl,uned,new
%O A172347 0,8
%A A172347 Roger L. Bagula and Gary W. Adamson (rlbagulatftn(AT)yahoo.com), Feb 01 2010

%I A172346
%S A172346 1,1,1,1,8,1,1,65,65,1,1,528,4290,528,1,1,4289,283074,283074,4289,1,1,
%T A172346 34840,18678595,151727664,18678595,34840,1,1,283009,1232504195,
%U A172346 81326315267,81326315267,1232504195,283009,1,1,2298912,81326598276
%N A172346 Beta integer triangle sequence: a=8;f(n,a)=a*f(n-1,a)+f(n-2,a);c(n,a)=If[n == 0, 1, Product[f(i, a), {i, 1, n}]];t(n,m,a)=c(n, a)/(c(m, a)*c(n - m, a))
%C A172346 a = 1 : A010048.
%C A172346 a = 2 : A099927.
%C A172346 a = 3 : A172339.
%C A172346 a = 4 : A034802.
%C A172346 These are in complete analogy to q-form combinations.
%C A172346 Row sums are:
%C A172346 {1, 2, 10, 132, 5348, 574728, 189154536, 165118204944, 441439547818768,
%C A172346 3130197658239760416, 67978275921898969849504,...}
%F A172346 a=8;
%F A172346 f(n,a)=a*f(n-1,a)+f(n-2,a);
%F A172346 c(n,a)=If[n == 0, 1, Product[f(i, a), {i, 1, n}]];
%F A172346 t(n,m,a)=c(n, a)/(c(m, a)*c(n - m, a))
%e A172346 {1},
%e A172346 {1, 1},
%e A172346 {1, 8, 1},
%e A172346 {1, 65, 65, 1},
%e A172346 {1, 528, 4290, 528, 1},
%e A172346 {1, 4289, 283074, 283074, 4289, 1},
%e A172346 {1, 34840, 18678595, 151727664, 18678595, 34840, 1},
%e A172346 {1, 283009, 1232504195, 81326315267, 81326315267, 1232504195, 283009, 1},
%e A172346 {1, 2298912, 81326598276, 43591056675936, 354094776672518, 43591056675936, 81326598276, 2298912, 1},
%e A172346 {1, 18674305, 5366322982020, 23364887704899972, 1541728575073323910, 1541728575073323910, 23364887704899972, 5366322982020, 18674305, 1},
%e A172346 {1, 151693352, 354095990215045, 12523623400880762016, 6712685861855802229898, 54527856243193320048880, 6712685861855802229898, 12523623400880762016, 354095990215045, 151693352, 1}
%t A172346 Clear[f, c, a, t];
%t A172346 f[0, a_] := 0; f[1, a_] := 1;
%t A172346 f[n_, a_] := f[n, a] = a*f[n - 1, a] + f[n - 2, a];
%t A172346 c[n_, a_] := If[n == 0, 1, Product[f[i, a], {i, 1, n}]];
%t A172346 t[n_, m_, a_] := c[n, a]/(c[m, a]*c[n - m, a]);
%t A172346 Table[Table[Table[t[n, m, a], {m, 0, n}], {n, 0, 10}], {a, 1, 10}];
%t A172346 Table[Flatten[Table[Table[t[n, m, a], {m, 0, n}], {n, 0, 10}]], {a, 1, 10}]
%Y A172346 A010048, A099927, A172339, A034802
%K A172346 nonn,tabl,uned,new
%O A172346 0,5
%A A172346 Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Feb 01 2010

%I A172345
%S A172345 1,1,1,1,7,1,1,50,50,1,1,357,2550,357,1,1,2549,129999,129999,2549,1,1,
%T A172345 18200,6627400,47319636,6627400,18200,1,1,129949,337867400,17224480052,
%U A172345 17224480052,337867400,129949,1,1,927843,17224610001,6269758040364
%N A172345 Beta integer triangle sequence: a=7;f(n,a)=a*f(n-1,a)+f(n-2,a);c(n,a)=If[n == 0, 1, Product[f(i, a), {i, 1, n}]];t(n,m,a)=c(n, a)/(c(m, a)*c(n - m, a))
%C A172345 a = 1 : A010048.
%C A172345 a = 2 : A099927.
%C A172345 a = 3 : A172339.
%C A172345 a = 4 : A034802.
%C A172345 These are in complete analogy to q-form combinations.
%C A172345 Row sums are:
%C A172345 {1, 2, 9, 102, 3266, 265098, 60610838, 35124954804, 57340390811566,
%C A172345 237262009585104396, 2765506241462282647452,...}
%F A172345 a=7;
%F A172345 f(n,a)=a*f(n-1,a)+f(n-2,a);
%F A172345 c(n,a)=If[n == 0, 1, Product[f(i, a), {i, 1, n}]];
%F A172345 t(n,m,a)=c(n, a)/(c(m, a)*c(n - m, a))
%e A172345 {1},
%e A172345 {1, 1},
%e A172345 {1, 7, 1},
%e A172345 {1, 50, 50, 1},
%e A172345 {1, 357, 2550, 357, 1},
%e A172345 {1, 2549, 129999, 129999, 2549, 1},
%e A172345 {1, 18200, 6627400, 47319636, 6627400, 18200, 1},
%e A172345 {1, 129949, 337867400, 17224480052, 17224480052, 337867400, 129949, 1},
%e A172345 {1, 927843, 17224610001, 6269758040364, 44766423655148, 6269758040364, 17224610001, 927843, 1},
%e A172345 {1, 6624850, 878117242650, 2282209151302497, 116347917517382200, 116347917517382200, 2282209151302497, 878117242650, 6624850, 1},
%e A172345 {1, 47301793, 44766754765150, 830730400831221429, 302388192878477292653, 2159068305370061485400, 302388192878477292653, 830730400831221429, 44766754765150, 47301793, 1}
%t A172345 Clear[f, c, a, t];
%t A172345 f[0, a_] := 0; f[1, a_] := 1;
%t A172345 f[n_, a_] := f[n, a] = a*f[n - 1, a] + f[n - 2, a];
%t A172345 c[n_, a_] := If[n == 0, 1, Product[f[i, a], {i, 1, n}]];
%t A172345 t[n_, m_, a_] := c[n, a]/(c[m, a]*c[n - m, a]);
%t A172345 Table[Table[Table[t[n, m, a], {m, 0, n}], {n, 0, 10}], {a, 1, 10}];
%t A172345 Table[Flatten[Table[Table[t[n, m, a], {m, 0, n}], {n, 0, 10}]], {a, 1, 10}]
%Y A172345 A010048, A099927, A172339, A034802
%K A172345 nonn,tabl,uned,new
%O A172345 0,5
%A A172345 Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Feb 01 2010

%I A172342
%S A172342 1,1,1,1,5,1,1,26,26,1,1,135,702,135,1,1,701,18927,18927,701,1,1,3640,
%T A172342 510328,2649780,510328,3640,1,1,18901,13759928,370988828,370988828,
%U A172342 13759928,18901,1,1,98145,371007729,51941082060,269708877956
%N A172342 Beta integer triangle sequence: a=5;f(n,a)=a*f(n-1,a)+f(n-2,a);c(n,a)=If[n == 0, 1, Product[f(i, a), {i, 1, n}]];t(n,m,a)=c(n, a)/(c(m, a)*c(n - m, a))
%C A172342 a = 1 : A010048.
%C A172342 a = 2 : A099927.
%C A172342 a = 3 : A172339.
%C A172342 a = 4 : A034802.
%C A172342 These are in complete analogy to q-form combinations.
%C A172342 Row sums are:
%C A172342 {1, 2, 7, 54, 974, 39258, 3677718, 769535316, 374333253826, 406720191959532,
%C A172342 1027328001602216932,...}
%F A172342 a=5;
%F A172342 f(n,a)=a*f(n-1,a)+f(n-2,a);
%F A172342 c(n,a)=If[n == 0, 1, Product[f(i, a), {i, 1, n}]];
%F A172342 t(n,m,a)=c(n, a)/(c(m, a)*c(n - m, a))
%e A172342 {1},
%e A172342 {1, 1},
%e A172342 {1, 5, 1},
%e A172342 {1, 26, 26, 1},
%e A172342 {1, 135, 702, 135, 1},
%e A172342 {1, 701, 18927, 18927, 701, 1},
%e A172342 {1, 3640, 510328, 2649780, 510328, 3640, 1},
%e A172342 {1, 18901, 13759928, 370988828, 370988828, 13759928, 18901, 1},
%e A172342 {1, 98145, 371007729, 51941082060, 269708877956, 51941082060, 371007729, 98145, 1},
%e A172342 {1, 509626, 10003448754, 7272122496129, 196077969525256, 196077969525256, 7272122496129, 10003448754, 509626, 1},
%e A172342 {1, 2646275, 269722108630, 1018149090441975, 142548414506990885, 740194334957841400, 142548414506990885, 1018149090441975, 269722108630, 2646275, 1}
%t A172342 Clear[f, c, a, t];
%t A172342 f[0, a_] := 0; f[1, a_] := 1;
%t A172342 f[n_, a_] := f[n, a] = a*f[n - 1, a] + f[n - 2, a];
%t A172342 c[n_, a_] := If[n == 0, 1, Product[f[i, a], {i, 1, n}]];
%t A172342 t[n_, m_, a_] := c[n, a]/(c[m, a]*c[n - m, a]);
%t A172342 Table[Table[Table[t[n, m, a], {m, 0, n}], {n, 0, 10}], {a, 1, 10}];
%t A172342 Table[Flatten[Table[Table[t[n, m, a], {m, 0, n}], {n, 0, 10}]], {a, 1, 10}]
%Y A172342 A010048, A099927, A172339, A034802
%K A172342 nonn,tabl,uned,new
%O A172342 0,5
%A A172342 Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Feb 01 2010

%I A172344
%S A172344 609,1827,4263,5481,12789,16443,17661,29841,38367,49329,52983,89523,
%T A172344 115101,123627,147987,158949,208887,268569,345303,370881,443961,476847,
%U A172344 512169,626661,805707,865389,1035909,1112643,1331883,1430541,1462209
%N A172344 Numbers n such that phi(n)/n = 16/29
%C A172344 Also, numbers n such that phi(n)/n = 16/29, where phi is the Euler totient function "A000010". - Lekraj Beedassy (blekraj(AT)yahoo.com), Jul 18 2008
%D A172344 M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 840.
%D A172344 T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 24.
%D A172344 J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 189, p. 57, Ellipses, Paris 2008.
%H A172344 Daniel Forgues, Table of n, <a href="http://www.research.att.com/~njas/sequences/b000010.txt">phi(n) for n=1..100000</a>
%H A172344 H. Fripertinger, <a href="http://www.uni-graz.at/~fripert/fga/k1euler.html">The Euler phi function</a>
%H A172344 Graeme McRae, <a href="http://2000clicks.com/MathHelp/NumberFactorsTotientFunction.htm">Euler's Totient Function</a>
%p A172344 with(numtheory):for n from 1 to 1500000 do; if evalf(phi(n)/n) = evalf(16/29) then print(n); else fi ; od;
%Y A172344 phi(n), "A000010" : <a href="http://www.research.att.com/~njas/sequences/Sindx_To.html#totient">see totient function phi(n)</a> Adjacent sequences: "A033850"
%K A172344 nonn,new
%O A172344 1,1
%A A172344 Michel Lagneau (mn.lagneau2(AT)orange.fr), Feb 01 2010

%I A172343
%S A172343 1,1,1,1,6,1,1,37,37,1,1,228,1406,228,1,1,1405,53390,53390,1405,1,1,
%T A172343 8658,2027415,12493260,2027415,8658,1,1,53353,76988379,2923477635,
%U A172343 2923477635,76988379,53353,1,1,328776,2923530988,684106251192
%N A172343 Beta integer triangle sequence: a=6;f(n,a)=a*f(n-1,a)+f(n-2,a);c(n,a)=If[n == 0, 1, Product[f(i, a), {i, 1, n}]];t(n,m,a)=c(n, a)/(c(m, a)*c(n - m, a))
%C A172343 a = 1 : A010048.
%C A172343 a = 2 : A099927.
%C A172343 a = 3 : A172339.
%C A172343 a = 4 : A034802.
%C A172343 a = 5 : A172342.
%C A172343 These are in complete analogy to q-form combinations. Row sums are:
%C A172343 {1, 2, 8, 76, 1864, 109592, 16565408, 6001038736, 5589714971584,
%C A172343 12478331908166432, 71624411004755875328,...}
%F A172343 a=6;
%F A172343 f(n,a)=a*f(n-1,a)+f(n-2,a);
%F A172343 c(n,a)=If[n == 0, 1, Product[f(i, a), {i, 1, n}]];
%F A172343 t(n,m,a)=c(n, a)/(c(m, a)*c(n - m, a))
%e A172343 {1},
%e A172343 {1, 1},
%e A172343 {1, 6, 1},
%e A172343 {1, 37, 37, 1},
%e A172343 {1, 228, 1406, 228, 1},
%e A172343 {1, 1405, 53390, 53390, 1405, 1},
%e A172343 {1, 8658, 2027415, 12493260, 2027415, 8658, 1},
%e A172343 {1, 53353, 76988379, 2923477635, 2923477635, 76988379, 53353, 1},
%e A172343 {1, 328776, 2923530988, 684106251192, 4215654749670, 684106251192, 2923530988, 328776, 1},
%e A172343 {1, 2026009, 111017189164, 160083786309916, 6078971148558126, 6078971148558126, 160083786309916, 111017189164, 2026009, 1}, {1, 12484830, 4215729657245, 37460290102442760, 8765872183489599010, 54017737626087507636, 8765872183489599010, 37460290102442760, 4215729657245, 12484830, 1}
%t A172343 Clear[f, c, a, t];
%t A172343 f[0, a_] := 0; f[1, a_] := 1;
%t A172343 f[n_, a_] := f[n, a] = a*f[n - 1, a] + f[n - 2, a];
%t A172343 c[n_, a_] := If[n == 0, 1, Product[f[i, a], {i, 1, n}]];
%t A172343 t[n_, m_, a_] := c[n, a]/(c[m, a]*c[n - m, a]);
%t A172343 Table[Table[Table[t[n, m, a], {m, 0, n}], {n, 0, 10}], {a, 1, 10}];
%t A172343 Table[Flatten[Table[Table[t[n, m, a], {m, 0, n}], {n, 0, 10}]], {a, 1, 10}]
%Y A172343 A010048, A099927, A172339, A034802, A172342
%K A172343 nonn,tabl,uned,new
%O A172343 0,5
%A A172343 Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Feb 01 2010

%I A172341
%S A172341 2,17,536,67073,33882272,68971196417,563671581328256,
%T A172341 18454625673706995713,2418183685055827278828032,
%U A172341 1267705002132225806674491867137,2658490852011206228820564373456050176
%N A172341 a(n)=((2^n+1)^n+(2^n-1)^n)/2
%t A172341 Table[((2^x + 1)^x + (2^x - 1)^x)/2, {x, 1, 15}]
%K A172341 nonn,new
%O A172341 1,1
%A A172341 Artur Jasinski (grafix(AT)csl.pl), Feb 01 2010

%I A172340
%S A172340 1,8,193,16448,5253121,6447694208,30795721162753,576522325894105088,
%T A172340 42502811563960266915841,12379542064132015743721023488,
%U A172340 14278867426031486089248841370959873
%N A172340 a(n)=((2^n+1)^n-(2^n-1)^n)/2
%t A172340 Table[((2^x + 1)^x - (2^x - 1)^x)/2, {x, 1, 15}]
%K A172340 nonn,new
%O A172340 1,2
%A A172340 Artur Jasinski (grafix(AT)csl.pl), Feb 01 2010

%I A172339
%S A172339 1,1,1,1,3,1,1,10,10,1,1,33,110,33,1,1,109,1199,1199,109,1,1,360,13080,
%T A172339 43164,13080,360,1,1,1189,142680,1555212,1555212,142680,1189,1,1,3927,
%U A172339 1556401,56030436,185070228,56030436,1556401,3927,1,1,12970,16977730
%N A172339 beta integer triangle sequence: a=3;f(n,a)=a*f(n-1,a)+f(n-2,a);c(n,a)=If[n == 0, 1, Product[f(i, a), {i, 1, n}]];t(n,m,a)=c(n, a)/(c(m, a)*c(n - m, a))
%C A172339 a = 1 : A010048.
%C A172339 a = 2 : A099927.
%C A172339 These are in complete analogy to q-form combinations. Row sums are:
%C A172339 {1, 2, 5, 22, 178, 2618, 70046, 3398164, 300251758, 48114604076,
%C A172339 14041125439724,...}
%F A172339 a=3;
%F A172339 f(n,a)=a*f(n-1,a)+f(n-2,a);
%F A172339 c(n,a)=If[n == 0, 1, Product[f(i, a), {i, 1, n}]];
%F A172339 t(n,m,a)=c(n, a)/(c(m, a)*c(n - m, a))
%e A172339 {1},
%e A172339 {1, 1},
%e A172339 {1, 3, 1},
%e A172339 {1, 10, 10, 1},
%e A172339 {1, 33, 110, 33, 1},
%e A172339 {1, 109, 1199, 1199, 109, 1},
%e A172339 {1, 360, 13080, 43164, 13080, 360, 1},
%e A172339 {1, 1189, 142680, 1555212, 1555212, 142680, 1189, 1},
%e A172339 {1, 3927, 1556401, 56030436, 185070228, 56030436, 1556401, 3927, 1},
%e A172339 {1, 12970, 16977730, 2018652097, 22021659240, 22021659240, 2018652097, 16977730, 12970, 1},
%e A172339 {1, 42837, 185198630, 72727502001, 2620393935733, 8654512081320, 2620393935733, 72727502001, 185198630, 42837, 1}
%t A172339 Clear[f, c, a, t];
%t A172339 f[0, a_] := 0; f[1, a_] := 1;
%t A172339 f[n_, a_] := f[n, a] = a*f[n - 1, a] + f[n - 2, a];
%t A172339 c[n_, a_] := If[n == 0, 1, Product[f[i, a], {i, 1, n}]];
%t A172339 t[n_, m_, a_] := c[n, a]/(c[m, a]*c[n - m, a]);
%t A172339 Table[Table[Table[t[n, m, a], {m, 0, n}], {n, 0, 10}], {a, 1, 10}];
%t A172339 Table[Flatten[Table[Table[t[n, m, a], {m, 0, n}], {n, 0, 10}]], {a, 1, 10}]
%Y A172339 A010048, A099927
%K A172339 nonn,tabl,uned,new
%O A172339 0,5
%A A172339 Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Feb 01 2010

%I A172338
%S A172338 0,3,7,11,15,19,23,27,31,35,39,43,47,51,55,59,63,67,71,75,79,83,87,91,
%T A172338 95,99,103,107,111,115,119,123,126,130,134,138,142,146,150,154,158,162,
%U A172338 166,170,174,178,182,186,190,194,198,202,206,210,214,218
%N A172338 Floor(n*(sqrt(5)+sqrt(3))).
%C A172338 a(n)= n*3,9681187850686669899366200102371....
%K A172338 nonn,new
%O A172338 0,2
%A A172338 Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Feb 01 2010

%I A172337
%S A172337 0,5,11,17,23,29,35,41,47,53,59,65,71,77,83,89,95,101,107,113,119,125,
%T A172337 131,137,143,149,155,160,166,172,178,184,190,196,202,208,214,220,226,
%U A172337 232,238,244,250,256,262,268,274,280,286,292,298,304,310,316,321,327
%N A172337 Floor(n*(sqrt(11)+sqrt(7))).
%C A172337 a(n)= n*5,9623761014199904396165484903099.....
%K A172337 nonn,new
%O A172337 0,2
%A A172337 Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Feb 01 2010

%I A172336
%S A172336 0,5,10,15,20,25,30,35,40,45,50,55,60,65,70,75,80,85,90,95,100,105,110,
%T A172336 115,120,125,130,135,140,145,150,155,160,165,170,175,180,185,190,195,
%U A172336 200,205,210,215,220,225,230,235,240,245,250,256,261,266,271,276
%N A172336 Floor(n*(sqrt(13)+sqrt(2))).
%C A172336 a(n)=n*5,0197648378370843419209099916802......
%K A172336 nonn,new
%O A172336 0,2
%A A172336 Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Feb 01 2010

%I A172335
%S A172335 142,238,418,429,598,622,2985,3502,16269,22678,23188,27778,38494,46761,
%T A172335 48489,62235,74188,98745,110170,120345,129448,151677,187822,190888,
%U A172335 194818,205185,223685,235438,246934,249166
%N A172335 Numbers n such that n and n+17 have same sum of divisors.
%D A172335 M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 840.
%D A172335 J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 62, p. 22, Ellipses, Paris 2008.
%D A172335 W. Sierpi\'{n}ski, A Selection of Problems in the Theory of Numbers. Macmillan, NY, 1964, p. 110.
%H A172335 M. Abramowitz and I. A. Stegun, eds., <a href="http://www.nrbook.com/abramowitz_and_stegun/">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
%p A172335 with(numtheory):for n from 1 to 250000 do;if sigma(n) = sigma(n+17) then print(n); else fi ; od;
%Y A172335 "A000203" (sigma function), Adjacent sequences: "A015861", "A002961", "A015865", "A015867", "A015858", "A015859", "A015860"
%K A172335 nonn,new
%O A172335 1,1
%A A172335 Michel Lagneau (mn.lagneau2(AT)orange.fr), Feb 01 2010

%I A172334
%S A172334 0,5,10,16,21,26,32,37,42,48,53,58,64,69,74,80,85,90,96,101,106,112,117,
%T A172334 122,128,133,138,144,149,154,160,165,170,176,181,186,192,197,202,208,
%U A172334 213,218,224,229,234,240,245,250,256,261,266
%N A172334 Floor(n*(sqrt(13)+sqrt(3))).
%C A172334 a(n)= n*5,3376020830328665866466676089764....
%K A172334 nonn,new
%O A172334 0,2
%A A172334 Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Feb 01 2010

%I A172333
%S A172333 57,85,213,224,354,476,568,594,812,1218,1235,1316,1484,2103,2470,2492,
%T A172333 2643,2840,2996,3836,3978,4026,4544,4810,4844,5012,6125,6356,6524,7364,
%U A172333 7532,7648,8876,9272,9328,10098,11107,11797,12572,12594,13412,13640
%N A172333 Numbers n such that n and n+22 have same sum of divisors.
%D A172333 M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 840.
%D A172333 J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 62, p. 22, Ellipses, Paris 2008.
%D A172333 W. Sierpi\'{n}ski, A Selection of Problems in the Theory of Numbers. Macmillan, NY, 1964, p. 110.
%H A172333 M. Abramowitz and I. A. Stegun, eds., <a href="http://www.nrbook.com/abramowitz_and_stegun/">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
%p A172333 with(numtheory):for n from 1 to 20000 do;if sigma(n) = sigma(n+22) then print(n); else fi ; od;
%Y A172333 "A000203" (sigma function), Adjacent sequences: "A015861", "A002961", "A015865", "A015867", "A015858", "A015859", "A015860"
%K A172333 nonn,new
%O A172333 1,1
%A A172333 Michel Lagneau (mn.lagneau2(AT)orange.fr), Feb 01 2010

%I A172332
%S A172332 0,5,11,17,23,29,35,40,46,52,58,64,70,75,81,87,93,99,105,110,116,122,
%T A172332 128,134,140,146,151,157,163,169,175,181,186,192,198,204,210,216,221,
%U A172332 227,233,239,245,251,257,262,268,274,280,286,292
%N A172332 Floor(n*(sqrt(13)+sqrt(5))).
%C A172332 a(n)= n*5,8416192529637789895283949362018...
%K A172332 nonn,new
%O A172332 0,2
%A A172332 Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Feb 01 2010

%I A172331
%S A172331 0,6,12,18,25,31,37,43,50,56,62,68,75,81,87,93,100,106,112,118,125,131,
%T A172331 137,143,150,156,162,168,175,181,187,193,200,206,212,218,225,231,237,
%U A172331 243,250,256,262,268,275,281,287,293,300,306,312
%N A172331 Floor(n*(sqrt(13)+sqrt(7))).
%C A172331 a)n)= n*6,2513025865285798836208370211098....
%K A172331 nonn,new
%O A172331 0,2
%A A172331 Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Feb 01 2010

%I A172330
%S A172330 0,6,13,20,27,34,41,48,55,62,69,76,83,89,96,103,110,117,124,131,138,145,
%T A172330 152,159,166,173,179,186,193,200,207,214,221,228,235,242,249,256,263,
%U A172330 269,276,283,290,297,304,311,318,325,332,339,346
%N A172330 Floor(n*(sqrt(13)+sqrt(11))).
%C A172330 a(n)= n*6,9221760658193891422341540041412....
%K A172330 nonn,new
%O A172330 0,2
%A A172330 Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Feb 01 2010

%I A172329
%S A172329 0,4,9,14,18,23,28,33,37,42,47,52,56,61,66,70,75,80,85,89,94,99,104,108,
%T A172329 113,118,123,127,132,137,141,146,151,156,160,165,170,175,179,184,189,
%U A172329 193,198,203,208,212,217,222,227,231,236
%N A172329 Floor(n*(sqrt(11)+sqrt(2))).
%C A172329 a(n)=n*4,7308383527284948979166214608804...
%K A172329 nonn,new
%O A172329 0,2
%A A172329 Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Feb 01 2010

%I A172328
%S A172328 0,5,10,15,20,25,30,35,40,45,50,55,60,65,70,75,80,85,90,95,100,106,111,
%T A172328 116,121,126,131,136,141,146,151,156,161,166,171,176,181,186,191,196,
%U A172328 201,206,212,217,222,227,232,237,242,247,252
%N A172328 Floor(n*(sqrt(11)+sqrt(3))).
%C A172328 a(n)=n*5,0486755979242771426423790781766....
%K A172328 nonn,new
%O A172328 0,2
%A A172328 Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Feb 01 2010

%I A172327
%S A172327 0,5,11,16,22,27,33,38,44,49,55,61,66,72,77,83,88,94,99,105,111,116,122,
%T A172327 127,133,138,144,149,155,161,166,172,177,183,188,194,199,205,211,216,
%U A172327 222,227,233,238,244,249,255,260,266,272,277
%N A172327 Floor(n*(sqrt(11)+sqrt(5))).
%C A172327 a(n)=n*5,552692767855189545524106405402....
%K A172327 nonn,new
%O A172327 0,2
%A A172327 Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Feb 01 2010

%I A172326
%S A172326 0,4,8,12,16,20,24,28,32,36,40,44,48,52,56,60,64,69,73,77,81,85,89,93,
%T A172326 97,101,105,109,113,117,121,125,129,133,138,142,146,150,154,158,162,166,
%U A172326 170,174,178,182,186,190,194,198,202
%N A172326 Floor(n*(sqrt(7)+sqrt(2))).
%C A172326 a(n)=n*4,059964873437685639303304477849....
%K A172326 nonn,new
%O A172326 0,2
%A A172326 Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Feb 01 2010

%I A172325
%S A172325 0,4,8,13,17,21,26,30,35,39,43,48,52,56,61,65,70,74,78,83,87,91,96,100,
%T A172325 105,109,113,118,122,126,131,135,140,144,148,153,157,161,166,170,175,
%U A172325 179,183,188,192,197,201,205,210,214,218
%N A172325 Floor(n*(sqrt(7)+sqrt(3))).
%C A172325 a(n)=n*4,3778021186334678840290620951451.....
%K A172325 nonn,new
%O A172325 0,2
%A A172325 Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Feb 01 2010

%I A172324
%S A172324 0,4,9,14,19,24,29,34,39,43,48,53,58,63,68,73,78,82,87,92,97,102,107,
%T A172324 112,117,122,126,131,136,141,146,151,156,161,165,170,175,180,185,190,
%U A172324 195,200,205,209,214,219,224,229,234,239,244
%N A172324 Floor(n*(sqrt(7)+sqrt(5))).
%C A172324 a(n)=n*4,8818192885643802869107894223705....
%K A172324 nonn,new
%O A172324 0,2
%A A172324 Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Feb 01 2010

%I A172323
%S A172323 0,3,7,10,14,18,21,25,29,32,36,40,43,47,51,54,58,62,65,69,73,76,80,83,
%T A172323 87,91,94,98,102,105,109,113,116,120,124,127,131,135,138,142,146,149,
%U A172323 153,156,160,164,167,171,175,178,182
%N A172323 Floor(n*(sqrt(5)+sqrt(2))).
%C A172323 a(n)=n*3,650281539872884745210862392941......
%K A172323 nonn,new
%O A172323 0,2
%A A172323 Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Feb 01 2010

%I A172322
%S A172322 0,15,4095,16777215,1099511627775,1152921504606846975,
%T A172322 19342813113834066795298815,5192296858534827628530496329220095,
%U A172322 22300745198530623141535718272648361505980415
%N A172322 Number of graphs with non-zero edge sets on an n x n square grid for n=(1,2,3,...).
%F A172322 a(n)=sum{i=1..2(n^2-1)}C(2(n^2-1),i), n = 1,2,...
%e A172322 For n=1, a(1)=0 since a minimum of a 2 x 2 grid is required to have edges.
%t A172322 Total[Table[Binomial[2(n^2-1), i], {i, 2(n^2-1)}]]
%K A172322 nonn,new
%O A172322 1,2
%A A172322 Alexander M Kerr (alexander.kerr(AT)aya.yale.edu), Jan 31 2010

%I A172319
%S A172319 1,2,4,8,16,32,64,128,256,512,1023,2044,4084,8160,16304,32576,65088,
%T A172319 130048,259840,519168,1037313,2072582,4141080,8274000,16531696,33030816,
%U A172319 65996544,131863040,263466240,526413312,1051789311
%N A172319 10-th column of A172119.
%F A172319 G.f f such that: f(z)=1/(1-2*z+z^(10)). a(n)=sum((-1)^j*binomial(n-k*j,n-(k+1)*j)*2^(n-(k+1)*j),j=0..floor(n/(k+1))). a(n+10)=2*a(n+9)-a(n).
%p A172319 for k from 0 to 20 do for n from 0 to 30 do b(n):=sum((-1)^j*binomial(n-k*j,n-(k+1)*j)*2^(n-(k+1)*j),j=0..floor(n/(k+1))):od:k: seq(b(n),n=0..30):od;
%Y A172319 Cf. A172318, A172317, A172316, A172119, A001949, A107066.
%K A172319 easy,nonn,new
%O A172319 0,2
%A A172319 Richard Choulet (richardchoulet(AT)yahoo.fr), Jan 31 2010

%I A172320
%S A172320 1,2,4,8,16,32,64,128,256,512,1024,2047,4092,8180,16352,32688,65344,
%T A172320 130624,261120,521984,1043456,2085888,4169729,8335366,16662552,33308752,
%U A172320 66584816,133104288,266077952,531894784,1063267584
%N A172320 11-th column of A172119.
%F A172320 a(n)=sum((-1)^j*binomial(n-k*j,n-(k+1)*j)*2^(n-(k+1)*j),j=0..floor(n/(k+1))) with k=10. G.f f such that: f(z)=1/(1-2*z+z^(11)). a(n+11)=2*a(n+10)-a(n).
%e A172320 a(12)=C(12,12)*2^12-C(2,1)*2^1=4092.
%p A172320 k:=10:taylor(1/(1-2*z+z^(k+1)),z=0,30); for k from 0 to 20 do for n from 0 to 30 do b(n):=sum((-1)^j*binomial(n-k*j,n-(k+1)*j)*2^(n-(k+1)*j),j=0..floor(n/(k+1))):od:k: seq(b(n),n=0..30):od;
%Y A172320 Cf. A172319, A172318, A172317, A172316, A172119, A001949, A107066.
%K A172320 easy,nonn,new
%O A172320 0,2
%A A172320 Richard Choulet (richardchoulet(AT)yahoo.fr), Jan 31 2010

%I A172318
%S A172318 1,2,4,8,16,32,64,128,256,511,1020,2036,4064,8112,16192,32320,64512,
%T A172318 128768,257025,513030,1024024,2043984,4079856,8143520,16254720,32444928,
%U A172318 64761088,129265151,258017272,515010520,1027977056
%N A172318 9-th column of the array A172119.
%F A172318 G.f f such that: f(z)=1/(1-2*z+z^9). a(n)=sum((-1)^j*binomial(n-k*j,n-(k+1)*j)*2^(n-(k+1)*j),j=0..floor(n/(k+1))) with k=8. Recurrence relation: a(n+9)=2*a(8)-a(n).
%e A172318 a(7)=C(7,7)*2^7=128. a(10)=C(10,10)*2^10-C(2,1)*2^1=1020.
%p A172318 for k from 0 to 20 do for n from 0 to 30 do b(n):=sum((-1)^j*binomial(n-k*j,n-(k+1)*j)*2^(n-(k+1)*j),j=0..floor(n/(k+1))):od:k: seq(b(n),n=0..30):od; k:=8:taylor(1/(1-2*z+z^(k+1)),z=0,30);
%Y A172318 Cf. A172317, A172316, A172119, A001949, A107066.
%K A172318 easy,nonn,new
%O A172318 0,2
%A A172318 Richard Choulet (richardchoulet(AT)yahoo.fr), Jan 31 2010

%I A172317
%S A172317 1,2,4,8,16,32,64,128,255,508,1012,2016,4016,8000,15936,31744,63233,
%T A172317 125958,250904,499792,995568,1983136,3950336,7868928,15674623,31223288,
%U A172317 62195672,123891552,246787536,491591936,979233536
%N A172317 8-th column of A172119.
%F A172317 The generating fonction is f such that: f(z)=1/(1-2*z+z^8). Recurrence relation: a(n+8)=2*a(n+7)-a(n). General term: a(n)=sum((-1)^j*binomial(n-k*j,n-(k+1)*j)*2^(n-(k+1)*j),j=0..floor(n/(k+1))) with k=7.
%e A172317 a(4)=C(4,4)*2^4=16. a(9)=C(9,9)*2^9-C(2,1)*2^1=512-4=508.
%p A172317 k:=7:taylor(1/(1-2*z+z^(k+1)),z=0,30); for k from 0 to 20 do for n from 0 to 30 do b(n):=sum((-1)^j*binomial(n-k*j,n-(k+1)*j)*2^(n-(k+1)*j),j=0..floor(n/(k+1))):od:k: seq(b(n),n=0..30):od;
%Y A172317 Cf. A172316, A001949, A107066, A008937, A172119.
%K A172317 easy,nonn,new
%O A172317 0,2
%A A172317 Richard Choulet (richardchoulet(AT)yahoo.fr), Jan 31 2010

%I A172316
%S A172316 1,2,4,8,16,32,64,127,252,500,992,1968,3904,7744,15361,30470,60440,
%T A172316 119888,237808,471712,935680,1855999,3681528,7302616,14485344,28732880,
%U A172316 56994048,113052416,224248833,444816138,882329660
%N A172316 7-th column of the array A172119
%F A172316 The generating fonction is f such that: f(z)=1/(1-2*z+z^7). Recurrence formula: a(n+7)=2*a(n+6)-a(n). a(n)=sum((-1)^j*binomial(n-k*j,n-(k+1)*j)*2^(n-(k+1)*j),j=0..floor(n/(k+1))) with k=6.
%e A172316 a(3)=C(3,3)*2^3=8. a(7)=C(7,7)*2^7-C(1,0)*2^0=127.
%p A172316 for k from 0 to 20 do for n from 0 to 30 do b(n):=sum((-1)^j*binomial(n-k*j,n-(k+1)*j)*2^(n-(k+1)*j),j=0..floor(n/(k+1))):od:k: seq(b(n),n=0..30):od; k:=6:taylor(1/(1-2*z+z^(k+1)),z=0,30);
%Y A172316 Cf. A172119, A008937, A107066, A001949.
%K A172316 easy,nonn,new
%O A172316 0,2
%A A172316 Richard Choulet (richardchoulet(AT)yahoo.fr), Jan 31 2010

%I A172315
%S A172315 8191,27647,62207,139967,314927,472391,1062881
%N A172315 Primes of the form 2^i*3^j - 1 with i + j = 13.
%C A172315 Note that bases 2 = prime(1), 3 = prime(2)
%C A172315 13 = prime(2 x 3) = prime(prime(1) x prime(2))
%C A172315 Smallest term 8191 is the 5th Mersenne prime
%C A172315 It is a finite "FUN" sequence with 7 = prime(4) terms
%D A172315 Helmut Kracke, Mathe-musische Knobelisken, Duemmler Bonn, 2. Auflage 1983
%e A172315 8191 = 2^13 - 1 = prime(1028)
%e A172315 27647 = 2^10 x 3^3 - 1 = prime(3016) = prime(2^3 x 13 x 29)
%e A172315 62207 = 2^8 x 3^5 - 1 = prime(6253) = prime(13^ 2 x 37)
%e A172315 139967 = 2^6 x 3^7 - 1 = prime(13005)
%e A172315 314927 = 2^4 x 3^9 - 1 = prime(27191), index is prime(2978)
%e A172315 472391 = 2^3 x 3^10 - 1 = prime(39419), index is prime(4150)
%e A172315 1062881 = 2 x 3^12 - 1 = prime(83024)
%Y A172315 A005105, A168385, A168349
%K A172315 fini,nonn,new
%O A172315 1,1
%A A172315 Eva-Maria Zschorn (e-m.zschorn(AT)zaschendorf.km3.de), Jan 31 2010

%I A143579
%S A143579 1,0,2,3,4,5,7,6,8,9,11,10,13,12,14,15,16,17,19,18,21,20,22,23,25,24,26,
%T A143579 27,28,29,31,30,32,33,35,34,37,36,38,39,41,40,42,43,44,45,47,46,49,48,
%U A143579 50,51,52,53,55,54,56,57,59,58,61,60,62,63,64,65,67,66,69,68,70,71,73
%N A143579 Permutation of the natural numbers (0,1,2,3,...): Odious numbers (A000069) interleaved with Evil numbers (A001969).
%C A143579 This is the lodumo_2 of the Thue-Morse sequence A010059, where lodumo_k of sequences is defined in A159970. [Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Apr 29 2009]
%F A143579 Antidiagonals of an array, Odious numbers (A000069) in row 1 and Evil numbers (A001969) in row 2: 1, 2, 4, 7, 8, 11, 13,... 0, 3, 5, 6, 9, 10, 22,...
%Y A143579 Cf. A000069, A001969.
%K A143579 nonn,new
%O A143579 0,3
%A A143579 Gary W. Adamson (qntmpkt(AT)yahoo.com), Aug 24 2008
%E A143579 Extended by R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Sep 05 2008
%E A143579 Edited by R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Feb 04 2010

%I A170929
%S A170929 1,4,5,3,7,17,17,7,6,13,13,13,32,55,45,15,6,13,13,13,31,51,41,20,25,39,
%T A170929 39,58,120,159,109,31,6,13,13,13,31,51,41,20,25,39,39,58,119,155,105,36,
%U A170929 25,39,39,57,113,143,102,65,89,117,136,236,400,431,253,63,6,13,13,13,31
%N A170929 When regarded as a triangle, the rows of A168131 converge to this sequence.
%H A170929 David Applegate, Omar E. Pol and N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/tooth.pdf">The Toothpick Sequence and Other Sequences from Cellular Automata</a>
%H A170929 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/toothlist.html">Catalog of Toothpick and Cellular Automata Sequences in the OEIS</a>
%K A170929 nonn,new
%O A170929 0,2
%A A170929 N. J. A. Sloane (njas(AT)research.att.com), Feb 04 2010

%I A170928
%S A170928 822,1195,1636,2472
%N A170928 Least magic constant of magic squares using Smith numbers.
%C A170928 The sequence is 822, 1195, 1636, 2472, ?, ?, ?, 12202, 16335, 21333, 27612, 35185, 43968, 54013, 65464, 78281, 92422, 107932, 126404, 147816, 171556, 197041, 224506, 253587, 285314, 320620, 359151, 400064, 442886, 487920, 536844, 589129, 644797, ...
%e A170928 Magic square of order 3: see the book: M. Gardner. From the Penrose tilings to securely encrypted, 1993:
%e A170928 94 382 346
%e A170928 526 274 22
%e A170928 202 166 454
%e A170928 The magic constant S = 822
%e A170928 Orders 4 to 6 are from participants of scientific forum at dxdy.ru.
%e A170928 The square of order 4: http://dxdy.ru/post226917.html#p226917
%e A170928 22 346 562 265
%e A170928 778 274 85 58
%e A170928 4 454 382 355
%e A170928 391 121 166 517
%e A170928 S = 1195
%e A170928 The square of order 5: http://dxdy.ru/post257980.html#p257980
%e A170928 355 576 4 319 382
%e A170928 454 85 391 648 58
%e A170928 27 535 346 526 202
%e A170928 706 166 378 121 265
%e A170928 94 274 517 22 729
%e A170928 S = 1636
%e A170928 The square of order 6: http://dxdy.ru/post258614.html#p258614
%e A170928 729 4 636 762 22 319
%e A170928 27 663 654 526 85 517
%e A170928 391 645 58 378 438 562
%e A170928 382 346 454 121 634 535
%e A170928 355 648 94 483 627 265
%e A170928 588 166 576 202 666 274
%e A170928 S = 2472.
%e A170928 Magic squares of order 10 - 35 are from Natalia Makarova and are located here: http://www.natalimak1.narod.ru/minsmit1.htm
%K A170928 nonn,new
%O A170928 3,1
%A A170928 Stefano Tognon, Feb 04 2010

%I A133933
%S A133933 1,2,6,15,0,1,0,1,28,1,0,25,0,1,16,33,0,1,0,41,64,1,0,49,76,1,28,57,
%T A133933 0,1,0,65,100,1,36,73,0,1,40,81,0,1,0,89,136,1,0,97,148,1,52,105,0,
%U A133933 1,56,113,172,1,0,121,0,1,64,129,196,1,0,137,208,1,0,145,0,1,76,153
%N A133933 a(n) = (1 + n (n - 2) + (n - 1)!) mod (4n).
%C A133933 a(n) = 0 if n >= 5 is prime.
%D A133933 Adalbert Kerber, APPLIED FINITE GROUP ACTIONS, Springer, 2nd Revised and Expanded Edition, p. 114.
%K A133933 nonn,new
%O A133933 1,2
%A A133933 Neven Juric, Feb 04 2010

%I A170927
%S A170927 1,2,5,12,21,44,89,180,362,728,1459,2921,5843,11690,23384,46770,
%T A170927 93544,187094,374193,748391,1496786,2993576,5987158,11974321,
%U A170927 23948647,47897300
%N A170927 Consider the 2^n values of A139250(i)/i^2 for 2^n <= i < 2^(n+1); a(n) = value of i where this quantity is minimized.
%H A170927 David Applegate, Omar E. Pol and N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/tooth.pdf">The Toothpick Sequence and Other Sequences from Cellular Automata</a>
%H A170927 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/toothlist.html">Catalog of Toothpick and Cellular Automata Sequences in the OEIS</a>
%e A170927 The values of A139250(i)/i^2 for i = 1 .., 15 are 1., 0.7500000000, 0.7777777778, 0.6875000000, 0.6000000000, 0.6388888889, 0.7142857143, 0.6718750000, 0.5802469136, 0.5500000000, 0.5537190083, 0.5486111111, 0.5621301775, 0.6275510204, 0.6888888889, 0.6679687500. The minimal value for 4 <= i <= 7 is 0.6000000000 at i=5.
%K A170927 nonn,new
%O A170927 0,2
%A A170927 Benoit Jubin (benoit.jubin(AT)gmail.com), Jan 22, 2010, Feb 06 2010

%I A170926
%S A170926 0,0,1,3,4,5,10,17,20,21,25,30,33,40,58,77,84,85,89,94,97,104,121,138,145,
%T A170926 151,164,177,190,222,278,325,340,341,345,350,353,360,377,394,401,407,420,
%U A170926 433,446,478,533,578,593,599,612,625,638,669,720,761,781,806,845,884,942
%N A170926 Total number of squares and rectangles at the n-th stage in the corner toothpick structure (see A152890, A153006).
%H A170926 David Applegate, Omar E. Pol and N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/tooth.pdf">The Toothpick Sequence and Other Sequences from Cellular Automata</a>
%H A170926 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/toothlist.html">Catalog of Toothpick and Cellular Automata Sequences in the OEIS</a>
%Y A170926 Partial sums of A168131.
%K A170926 nonn,new
%O A170926 0,4
%A A170926 N. J. A. Sloane (njas(AT)research.att.com), Feb 01 2010

%I A168131
%S A168131 0,0,1,2,1,1,5,7,3,1,4,5,3,7,18,19,7,1,4,5,3,7,17,17,7,6,13,13,13,32,56,
%T A168131 47,15,1,4,5,3,7,17,17,7,6,13,13,13,32,55,45,15,6,13,13,13,31,51,41,20,
%U A168131 25,39,39,58,120,160,111,31,1,4,5,3,7,17,17,7,6,13,13,13,32,55,45,15,6
%N A168131 Number of squares and rectangles that are created at the n-th stage in the corner toothpick structure (see A152980, A153006).
%H A168131 David Applegate, Omar E. Pol and N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/tooth.pdf">The Toothpick Sequence and Other Sequences from Cellular Automata</a>
%H A168131 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/toothlist.html">Catalog of Toothpick and Cellular Automata Sequences in the OEIS</a>
%F A168131 See Maple program for recurrence.
%e A168131 If written as a triangle:
%e A168131 0,
%e A168131 0,
%e A168131 1,2,
%e A168131 1,1,5,7,
%e A168131 3,1,4,5,3,7,18,19,
%e A168131 7,1,4,5,3,7,17,17,7,6,13,13,13,32,56,47,
%e A168131 15,1,4,5,3,7,17,17,7,6,13,13,13,32,55,45,15,6,13,13,13,31,51,41,20,...
%e A168131 The rows (omitting the first term) converge to A170929.
%p A168131 w := proc(n) option remember; local k,i;
%p A168131 if (n=0) then RETURN(0)
%p A168131 elif (n <= 3) then RETURN(n-1)
%p A168131 else
%p A168131 k:=floor(log(n)/log(2));
%p A168131 i:=n-2^k;
%p A168131 if (i=0) then RETURN(2^(k-1)-1)
%p A168131 elif (i<2^k-2) then RETURN(2*w(i)+w(i+1));
%p A168131 elif (i=2^k-2) then RETURN(2*w(i)+w(i+1)+1);
%p A168131 else RETURN(2*w(i)+w(i+1)+2);
%p A168131 fi;
%p A168131 fi;
%p A168131 end;
%p A168131 [seq(w(n),n=0..256)];
%Y A168131 Cf. A152980, A153006, A170926, A160124, A160125, A139250.
%K A168131 nonn,new
%O A168131 0,4
%A A168131 N. J. A. Sloane (njas(AT)research.att.com), Feb 01 2010

%I A170925
%S A170925 2,8,8,128,32,512,128,32768,128,32768,2048,2097152,8192,2097152,32768,2147483648,
%T A170925 131072,16777216,524288,34359738368,2097152,8589934592,8388608,35184372088832,
%U A170925 524288,549755813888,33554432,562949953421312,536870912,35184372088832
%N A170925 Write 1/sqrt(1-x) = Prod_{n>=1} (1 + g_n x^2n); a(n) = denominator(g_n).
%Y A170925 Cf. A170924.
%K A170925 nonn,frac,new
%O A170925 1,1
%A A170925 N. J. A. Sloane (njas(AT)research.att.com), Jan 31 2010

%I A170924
%S A170924 1,3,1,27,3,39,9,2955,7,1737,93,88047,315,79779,1083,77010795,3855,488391,
%T A170924 13797,905252529,49689,204066351,182361,756251509503,10485,10978530465,
%U A170924 619549,10462007147787,9256395,603860858253,34636833,150202954242966315
%N A170924 Write 1/sqrt(1-x) = Prod_{n>=1} (1 + g_n x^2n); a(n) = numerator(g_n).
%e A170924 1/2, 3/8, 1/8, 27/128, 3/32, 39/512, 9/128, 2955/32768, 7/128, ...
%Y A170924 Cf. A170925.
%K A170924 nonn,frac,new
%O A170924 1,2
%A A170924 N. J. A. Sloane (njas(AT)research.att.com), Jan 31 2010

%I A170923
%S A170923 2,8,8,128,32,512,128,32768,128,32768,2048,2097152,8192,2097152,32768,2147483648,
%T A170923 131072,16777216,524288,34359738368,2097152,8589934592,8388608,35184372088832,
%U A170923 524288,549755813888,33554432,562949953421312,536870912,35184372088832
%N A170923 Write sqrt(1+x) = Prod_{n>=1} (1 + g_n x^2n); a(n) = denominator(g_n).
%Y A170923 Cf. A170922.
%K A170923 nonn,frac,new
%O A170923 1,1
%A A170923 N. J. A. Sloane (njas(AT)research.att.com), Jan 31 2010

%I A170922
%S A170922 1,1,1,13,3,37,9,1861,7,1491,93,81001,315,69705,1083,63586357,3855,438821,
%T A170922 13797,822684711,49689,186369117,182361,704368012465,10485,10165801275,
%U A170922 619549,9738266477517,9256395,566066862375,34636833,140047960975823893
%V A170922 1,-1,1,-13,3,-37,9,-1861,7,-1491,93,-81001,315,-69705,1083,-63586357,3855,-438821,
%W A170922 13797,-822684711,49689,-186369117,182361,-704368012465,10485,-10165801275,
%X A170922 619549,-9738266477517,9256395,-566066862375,34636833,-140047960975823893
%N A170922 Write sqrt(1+x) = Prod_{n>=1} (1 + g_n x^2n); a(n) = numerator(g_n).
%e A170922 1/2, -1/8, 1/8, -13/128, 3/32, -37/512, 9/128, -1861/32768, ...
%Y A170922 Cf. A170923.
%K A170922 sign,frac,new
%O A170922 1,4
%A A170922 N. J. A. Sloane (njas(AT)research.att.com), Jan 31 2010

%I A170921
%S A170921 3,27,3645,98415,93002175,279006525,112997642625,9152809052625,31714483367345625,
%T A170921 513774630550999125,644053197583573903125,9363542641791959053125,8848547796493401305203125,
%U A170921 3105840276569183858126296875,215997073779584150133328828125,38490678547521895553759197171875
%N A170921 Write cot(x) = Prod_{n>=1} (1 + g_n x^2n); a(n) = denominator(g_n).
%Y A170921 Cf. A170920.
%K A170921 nonn,frac,new
%O A170921 1,1
%A A170921 N. J. A. Sloane (njas(AT)research.att.com), Jan 31 2010

%I A170920
%S A170920 1,2,211,2743,1418638,2268068,550669394,23861846102,48726978122069,461240499534601,
%T A170920 336080282371649483,2842820955735934463,1628158082048867402632,331277549738199913530049,
%U A170920 14001180998026101838535141,1500970948062470951644822898,889277588648296174667504384505014
%V A170920 -1,-2,-211,-2743,-1418638,-2268068,-550669394,-23861846102,-48726978122069,-461240499534601,
%W A170920 -336080282371649483,-2842820955735934463,-1628158082048867402632,-331277549738199913530049,
%X A170920 -14001180998026101838535141,-1500970948062470951644822898,-889277588648296174667504384505014
%N A170920 Write cot(x) = Prod_{n>=1} (1 + g_n x^2n); a(n) = numerator(g_n).
%e A170920 -1/3, -2/27, -211/3645, -2743/98415, -1418638/93002175, ...
%p A170920 t1:=cot(x);
%p A170920 L:=100;
%p A170920 t0:=series(t1,x,L):
%p A170920 g:=[];
%p A170920 M:=20; # number of terms to get
%p A170920 t2:=t0:
%p A170920 for n from 1 to M do
%p A170920 t3:=coeff(t2,x,n); t2:=series(t2/(1+t3*x^n),x,L); g:=[op(g),t3];
%p A170920 od:
%p A170920 g;
%p A170920 g1:=map(numer,g);
%p A170920 g2:=map(denom,g);
%Y A170920 Cf. A170921, A170908-A170919.
%K A170920 sign,frac,new
%O A170920 1,2
%A A170920 N. J. A. Sloane (njas(AT)research.att.com), Jan 31 2010

%I A172314
%S A172314 1260,13650,17556,18720,24510,42120,113610,244530,266070,712080,749910,
%T A172314 795690,992250,1080720,1286730,1458270,1849470,2271060,2457690,3295380,
%U A172314 3370770,3414840,3714750
%N A172314 Numbers n such that phi(n+1) = 4 phi(n)
%D A172314 A. Shinzel, Sur l'equation phi(x+k) = phi(x), Acta Arith. 4 (1958), 181-184
%D A172314 J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 51, p. 19, Ellipses, Paris 2008.
%D A172314 R. K. Guy, Unsolved Problems Number Theory, Sect. B36. V. L. Klee, Jr., Some remarks on Euler's totient function, Amer. Math. Monthly, 54 (1947), 332. M. Lal and P. Gillard, On the equation phi(n) = phi(n+k), Math. Comp., 26 (1972), 579-583. K. Miller, Solutions of phi(n) = phi(n+1) for 1 <= n <= 500000. Unpublished, 1972. [ Cf. Math. Comp., Vol. 27, p. 447, 1973 ]. L. Moser, Some equations involving Euler's totient function, Amer. Math. Monthly, 56 (1949), 22-23.
%e A172314 phi(1260) = 288 ; phi(1261) = 1152 ; 4 phi(1260) = phi(1261)
%p A172314 with(numtheory): for n from 1 to 4000000 do; if 4*phi(n) = phi(n+1) then print(n); else fi ; od;
%K A172314 nonn,new
%O A172314 1,1
%A A172314 Michel Lagneau (mn.lagneau2(AT)orange.fr), Jan 31 2010

%I A175094
%S A175094 1,3,7,17,41,181,1489,18049,218077,1326511,69196649,3045979067,
%T A175094 67080736123
%N A175094 a(1) = 1, a(2) = 3, for n >= 3, a(n) = smallest primes such that a(n) mod a(n-1) = a(n-2).
%K A175094 nonn,new
%O A175094 1,2
%A A175094 Jaroslav Krizek (jaroslav.krizek(AT)atlas.cz), Jan 31 2010

%I A175093
%S A175093 1,2,3,5,13,31,137,853,6961,28697,179143,6836131,68540453,966402473,
%T A175093 15530980021
%N A175093 a(1) = 1, a(2) = 2, for n >= 3, a(n) = smallest primes such that a(n) mod a(n-1) = a(n-2).
%C A175093 Essentially the same as A072999. [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Feb 01 2010]
%K A175093 nonn,new
%O A175093 1,2
%A A175093 Jaroslav Krizek (jaroslav.krizek(AT)atlas.cz), Jan 31 2010

%I A172312
%S A172312 1,2,3,4,6,7,7
%N A172312 a(n) = A172311(n+1)/2.
%H A172312 David Applegate, Omar E. Pol and N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/tooth.pdf">The Toothpick Sequence and Other Sequences from Cellular Automata</a>
%H A172312 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/toothlist.html">Catalog of Toothpick and Cellular Automata Sequences in the OEIS</a>
%Y A172312 Cf. A139250, A139251, A172310, A172311.
%K A172312 more,nonn,new
%O A172312 1,2
%A A172312 Omar E. Pol (info(AT)polprimos.com), Jan 31 2010

%I A172311
%S A172311 0,1,2,4,6,8,12,14,14
%N A172311 First differences of A172310.
%C A172311 Number of L-toothpicks added to the L-toothpick structure of A172310 at the nth stage.
%H A172311 David Applegate, Omar E. Pol and N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/tooth.pdf">The Toothpick Sequence and Other Sequences from Cellular Automata</a>
%H A172311 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/toothlist.html">Catalog of Toothpick and Cellular Automata Sequences in the OEIS</a>
%Y A172311 Cf. A139250, A139251, A160121, A160171, A160173, A161207, A161329, A172310.
%K A172311 more,nonn,new
%O A172311 0,3
%A A172311 Omar E. Pol (info(AT)polprimos.com), Jan 31 2010

%I A172303
%S A172303 0,1,0,1,0,1,0,1,0,0,0,1,1,0,1,0,0,0,0,1,0,1,1,1,0,1,0,0,0,0,0,0,0,1,1,
%T A172303 1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,0,1,1,2,1,2,1,1,1,1,0,1,0,0,0,0,0,0,
%U A172303 0,0,0,0,0,0,0,1,1,1,2,1,2,2,2,2,2,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0
%N A172303 Table T(n,k) with coefficients [x^k] of the polynomials p(x,n) = x^(n-1)*p(x,n-1) + x^(n-2)*p(x,n-2), recurrence starting p(x,0)=0, p(x,1)=1.
%C A172303 Row sums are sum_{k>=0} T(n,k) = A000045(n).
%C A172303 Lengths of the rows (1 + the degrees of the polynomials) are: 0, 1, 2, 4, 7, 11, 16, 22, 29, 37, 46,..., A000124
%F A172303 T(n,k)= [x^k] p(x,n). p(x,n)=x^(n - 1)*p(x, n - 1) + x^(n - 2)*p(x, n - 2)
%e A172303 0;
%e A172303 1;
%e A172303 0,1;
%e A172303 0,1,0,1;
%e A172303 0,0,0,1,1,0,1;
%e A172303 0,0,0,0,1,0,1,1,1,0,1;
%e A172303 0,0,0,0,0,0,0,1,1,1,1,1,1,1,0,1;
%t A172303 p[x,0]=0;
%t A172303 p[x,1]=1;
%t A172303 p[x_,n_]:=p[x,n]=x^(n-1)*p[x,n-1]+x^(n-1)*p[x,n-2];
%t A172303 Flatten[Table[CoefficientList[p[x,n],x],{n,0,10}]]
%K A172303 nonn,tabf,new
%O A172303 0,56
%A A172303 Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Jan 31 2010
%E A172303 Keyword tabf added; A-sequences of row sums and lengths identified. The Assoc. Editors of the OEIS - Feb 02 2010

%I A172300
%S A172300 1,1,1,1,13,1,1,130,130,1,1,1210,12100,1210,1,1,11011,1024870,1024870,
%T A172300 11011,1,1,99463,84245161,784128037,84245161,99463,1,1,896260,
%U A172300 6857285260,580812061522,580812061522,6857285260,896260,1,1,8069620
%N A172300 q-form Narayana triangle sequence:q=3:c(n,q)=Product[(1 - q^i), {i, 1, n}]; w(n,m,q)=-(q - 1)*c(n - 1, q)*c(n, q)/(c(m - 1, q)* c(n - m, q)*c(m - 1, q)*c(n - m + 1, q)*(1 - q^m ))
%C A172300 q=2 is A156916.
%C A172300 Row sums are:
%C A172300 {1, 2, 15, 262, 14522, 2071764, 952817287, 1175340486086, 4725928995048606,
%C A172300 51726515551546238332,...}.
%F A172300 q=3:
%F A172300 c(n,q)=Product[(1 - q^i), {i, 1, n}];
%F A172300 w(n,m,q)=-(q - 1)*c(n - 1, q)*c(n, q)/(c(m - 1, q)* c(n - m, q)*c(m - 1, q)*c(n - m + 1, q)*(1 - q^m ))
%e A172300 {1},
%e A172300 {1, 1},
%e A172300 {1, 13, 1},
%e A172300 {1, 130, 130, 1},
%e A172300 {1, 1210, 12100, 1210, 1},
%e A172300 {1, 11011, 1024870, 1024870, 11011, 1},
%e A172300 {1, 99463, 84245161, 784128037, 84245161, 99463, 1},
%e A172300 {1, 896260, 6857285260, 580812061522, 580812061522, 6857285260, 896260, 1},
%e A172300 {1, 8069620, 556344432400, 425659125229240, 3873498039586084, 425659125229240, 556344432400, 8069620, 1},
%e A172300 {1, 72636421, 45088331971540, 310852833944711080, 25552359853423800124, 25552359853423800124, 310852833944711080, 45088331971540, 72636421, 1}
%t A172300 c[n_, q_] = Product[(1 - q^i), {i, 1, n}];
%t A172300 w[n_, m_, q_] = -(q - 1)*c[n - 1, q]*c[n, q]/(c[m - 1, q]*c[n - m, q]*c[m - 1, q]*c[n - m + 1, q]*(1 - q^m ));
%t A172300 Table[Table[Table[w[n, m, q], {m, 1, n}], {n, 1, 10}], {q, 2, 12}]
%t A172300 Table[Flatten[Table[Table[w[n, m, q], {m, 1, n}], {n, 1, 10}]], {q, 2, 12}]
%Y A172300 A156916
%K A172300 nonn,tabl,uned,new
%O A172300 1,5
%A A172300 Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Jan 31 2010

%I A172302
%S A172302 1,1,1,1,31,1,1,806,806,1,1,20306,527956,20306,1,1,508431,333038706,
%T A172302 333038706,508431,1,1,12714681,208533483081,5253698396331,208533483081,
%U A172302 12714681,1,1,317886556,130381488829956,82245646088465706
%N A172302 q-form Narayana triangle sequence:q=5:c(n,q)=Product[(1 - q^i), {i, 1, n}]; w(n,m,q)=-(q - 1)*c(n - 1, q)*c(n, q)/(c(m - 1, q)* c(n - m, q)*c(m - 1, q)*c(n - m + 1, q)*(1 - q^m ))
%C A172302 q=2 is A156916.
%C A172302 Row sums are:
%C A172302 {1, 2, 33, 1614, 568570, 667094276, 5670790791857, 164752055790364438,
%C A172302 34760211807180756108894, 25197394536008396774892947100,...}
%F A172302 q=5:
%F A172302 c(n,q)=Product[(1 - q^i), {i, 1, n}];
%F A172302 w(n,m,q)=-(q - 1)*c(n - 1, q)*c(n, q)/(c(m - 1, q)* c(n - m, q)*c(m - 1, q)*c(n - m + 1, q)*(1 - q^m ))
%e A172302 {1},
%e A172302 {1, 1},
%e A172302 {1, 31, 1},
%e A172302 {1, 806, 806, 1},
%e A172302 {1, 20306, 527956, 20306, 1},
%e A172302 {1, 508431, 333038706, 333038706, 508431, 1},
%e A172302 {1, 12714681, 208533483081, 5253698396331, 208533483081, 12714681, 1},
%e A172302 {1, 317886556, 130381488829956, 82245646088465706, 82245646088465706, 130381488829956, 317886556, 1},
%e A172302 {1, 7947261556, 81494438892517456, 1285577907930958734456, 32188893002425159081956, 1285577907930958734456, 81494438892517456, 7947261556, 1},
%e A172302 {1, 198682027181, 50934775370442204956, 20088685264446025720234456, 12578608531804976792602006956, 12578608531804976792602006956, 20088685264446025720234456, 50934775370442204956, 198682027181, 1}
%t A172302 c[n_, q_] = Product[(1 - q^i), {i, 1, n}];
%t A172302 w[n_, m_, q_] = -(q - 1)*c[n - 1, q]*c[n, q]/(c[m - 1, q]*c[n - m, q]*c[m - 1, q]*c[n - m + 1, q]*(1 - q^m ));
%t A172302 Table[Table[Table[w[n, m, q], {m, 1, n}], {n, 1, 10}], {q, 2, 12}]
%t A172302 Table[Flatten[Table[Table[w[n, m, q], {m, 1, n}], {n, 1, 10}]], {q, 2, 12}]
%Y A172302 A156916
%K A172302 nonn,tabl,uned,new
%O A172302 1,5
%A A172302 Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Jan 31 2010

%I A172301
%S A172301 1,1,1,1,21,1,1,357,357,1,1,5797,98549,5797,1,1,93093,25698101,25698101,
%T A172301 93093,1,1,1490853,6608951349,107316781429,6608951349,1490853,1,1,
%U A172301 23859109,1693829725237,441691010116213,441691010116213,1693829725237
%N A172301 q-form Narayana triangle sequence:q=4:c(n,q)=Product[(1 - q^i), {i, 1, n}]; w(n,m,q)=-(q - 1)*c(n - 1, q)*c(n, q)/(c(m - 1, q)* c(n - m, q)*c(m - 1, q)*c(n - m + 1, q)*(1 - q^m ))
%C A172301 q=2 is A156916.
%C A172301 Row sums are:
%C A172301 {1, 2, 23, 716, 110145, 51582390, 120537665835,
%C A172301 886769727401120, 32711582953060812821, 3832110174372723393438602,...}
%F A172301 q=4:
%F A172301 c(n,q)=Product[(1 - q^i), {i, 1, n}];
%F A172301 w(n,m,q)=-(q - 1)*c(n - 1, q)*c(n, q)/(c(m - 1, q)* c(n - m, q)*c(m - 1, q)*c(n - m + 1, q)*(1 - q^m ))
%e A172301 {1},
%e A172301 {1, 1},
%e A172301 {1, 21, 1},
%e A172301 {1, 357, 357, 1},
%e A172301 {1, 5797, 98549, 5797, 1},
%e A172301 {1, 93093, 25698101, 25698101, 93093, 1},
%e A172301 {1, 1490853, 6608951349, 107316781429, 6608951349, 1490853, 1},
%e A172301 {1, 23859109, 1693829725237, 441691010116213, 441691010116213, 1693829725237, 23859109, 1},
%e A172301 {1, 381767589, 433744500886581, 1811342550084767349, 29088030363125969781, 1811342550084767349, 433744500886581, 381767589, 1},
%e A172301 {1, 6108368805, 111046534828936245, 7421488455338616872565, 1908633487684482142541685, 1908633487684482142541685, 7421488455338616872565, 111046534828936245, 6108368805, 1}
%t A172301 c[n_, q_] = Product[(1 - q^i), {i, 1, n}];
%t A172301 w[n_, m_, q_] = -(q - 1)*c[n - 1, q]*c[n, q]/(c[m - 1, q]*c[n - m, q]*c[m - 1, q]*c[n - m + 1, q]*(1 - q^m ));
%t A172301 Table[Table[Table[w[n, m, q], {m, 1, n}], {n, 1, 10}], {q, 2, 12}]
%t A172301 Table[Flatten[Table[Table[w[n, m, q], {m, 1, n}], {n, 1, 10}]], {q, 2, 12}]
%Y A172301 A156916
%K A172301 nonn,tabl,uned,new
%O A172301 1,5
%A A172301 Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Jan 31 2010

%I A172299
%S A172299 2,2,1,1,1,1,1,1,1,25,25,477481,477481,49,49,13082689,13082689,1924313689,
%T A172299 1924313689,30489001321,30489001321,730192467169,730192467169,55867983514256281,
%U A172299 55867983514256281,73155570928609,73155570928609,564036899167989738841
%V A172299 -2,2,-1,1,-1,1,-1,1,-1,25,-25,477481,-477481,49,-49,13082689,-13082689,
%W A172299 1924313689,-1924313689,30489001321,-30489001321,730192467169,-730192467169,
%X A172299 55867983514256281,-55867983514256281,73155570928609,-73155570928609
%N A172299 First differences of A172298.
%C A172299 Differences between squares of Bernoulli number denominators.
%F A172299 a(n) = A172298(n+1)-A172298(n).
%F A172299 a(2n) = -a(2n-1), n>=2 .
%K A172299 sign,new
%O A172299 0,1
%A A172299 Paul Curtz (bpcrtz(AT)free.fr), Jan 31 2010
%E A172299 Keyword:sign added, sequence extended by R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Feb 02 2010

%I A172298
%S A172298 1,1,1,0,1,0,1,0,1,0,25,0,477481,0,49,0,13082689,0,1924313689,0,30489001321,
%T A172298 0,730192467169,0,55867983514256281,0,73155570928609,0,564036899167989738841,
%U A172298 0,74232720893311466588760025,0,59433630916551169012841089,0,6644474695172651051906689
%V A172298 1,-1,1,0,1,0,1,0,1,0,25,0,477481,0,49,0,13082689,0,1924313689,0,30489001321,
%W A172298 0,730192467169,0,55867983514256281,0,73155570928609,0,564036899167989738841,
%X A172298 0,74232720893311466588760025,0,59433630916551169012841089,0,6644474695172651051906689
%N A172298 A027641(n) * A164555(n) .
%C A172298 Squares of Bernoulli number numerators (apart from the sign flipped aa(1)).
%C A172298 The associated denominators of the squared Bernoulli numbers are in A172282.
%K A172298 sign,easy,new
%O A172298 0,11
%A A172298 Paul Curtz (bpcrtz(AT)free.fr), Jan 31 2010
%E A172298 Edited and extended by R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Feb 02 2010

%I A172297
%S A172297 2,5,11,41,116,197,312,435,684,1046,1430,1892,2404,3155,3977,9289,17044,
%T A172297 26575,38954,54776,73661,96632,119637,218363,361381,512404,1179475,
%U A172297 2374678,3643657,5111420,7125412,9493318,13246266
%N A172297 Partial sums of A002234.
%C A172297 The subsequence of primes in this sequence begin: 2, 5, 11, 41, 197, 218363, 3643657.
%F A172297 a(n) = SUM[i=1..n] {i such that the Woodall number i*2^i - 1 is prime}.
%e A172297 a(6) = 2 + 3 + 6 + 30 + 75 + 81 = 197 is prime. a(33) = 2 + 3 + 6 + 30 + 75 + 81 + 115 + 123 + 249 + 362 + 384 + 462 + 512 + 751 + 822 + 5312 + 7755 + 9531 + 12379 + 15822 + 18885 + 22971 + 23005 + 98726 + 143018 + 151023 + 667071 + 1195203 + 1268979 + 1467763 + 2013992 + 2367906 + 3752948.
%Y A172297 Cf. A000040, A002234, A050918.
%K A172297 hard,nonn,new
%O A172297 1,1
%A A172297 Jonathan Vos Post (jvospost3(AT)gmail.com), Jan 30 2010

%I A172296
%S A172296 3,8,15,26,39,56,75,98,129,172,233,312,413,540,707,898,1097,1410,1757,
%T A172296 2458,4167,6784,10323,16130,26631,37322,48601,60992,75471,118208,201547,
%U A172296 296916,414155,541186,680123,821202,1088219,1358206,1732527,2718718
%N A172296 Partial sums of Wagstaff numbers A000978.
%C A172296 The prime partial sums of Wagstaff numbers begin 3, 233, 1097, 201547, 680123. None of these but 3 is itself a Wagstaff number.
%F A172296 a(n) = SUM[i=1..n] {integers i such that (2^i + 1)/3 is prime}.
%e A172296 a(40) = 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 31 + 43 + 61 + 79 + 101 + 127 + 167 + 191 + 199 + 313 + 347 + 701 + 1709 + 2617 + 3539 + 5807 + 10501 + 10691 + 11279 + 12391 + 14479 + 42737 + 83339 + 95369 + 117239 + 127031 + 138937 + 141079 + 267017 + 269987 + 374321 + 986191.
%Y A172296 Cf. A000040, A000978.
%K A172296 hard,nonn,new
%O A172296 1,1
%A A172296 Jonathan Vos Post (jvospost3(AT)gmail.com), Jan 30 2010

%I A172295
%S A172295 5,12,23,36,53,76,107,144,185,232,285,346,413,486,569,666,767,870,977,
%T A172295 1108,1259,1416,1583,1756,1947,2140,2363,2590,2823,3074,3331
%N A172295 Partial sums of A023201.
%C A172295 A023201 is also known as the smaller numbers of pairs of sexy primes. The subsequence of prime partial sums of smaller numbers of pairs of sexy primes begins 5, 23, 53, 107, 569, 977, 1259, 1583, 3331. The subsubsequence of smaller numbers of pairs of sexy prime partial sums of smaller numbers of pairs of sexy primes begins 5, 107, 977. This is to smaller members of sexy prime pairs as A172112 is to A023200 smaller member p of cousin prime pairs (p, p+4)
%e A172295 a(19) = 5 + 7 + 11 + 13 + 17 + 23 + 31 + 37 + 41 + 47 + 53 + 61 + 67 + 73 + 83 + 97 + 101 + 103 + 107 = 977, which is A023201(73).
%Y A172295 Cf. A000040, A023201, A172112.
%K A172295 easy,more,nonn,new
%O A172295 1,1
%A A172295 Jonathan Vos Post (jvospost3(AT)gmail.com), Jan 30 2010

%I A172294
%S A172294 12,42,1152,1452,1950,3672,5520,6660,8232,10890,13218,15288,15360,16062,
%T A172294 20898,21018,23628
%N A172294 This is the sequence of the "Natural Jewels": a natural jewel is a number that is totally enclosed by prime numbers in a version of Ulam Spiral.
%C A172294 There is no formula to calculate this numbers. If the sequence is infinite, then there are infinitely many primes p such that p + 2 is also prime (twin prime conjecture).
%e A172294 11-(top)->12, 13-(left)->12, 29-(right)->12, 31-(bottom)->12
%Y A172294 Cf. A156859
%K A172294 nonn,new
%O A172294 1,1
%A A172294 Emilio Apricena (emilioapricena(AT)yahoo.it), Jan 30 2010

%I A172291
%S A172291 3,7,13,1093
%N A172291 Numbers which squares divided 2^1092-1
%C A172291 Up to now existed only two primes p such that p^2 divide 2^(p-1)-1 (these two are Wieferich primes see A001220).
%Y A172291 A001220, A172290, A172292, A172293
%K A172291 fini,nonn,new
%O A172291 1,1
%A A172291 Artur Jasinski (grafix(AT)csl.pl), Jan 30 2010

%I A172290
%S A172290 3,3,5,7,7,13,13,29,43,53,79,113,127,157,313,337,547,911,1093,1093,1249,
%T A172290 1429,1613,2731,3121,4733,5419,8191,14449,21841,121369,224771,503413,
%U A172290 22366891,108749551,112901153,23140471537,25829691707,105310750819
%N A172290 Prime divisors of 2^1092-1
%C A172290 Up to now existed only two primes p such that p^2 divide 2^(p-1)-1 (these two are Wieferich primes see A001220).
%Y A172290 A001220, A172291, A172292, A172293
%K A172290 fini,nonn,new
%O A172290 1,1
%A A172290 Artur Jasinski (grafix(AT)csl.pl), Jan 30 2010

%I A172288
%S A172288 19,29,59,73,79,89,109,131,139,149,151,173,179,197,199,229,239,269,281,
%T A172288 313,349,353,359,367,379,383,389,397
%N A172288 Primes p such that trivial prime*p-trivial prime are all nonprimes.
%C A172288 Trivial primes (2 and 3): primes such that either p+-1 is prime.
%e A172288 a(1)=19 because 2*19-2=36(nonprime), 2*19-3=35(nonprime), 3*19-3=54(nonprime) and 3*19-2=55(nonprime).
%Y A172288 Cf. A131426.
%K A172288 nonn,new
%O A172288 1,1
%A A172288 Juri-Stepan Gerasimov (2stepan(AT)rambler.ru), Jan 30 2010

%I A172287
%S A172287 17,31,41,47,61,83,97,101,103,107,157,163,223,233,241,257,271,277,283,
%T A172287 293,307,311,317,337,373,401
%N A172287 Primes p such that either 2*p-3 and 3*p-2 is prime.
%e A172287 a(1)=17 because 2*17-3=31(prime) and 3*17-2=49(nonprime).
%Y A172287 Cf. A131426.
%K A172287 nonn,new
%O A172287 1,1
%A A172287 Juri-Stepan Gerasimov (2stepan(AT)rambler.ru), Jan 30 2010

%I A172289
%S A172289 3,8,15,26,39,56,75,98,127,158,199,242,289,342,403,474,547,626,709,798,
%T A172289 895,1002,1111,1224,1351,1488,1627,1778,1941,2108,2281,2460,2641,2832,
%U A172289 3025,3222,3421,3632,3855,4082
%N A172289 Partial sums of regular primes A007703.
%C A172289 First 10 terms identical to partial sum of odd primes A071148 because the first irregular prime A000928 is 37. The subsequence of regular primes in this partial sum of regular primes begins: 3, 127, 199, 709, 1627, 2281. The subsequence of irregular primes in this partial sum of regular primes begins: 127, 547.
%e A172289 a(20) = 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 41 + 43 + 47 + 53 + 61 + 71 + 73 + 79 + 83 + 89 = 798.
%Y A172289 Cf. A000040, A000367, A000928, A007703.
%K A172289 easy,more,nonn,new
%O A172289 1,1
%A A172289 Jonathan Vos Post (jvospost3(AT)gmail.com), Jan 30 2010

%I A172286
%S A172286 2,32,1458,131072,19531250,4353564672,1356446145698,562949953421312,
%T A172286 300189270593998272,200000000000000000000
%N A172286 Numbers of circuits (cycles) of length 2n in K_{n,n} (the complete bipartite graph on 2n vertices).
%F A172286 a(n) = 2*n^(2*n)
%e A172286 For n=2, a(2) = 32, that is : there are 32 cycles of length 4 in the complete bipartite graph K2,2.
%o A172286 (Other) % Matlab code
%o A172286 nmax = 10;
%o A172286 for k=1:nmax
%o A172286 an = 2*k^(2*k);
%o A172286 fprintf('%3.0f ', an);
%o A172286 end
%Y A172286 Cf. A118537
%K A172286 easy,nonn,new
%O A172286 1,1
%A A172286 Thibaut Lienart (syncthib(AT)gmail.com), Jan 30 2010

%I A172285
%S A172285 0,2,1,6,7,20,33,74,139,288,565,1142,2271,4556,9097,18210,36403,72824,
%T A172285 145629,291278,582535,1165092,2330161,4660346,9320667,18641360,37282693,
%U A172285 74565414,149130799,298261628,596523225,1193046482,2386092931,4772185896
%N A172285 (5*2^n -5*(-1)^n - 3*n*(-1)^n ) / 9.
%H A172285 <a href="Sindx_Rea.html#recLCC">Index to sequences with linear recurrences with constant coefficients</a>, signature (0,3,2).
%F A172285 a(n) = 3*a(n-2)+2*a(n-3).
%F A172285 a(n+1) = 2*a(n) +(-1)^n*(2+n).
%F A172285 a(n) + A001045() = A053088(n+1).
%F A172285 A053088(n)+a(n) = A000079(n).
%F A172285 a(2*n) = A141291(n). a(2*n+1) = 2*A164044(n).
%F A172285 G.f.: x*(2+x)/( (1-2*x)*(1+x)^2 ).
%p A172285 A172295 := proc(n) (5*2^n -5*(-1)^n - 3*n*(-1)^n ) / 9 ; end proc: seq(A172295(n),n=0..100) ; # R. J. Mathar, Feb 02 2010
%K A172285 nonn,new
%O A172285 0,2
%A A172285 Paul Curtz (bpcrtz(AT)free.fr), Jan 30 2010
%E A172285 Definition replaced by explicit formula; g.f. added - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Feb 02 2010

%I A172284
%S A172284 2,4,5,6,8,9,10,11,12,13,14,16,17,18,19,20,21,22,23,24,25,26,28,29,30,
%T A172284 31,32,33,34,35,36,37,38,39,40,41,42,43,45,46,47,48,49,50,51,52,53,54,
%U A172284 55,56,57,58,59,60,61,62,63,64,65,66,68,69,70,71,72,73,74,75,76,77,78
%N A172284 First positive zeros of Bessel function of order n rounded to nearest integer
%D A172284 M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 409.
%D A172284 British Association Mathematical Tables, Vol. 6, Bessel Functions, Part 1, Functions of Order Zero and Unity. Cambridge Univ. Press, 1937, p. 171.
%H A172284 M. Abramowitz and I. A. Stegun, eds.,<a href="http://www.nrbook.com/abramowitz_and_stegun/"> Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
%e A172284 BesselJzeros(0,1) = 2,40482555, a(0) = 2 BesselJzeros(1,1) = 3,83170597, a(1) = 4 BesselJzeros(2,1) =5,135622302, a(2) = 5
%p A172284 for n from 0 to 10000 do; x1:= evalf(BesselJZeros(n,1));evalf(%); od;
%Y A172284 Cf. A000134.
%K A172284 nonn,new
%O A172284 1,1
%A A172284 Michel Lagneau (mn.lagneau2(AT)orange.fr), Jan 30 2010

%I A172283
%S A172283 1,9,11,9,2,11,9,7,13,11,9,16,6,24,11,9,25,10,30,35,11,9,34,
%T A172283 35,20,65,46,11,9,43,69,15,85,111,57,11
%V A172283 1,-9,11,-9,2,11,-9,-7,13,11,-9,-16,6,24,11,-9,-25,-10,30,35,11,-9,-34,
%W A172283 -35,20,65,46,11,-9,-43,-69,-15,85,111,57,11
%N A172283 (-9,11) Pascal Triangle
%C A172283 A000984
%C A172283 With offset 0, triangle given by [ -9,10,0,0,0,0,0,0,0,...] DELTA [11,-10,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938. [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Feb 01 2010]
%F A172283 With offset 0 : Sum_{k, 0<=k<=n}T(n,k) = 2^n. [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Feb 01 2010]
%e A172283 Triangle begins:
%e A172283 .......1
%e A172283 .....-9,11
%e A172283 ....-9,2,11
%e A172283 ..-9,-7,13,11
%e A172283 -9,-16,6,24,11
%Y A172283 Cf. A093644, A172179, A022114, A000984
%K A172283 nonn,tabl,new
%O A172283 1,2
%A A172283 M. Dols (markdols99(AT)yahoo.com), Jan 30 2010

%I A172282
%S A172282 1,4,36,1,900,1,1764,1,900,1,4356,1,7452900,1,36,1,260100,1,636804,1,108900,
%T A172282 1,19044,1,7452900,1,36,1,756900,1,205119684,1,260100,1,36,1,3683290256100,
%U A172282 1,36,1,183060900,1,3261636,1,476100,1,79524,1,2153888100,1,4356,1,2528100
%N A172282 Squares of Bernoulli number denominators A027642.
%C A172282 Compare the sequence for example with A120083.
%F A172282 a(n) = A027642(n)^2 .
%Y A172282 Cf. A172298.
%K A172282 nonn,easy,new
%O A172282 0,2
%A A172282 Paul Curtz (bpcrtz(AT)free.fr), Jan 30 2010
%E A172282 Edited and extended by R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Feb 02 2010

%I A172281
%S A172281 1,1,1,1,1,1,1,2,2,1,1,2,3,2,1,1,2,5,4,2,1,1,2,6,10,6,2,1,1,2,9,18,14,6,
%T A172281 2,1,1,2,7,23,35,23,7,2,1,1,2,9,34,63,51,21,6,1,1,1,1,7,30,84,126,84,30,
%U A172281 7,1,1
%N A172281 Asymmetrical triangle form:l=1;t(n,k,l)=Ceiling[Binomial[n, k]*(l + 1)/((1 + l)^2 + (k - Floor[n/2])^2)]
%C A172281 Row sums are:
%C A172281 {1, 2, 3, 6, 9, 15, 28, 53, 101, 189, 372,...}.
%C A172281 This function was constructed to impose a "S" shape on a binomial triangle.
%F A172281 l=1;
%F A172281 t(n,k,l)=Ceiling[Binomial[n, k]*(l + 1)/((1 + l)^2 + (k - Floor[n/2])^2)]
%e A172281 {1},
%e A172281 {1, 1},
%e A172281 {1, 1, 1},
%e A172281 {1, 2, 2, 1},
%e A172281 {1, 2, 3, 2, 1},
%e A172281 {1, 2, 5, 4, 2, 1},
%e A172281 {1, 2, 6, 10, 6, 2, 1},
%e A172281 {1, 2, 9, 18, 14, 6, 2, 1},
%e A172281 {1, 2, 7, 23, 35, 23, 7, 2, 1},
%e A172281 {1, 2, 9, 34, 63, 51, 21, 6, 1, 1},
%e A172281 {1, 1, 7, 30, 84, 126, 84, 30, 7, 1, 1}
%t A172281 T[n_,k_,l_]=Ceiling[Binomial[n,k]*(l+1)/((1+l)^2+(k-Floor[n/2])^2)];
%t A172281 Table[Table[Table[T[n, k, l], {k, 0, n}], {n, 0, 10}], {l, 0, 5}];
%t A172281 Table[Flatten[Table[Table[T[n, k, l], {k, 0, n}], {n, 0, 10}]], {l, 0, 5}]
%K A172281 nonn,tabl,uned,new
%O A172281 0,8
%A A172281 Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Jan 30 2010

%I A172280
%S A172280 17,41,73,89,97,113,137,193,233,241,251,257,281,307,313,337,353,401,409,
%T A172280 433,439,449,457,499,521,569,577,593,601,617,641,643,673,727,761,769,
%U A172280 809,857,881,919,929,937,953,977,997,1009,1013,1033,1049,1097,1129,1153
%N A172280 Primes p with the property that no divisor of p-1 is a primitive root modulo p.
%C A172280 The sequence is probably infinite.
%C A172280 No element of A001122 nor of A172058 can be in this list.
%t A172280 m = 2; t = {}; While[m < bound, m = m + 1; p = Prime[m]; dp = Divisors[p - 1]; L = Length[dp]; j = 1; While[j < L - 1, j = j + 1; b = MultiplicativeOrder[dp[[j]], p]; If[b == p - 1, j = L + 1,] ]; If[j == L + 1, , t = {t, p}] ]; t = Flatten[t]
%Y A172280 Cf. A172058
%K A172280 easy,nonn,new
%O A172280 1,1
%A A172280 Emmanuel Vantieghem (manuvti(AT)hotmail.com), Jan 30 2010

%I A172279
%S A172279 1,1,1,1,2,1,1,3,2,1,1,2,6,2,1,1,3,10,5,1,1,1,2,8,20,8,2,1,1,2,11,35,18,
%T A172279 5,1,1,1,1,6,28,70,28,6,1,1,1,1,8,42,126,63,17,4,1,1,1,1,5,24,105,252,
%U A172279 105,24,5,1,1
%N A172279 Asymmetrical triangle form:l=0;t(n,k,l)=Ceiling[Binomial[n, k]*(l + 1)/((1 + l)^2 + (k - Floor[n/2])^2)]
%C A172279 Row sums are:
%C A172279 {1, 2, 4, 7, 12, 21, 42, 74, 142, 264, 524,...}.
%C A172279 This function was constructed to impose an "S" shape on a binomial triangle.
%F A172279 l=0;
%F A172279 t(n,k,l)=Ceiling[Binomial[n, k]*(l + 1)/((1 + l)^2 + (k - Floor[n/2])^2)]
%e A172279 {1},
%e A172279 {1, 1},
%e A172279 {1, 2, 1},
%e A172279 {1, 3, 2, 1},
%e A172279 {1, 2, 6, 2, 1},
%e A172279 {1, 3, 10, 5, 1, 1},
%e A172279 {1, 2, 8, 20, 8, 2, 1},
%e A172279 {1, 2, 11, 35, 18, 5, 1, 1},
%e A172279 {1, 1, 6, 28, 70, 28, 6, 1, 1},
%e A172279 {1, 1, 8, 42, 126, 63, 17, 4, 1, 1},
%e A172279 {1, 1, 5, 24, 105, 252, 105, 24, 5, 1, 1}
%t A172279 T[n_,k_,l_]=Ceiling[Binomial[n,k]*(l+1)/((1+l)^2+(k-Floor[n/2])^2)];
%t A172279 Table[Table[Table[T[n, k, l], {k, 0, n}], {n, 0, 10}], {l, 0, 5}];
%t A172279 Table[Flatten[Table[Table[T[n, k, l], {k, 0, n}], {n, 0, 10}]], {l, 0, 5}]
%K A172279 nonn,tabl,uned,new
%O A172279 0,5
%A A172279 Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Jan 30 2010

%I A172278
%S A172278 0,2,4,6,8,10,13,15,17,19,21,24,26,28,30,32,35,37,39,41,43,46,48,50,52,
%T A172278 54,56,59,61,63,65,67,70,72,74,76,78,81,83,85,87,89,92,94,96,98,100,102,
%U A172278 105,107,109,111,113,116,118,120,122,124,127,129,131,133,135,138,140
%N A172278 a(n)=floor(n*[sqrt(13)-sqrt(2)]
%K A172278 nonn,new
%O A172278 0,2
%A A172278 Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Jan 30 2010

%I A172277
%S A172277 0,1,3,5,7,9,11,13,14,16,18,20,22,24,26,28,29,31,33,35,37,39,41,43,44,
%T A172277 46,48,50,52,54,56,58,59,61,63,65,67,69,71,73,74,76,78,80,82,84,86,88,
%U A172277 89,91,93,95,97,99,101,103,104,106,108,110,112,114,116,118,119,121,123
%N A172277 a(n)=floor(n*[sqrt(13)-sqrt(3)])
%K A172277 nonn,new
%O A172277 0,3
%A A172277 Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Jan 30 2010

%I A172276
%S A172276 0,1,2,4,5,6,8,9,10,12,13,15,16,17,19,20,21,23,24,26,27,28,30,31,32,34,
%T A172276 35,36,38,39,41,42,43,45,46,47,49,50,52,53,54,56,57,58,60,61,62,64,65,
%U A172276 67,68,69,71,72,73,75,76,78,79,80,82,83,84,86,87,89,90,91,93,94,95,97
%N A172276 a(n)=floor(n*[sqrt(13)-sqrt(5)])
%K A172276 nonn,new
%O A172276 0,3
%A A172276 Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Jan 30 2010

%I A172275
%S A172275 0,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,23,24,
%T A172275 25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,
%U A172275 47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69
%N A172275 a(n)=floor(n*[sqrt(13)-sqrt(7)])
%K A172275 nonn,new
%O A172275 0,4
%A A172275 Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Jan 30 2010

%I A172274
%S A172274 0,0,0,0,1,1,1,2,2,2,2,3,3,3,4,4,4,4,5,5,5,6,6,6,6,7,7,7,8,8,8,8,9,9,9,
%T A172274 10,10,10,10,11,11,11,12,12,12,13,13,13,13,14,14,14,15,15,15,15,16,16,
%U A172274 16,17,17,17,17,18,18,18,19,19,19,19,20,20,20,21,21,21,21,22,22,22,23
%N A172274 a(n)=floor(n*[sqrt(13)-sqrt(11)])
%K A172274 nonn,new
%O A172274 0,8
%A A172274 Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Jan 30 2010

%I A172273
%S A172273 0,1,3,5,7,9,11,13,15,17,19,20,22,24,26,28,30,32,34,36,38,39,41,43,45,
%T A172273 47,49,51,53,55,57,58,60,62,64,66,68,70,72,74,76,77,79,81,83,85,87,89,
%U A172273 91,93,95,97,98,100,102,104,106,108,110,112,114,116,117,119,121,123,125
%N A172273 a(n)=floor(n*[sqrt(11)-sqrt(2)])
%C A172273 Initially similar to A038124 because sqrt(11)-sqrt(2) = 1.90241122... is close to A065421. [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Feb 05 2010]
%K A172273 nonn,new
%O A172273 0,3
%A A172273 Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Jan 30 2010

%I A172272
%S A172272 0,1,3,4,6,7,9,11,12,14,15,17,19,20,22,23,25,26,28,30,31,33,34,36,38,39,
%T A172272 41,42,44,45,47,49,50,52,53,55,57,58,60,61,63,64,66,68,69,71,72,74,76,
%U A172272 77,79,80,82,83,85,87,88,90,91,93,95,96,98,99,101,102,104,106,107
%N A172272 a(n)=floor(n*[sqrt(11)-sqrt(3)])
%K A172272 nonn,new
%O A172272 0,3
%A A172272 Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Jan 30 2010

%I A172270
%S A172270 0,1,2,3,4,5,6,7,8,9,10,11,12,14,15,16,17,18,19,20,21,22,23,24,25,27,28,
%T A172270 29,30,31,32,33,34,35,36,37,38,39,41,42,43,44,45,46,47,48,49,50,51,52,
%U A172270 54,55,56,57,58,59,60,61,62,63,64,65,66,68,69,70,71,72,73,74,75,76,77
%N A172270 a(n)=floor(n*[sqrt(11)-sqrt(5)])
%K A172270 nonn,new
%O A172270 0,3
%A A172270 Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Jan 30 2010

%I A172271
%S A172271 3,107,2634011,29659499,57395627,104792291,271669247,485149499,
%T A172271 568946591,588791807,752530067,863999999,2032678367,2772616499,
%U A172271 2945257307,3505869971,4473547487,4670303507,5470523999,6911999999
%N A172271 Smaller member p of a twin prime pair (p,p+2) with a cube sum N^3.
%C A172271 It is conjectured that the number of twin prime pairs is infinite, one of the great open questions in number theory
%C A172271 It is conjectured that this sequence is infinite
%C A172271 Necessarily the cube base is even: N = 2 x n => p = (2 x n)^3/2 - 1
%C A172271 For n>1: necessarily n = 3 x k as for n = 3 x k+1 resp. n = 3 x k+2: (2 x n)^3/2 - 1 resp. (2 x n)^3/2 + 1 has divisor 3
%C A172271 It has been proved that the pair (p,p+2) is a twin prime couple iff 4 x ((p - 1)! + 1) == -p (mod p x (p+2))
%D A172271 G. H. Hardy, E. M. Wright, E. M., An Introduction to the Theory of Numbers (Fifth edition), Oxford University Press, 1980
%D A172271 N. J. A. Sloane, S. Plouffe: The Encyclopedia of Integer Sequences, Academic Press, 1995
%e A172271 3 + 5 = 2^3, 3 = prime(2), 5 = prime(3)
%e A172271 107 + 109 = (2 x 3)^3, 107 = prime(28), 109 = prime(29)
%e A172271 2634011 + 2634013 = (2 x 87)^3, 2634011 = prime(192181), 2634013 = prime(192182)
%Y A172271 Cf. A001359, A061308, A069496, A061308
%K A172271 base,nonn,new
%O A172271 1,1
%A A172271 Eva-Maria Zschorn (e-m.zschorn(AT)zaschendorf.km3.de), Jan 30 2010

%I A172269
%S A172269 0,1,2,3,4,6,7,8,9,11,12,13,14,16,17,18,19,20,22,23,24,25,27,28,29,30,
%T A172269 32,33,34,35,36,38,39,40,41,43,44,45,46,48,49,50,51,52,54,55,56,57,59,
%U A172269 60,61,62,64,65,66,67,68,70,71,72,73,75,76,77,78,80,81,82,83,84,86,87
%N A172269 a(n)=floor(n*[sqrt(7)-sqrt(2)])
%K A172269 nonn,new
%O A172269 0,3
%A A172269 Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Jan 30 2010

%I A172268
%S A172268 0,0,1,2,3,4,5,6,7,8,9,10,10,11,12,13,14,15,16,17,18,19,20,21,21,22,23,
%T A172268 24,25,26,27,28,29,30,31,31,32,33,34,35,36,37,38,39,40,41,42,42,43,44,
%U A172268 45,46,47,48,49,50,51,52,52,53,54,55,56,57,58,59,60,61,62,63,63,64,65
%N A172268 a(n)=floor(n*[sqrt(7)-sqrt(3)])
%K A172268 nonn,new
%O A172268 0,4
%A A172268 Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Jan 30 2010

%I A172267
%S A172267 0,0,0,1,1,2,2,2,3,3,4,4,4,5,5,6,6,6,7,7,8,8,9,9,9,10,10,11,11,11,12,12,
%T A172267 13,13,13,14,14,15,15,15,16,16,17,17,18,18,18,19,19,20,20,20,21,21,22,
%U A172267 22,22,23,23,24,24,24,25,25,26,26,27,27,27,28,28,29,29,29,30,30,31,31
%N A172267 a(n)=floor(n*[sqrt(7)-sqrt(5)])
%K A172267 nonn,new
%O A172267 0,6
%A A172267 Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Jan 30 2010

%I A172266
%S A172266 0,0,1,2,3,4,4,5,6,7,8,9,9,10,11,12,13,13,14,15,16,17,18,18,19,20,21,22,
%T A172266 23,23,24,25,26,27,27,28,29,30,31,32,32,33,34,35,36,36,37,38,39,40,41,
%U A172266 41,42,43,44,45,46,46,47,48,49,50,50,51,52,53,54,55,55,56,57,58,59,59
%N A172266 a(n)=floor(n*[sqrt(5)-sqrt(2)])
%K A172266 nonn,new
%O A172266 0,4
%A A172266 Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Jan 30 2010

%I A172265
%S A172265 1,3,8,18,38,78,159,321,646,1297,2600,5207,10422,20852,41712,83433,
%T A172265 166876,333762,667534,1335078,2670166,5340342,10680695,21361402,
%U A172265 21361402,42722816,85445645,171891304,341782622,683565259,1367130525
%N A172265 Roots of sin(1/2^k) = 0 where 1/2^k < x < 1 (sum of A024810).
%C A172265 a(k) is the number of x in interval (10^(-k) , 1) defined as f(x) = sin(x^-1)) = 0, for k > =1. It is well known that the function f(x) oscillate indefinitely around 0 when x tend towards zero
%e A172265 a(1) = 1 in interval (0.5 , 1) , a(2) = 3 in interval (0.25 , 1) , a(3) = 8 in interval (0.125 , 1) , etc.
%Y A172265 Cf. A024810.
%K A172265 nonn,new
%O A172265 1,2
%A A172265 Michel Lagneau (mn.lagneau2(AT)orange.fr), Jan 30 2010

%I A172264
%S A172264 0,0,0,0,1,1,1,2,2,2,3,3,3,4,4,4,5,5,5,6,6,6,6,7,7,7,8,8,8,9,9,9,10,10,
%T A172264 10,11,11,11,12,12,12,13,13,13,13,14,14,14,15,15,15,16,16,16,17,17,17,
%U A172264 18,18,18,19,19,19,20,20,20,20,21,21,21,22,22,22,23,23,23,24,24,24,25
%N A172264 a(n)=floor(n*[sqrt(3)-sqrt(2)])
%K A172264 nonn,new
%O A172264 0,8
%A A172264 Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Jan 30 2010

%I A172263
%S A172263 0,1,1,1,2,2,2,3,3,3,3,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,6,6,7,7,7,7,7,7,7,
%T A172263 8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,11,
%U A172263 11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,12,12,12,12,12
%N A172263 a(n) is the greatest zero of Hermite polynomial H(n,x) to nearest integer
%D A172263 M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 801.
%H A172263 M. Abramowitz and I. A. Stegun, eds.,<a href="http://www.nrbook.com/abramowitz_and_stegun/"> Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
%H A172263 <a href="http://www.research.att.com/~njas/sequences/Sindx_He.html#Hermite">Index entries for sequences related to Hermite polynomials</a>
%F A172263 HermiteH(0,x) = 1, HermiteH(1,x) = 2*x,HermiteH(n,x) = 2*x*HermiteH(n-1,x) - 2*(n-1)*HermiteH(n-2,x), for n>1.
%e A172263 H(1,x) = 2x , a(1) = 0 ; H(2,x) = 4*x^2 - 2, a(2) = 1, etc.
%p A172263 for p from 2 to 1000 do; a:= realroot( expand(HermiteH(p,x)), 1/1000000); print (a);od;
%K A172263 nonn,new
%O A172263 0,5
%A A172263 Michel Lagneau (mn.lagneau2(AT)orange.fr), Jan 30 2010

%I A175092
%S A175092 1,2,3,5,7,10,13,17,20,26,28,33,35,41,43,45,52,57,60,69,83,89,98,104,
%T A175092 109,113,116,120,140,142
%N A175092 1 and the non-squares in A167761, sorted to natural order.
%C A175092 The sorted list after removing the entries of A000290 from A167761.
%Y A175092 Cf. A167761.
%K A175092 nonn,new
%O A175092 1,2
%A A175092 Giovanni Teofilatto (g.teofilatto(AT)tiscalinet.it), Jan 30 2010
%E A175092 14 removed by R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Feb 02 2010

%I A172262
%S A172262 2,5,8,11,14,18,21,25,28,32,35,39,43,46,50,54,57,61,65,68,72,76,80,83,
%T A172262 87,91,95,98,102,106,110,114,117,121,125,129,133,137,140,144,148,152,
%U A172262 156,160,163,168,171,175,179,183,186,190,194,198,202,206,210,214,217
%N A172262 a(n) is the greatest root of the Laguerre polynomial L_n(x) to the nearest integer
%D A172262 M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 799.
%H A172262 M. Abramowitz and I. A. Stegun, eds.,<a href="http://www.nrbook.com/abramowitz_and_stegun/"> Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
%H A172262 Eric Weisstein's World of Mathematics,<a href="http://mathworld.wolfram.com/LaguerrePolynomial.html"> Laguerre Polynomial</a>
%e A172262 L_n(1)=2;a(1)=2;L_n(2)=4,7320556;a(2)=5, etc.
%p A172262 for p from 1 to 10000 do; a:=realroot( expand(LaguerreL(p,1,x)), 1/10000); evalf(%);od;
%K A172262 nonn,new
%O A172262 2,1
%A A172262 Michel LAGNEAU (mn.lagneau2(AT)orange.fr), Jan 30 2010

%I A172261
%S A172261 0,0,25,1847,162531,2501726,21243084,119138166,502726650,1724809105,
%T A172261 5059647669,13132889249,30905051345,67124176002,136380034610,
%U A172261 261909043488,479315827404,841394145399,1424246670499,2334919892115
%N A172261 Number of ways to place 8 nonattacking kings on an 8 X n board
%H A172261 V. Kotesovec, <a href="http://web.telecom.cz/vaclav.kotesovec/math.htm">Number of ways of placing non-attacking queens and kings on boards of various sizes</a>
%F A172261 Explicit formula (Vaclav Kotesovec, 30.1.2010): a(n) = (1048576n^8-30277632n^7+406210560n^6-3319585920n^5+18136811049n^4-68048382318n^3+171628664735n^2-266425935930n+194935658400)/2520, n>=7
%Y A172261 A172202, A172203, A172204, A172205, A172206
%K A172261 nonn,new
%O A172261 1,3
%A A172261 Vaclav Kotesovec (kotesovec(AT)chello.cz), Jan 30 2010

%I A172260
%S A172260 0,1,3,5,6,8,10,12,13,15,17,19,20,22,24,25,27,29,31,32,34,36,38,39,41,
%T A172260 43,45,46,48,50,51,53,55,57,58,60,62,64,65,67,69,71,72,74,76,77,79,81,
%U A172260 83,84,86,88,90,91,93,95,96,98,100,102,103,105,107,109,110,112,114,116
%N A172260 A Beatty sequence: a(n) = floor[n*sqrt 3].
%F A172260 a(n) = A022838(n). [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Feb 05 2010]
%K A172260 nonn,new
%O A172260 0,3
%A A172260 Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Jan 30 2010

%I A172259
%S A172259 1,2,5,14,38,101,275,746,2026,5507,14969,40689,110604,300652,817255,
%T A172259 2221528,6038739,16414993,44620576,121291299
%N A172259 Let CK(m) denote the complete elliptic integral of the first kind. a(n) is the nth smallest integer k such that [(CK(1/k)] = [CK(1/(k-1)] + 1.
%C A172259 F(z,k) = int(1/sqrt(1-t^2)/sqrt(1-k^2*t^2),t=0..z)and the complete elliptic integral CK is defined by CK(k) = F(1,sqrt(1-k^2)). We calculate the values of CK(k)with k = 1/p, p = 1,2,3, ...and we purpose an very interessant property: a(n+1) / a(n) tend toward e = 2.7182818... when n tend into infinity. For example, a(8) / a(7) = 2.718281581 ; a(9) / a(8) = 2.7182817562 ; etc.
%D A172259 M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math.Series 55, Tenth Printing, 1972, p. 575, Eq. 16.22.1 and 16.22.2.
%D A172259 Chapter 17, "Elliptic Integrals" of M. Abramowitz and I. Stegun, Handbook of Mathematical Functions. Dover Publications Inc., New York, 1046 p., (1965).
%D A172259 A. Cayley, A memoir on the transformation of elliptic functions, Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 9, p. 128.
%H A172259 A. Cayley, <a href="http://digital.library.cornell.edu/cgi/t/text/text-idx?c=math;idno=cayl005">An Elementary Treatise on Elliptic Functions</a>, G. Bell and Sons, London, 1895, p. 56.
%H A172259 F. Clarke,<a href="http://www-maths.swan.ac.uk/staff/fwc/Gregynog-slides.pdf">The Taylor Series Coefficients of the Jacobi Elliptic Functions</a>, slides.
%F A172259 F(z,k) = int(1/sqrt(1-t^2)/sqrt(1-k^2*t^2),t=0..z) CK is defined by CK(k)= F(1,sqrt(1-k^2)), n = nth integer where a(n) = k such that [(CK(1/k)] = [CK(1/(k-1)] + 1
%e A172259 floor(CK(1) = floor(1,570796327 ) = 1 ; a(1) = 1 floor(CK(1/2)) = floor(2.156515647) = 2 ; floor (CK(1)) = floor(1.570796327) = 1 ; a(2) = 2 floor(CK(1/5)) = floor (3.016112492 = 3 ; floor (CK(1/4)) = floor(2.801206085) = 2 ; a(3) = 5 floor(CK(1/14)) = floor(4.029221058) = 4 ; floor(CK(1/13)) = floor (3,955623229) = 3 ; a(4) = 14 floor(CK(1/38)) = floor(5,024577436) = 5 ; floor (CK(1/37)) = floor(4,997942512) = 4 ; a(5) = 38
%p A172259 for p from 1 to 10000 do; a:= expand(EllipticCK(1/p));evalf(%); od;
%Y A172259 elliptic functions: A001936 , A002318 , A001937 , A001934 , A001938 , A002754 , A001939 , A001940 , A001941 , A002753 , A006089 , A004005
%K A172259 nonn,new
%O A172259 1,2
%A A172259 Michel LAGNEAU (mn.lagneau2(AT)orange.fr), Jan 30 2010

%I A172258
%S A172258 2,29,47,73,79,89,139,173,197,199,227,229,269,277,307,337,349,353,379,
%T A172258 383,397,409,439,463,467,503,509,523
%N A172258 Primes p such that either 2*p-+3 is prime.
%e A172258 a(1)=2 because 2*2-3=1(nonprime) and 2*2+3=7(prime); a(2)=29 because 2*29-3=55(nonprime) and 2*29+3=61(prime).
%Y A172258 Cf. A000040, A131426.
%K A172258 nonn,new
%O A172258 1,1
%A A172258 Juri-Stepan Gerasimov (2stepan(AT)rambler.ru), Jan 30 2010

%I A172256
%S A172256 17,59,61,103,109,131,149,151,163,179,239,257,271,281,293,313,359,367,
%T A172256 373,389,401,419,449,479,491,499,541
%N A172256 Primes p such that 2*p+-3 are both nonprimes.
%Y A172256 Cf. A000040, A131426, A141468.
%K A172256 nonn,new
%O A172256 1,1
%A A172256 Juri-Stepan Gerasimov (2stepan(AT)rambler.ru), Jan 30 2010

%I A172257
%S A172257 7,73,1476193,10087249723
%N A172257 The prime that gives n primes via concatenation of decremented numbers in sequence as quickly as possible
%C A172257 4th term and motivation from Jens Kruse Andersen. A152396 is similar,
%C A172257 but starts with concatenation of two numbers being prime.
%e A172257 For n=1, 76543 is prime; for n=2, 7372717069 and 73727170696867 are
%e A172257 both prime; and the concatenation of 11 and then 13 numbers give the
%e A172257 next two terms.
%Y A172257 A152396
%K A172257 base,nonn,uned,new
%O A172257 1,1
%A A172257 James G. Merickel (merk7(AT)verizon.net), Jan 30 2010

%I A171793
%S A171793 1,1,0,3,0,0,3,9,0,0,0,1,18,9,27,0,0,0,0,0,9,45,57,54,27,81,0,0,0,0,0,0,
%T A171793 0,36,87,270,81,297,171,162,81,243,0,0,0,0,0,0,0,0,0,84,261,567,756,936,
%U A171793 585,972,729,891,513,486,243,729,0,0,0,0,0,0,0,0,0,0,0,126,774,1080
%N A171793 Triangle read by rows: T(n,k) is the number of ternary trees with n edges and path length k; 0<=k<=n(n-1)/2.
%F A171793 G.f. satisfies: A(x,q) = 1 + x*A(q*x,q)^3.
%F A171793 Row sums equal A001764, which enumerates ternary trees and has g.f.: G(x) = 1 + x*G(x)^3.
%F A171793 Column sums equal A132331(k), which is the number of ternary trees of path length k.
%e A171793 G.f.: A(x,q) = 1 + x + (3*q)*x^2 + (3*q^2 + 9*q^3)*x^3 + (q^3 + 18*q^4 + 9*q^5 + 27*q^6)*x^4 +...
%e A171793 A(x,q)^3 = 1 + 3*x + (3 + 9*q)*x^2 + (1 + 18*q + 9*q^2 + 27*q^3)*x^3 +...
%e A171793 Triangle begins:
%e A171793 1;
%e A171793 1;
%e A171793 0,3;
%e A171793 0,0,3,9;
%e A171793 0,0,0,1,18,9,27;
%e A171793 0,0,0,0,0,9,45,57,54,27,81;
%e A171793 0,0,0,0,0,0,0,36,87,270,81,297,171,162,81,243;
%e A171793 0,0,0,0,0,0,0,0,0,84,261,567,756,936,585,972,729,891,513,486,243,729;
%e A171793 0,0,0,0,0,0,0,0,0,0,0,126,774,1080,2817,2682,4383,1998,4941,3294,3780,2241,4374,2187,2673,1539,1458,729,2187; ...
%o A171793 (PARI) {T(n,k)=local(A=1+x);for(i=1,n,A=1+x*subst(A,x,q*x+x*O(x^n))^3); polcoeff(polcoeff(A,n,x)+O(q^(n*(n-1)/2+1)),k,q)}
%Y A171793 Cf. A001764 (row sums), A132331 (column sums), A138157 (variant).
%K A171793 nonn,tabl,new
%O A171793 0,4
%A A171793 Paul D. Hanna (pauldhanna(AT)juno.com), Jan 29 2010

%I A172255
%S A172255 341,902,1547,2652,4039,5768,7673,9720,12185,14886,17707,20984,25017,
%T A172255 29386,33757,38438,43899,50500,58457,66778,75259,84170,94431,105016,
%U A172255 116321,129122,142863,156610,170591,185082,200791,216632,233337,252042
%N A172255 Partial sums of pseudoprimes A001567.
%C A172255 An odd composite number n is a Fermat pseudoprime to base b iff b^(n-1) == 1 mod n. Fermat pseudoprimes to base 2 are often simply called pseudoprimes, or Sarrus numbers. The subsequence of pseudoprime partial sum of pseudoprimes begins 341, and the next exceeds a(40). The subsequence of prime partial sum of pseudoprimes begins 7673, 17707, 33757, 270763.
%F A172255 a(n) = SUM[i=1..n] {odd composite numbers n such that 2^(n-1) == 1 mod n}.
%e A172255 a(15) = 341 + 561 + 645 + 1105 + 1387 + 1729 + 1905 + 2047 + 2465 + 2701 + 2821 + 3277 + 4033 + 4369 + 4371 = 33757 is prime.
%Y A172255 Cf. A000040, A001567.
%K A172255 easy,nonn,new
%O A172255 1,1
%A A172255 Jonathan Vos Post (jvospost3(AT)gmail.com), Jan 29 2010

%I A172254
%S A172254 1,1,3,2,1,1,3,2,1,1,3,2,1,1,3,2,1,1,3,2,1,1,3,2,1,1,
%T A172254 3,2,1,1,3,2,1,1,3,2,1,1,3,2,1,1,3,2,1,1,3,2,1,1,3,2,
%U A172254 1,1,3,2,1,1,3,2,1,1,3,2,1,1,3,2,1,1,3,2,1,1,3,2,1
%V A172254 1,-1,3,-2,-1,-1,3,-2,-1,-1,3,-2,-1,-1,3,-2,-1,-1,3,-2,-1,-1,3,-2,-1,-1,
%W A172254 3,-2,-1,-1,3,-2,-1,-1,3,-2,-1,-1,3,-2,-1,-1,3,-2,-1,-1,3,-2,-1,-1,3,-2,
%X A172254 -1,-1,3,-2,-1,-1,3,-2,-1,-1,3,-2,-1,-1,3,-2,-1,-1,3,-2,-1,-1,3,-2,-1
%N A172254 Value of Boubaker polynomial B_{4n}(1). The Boubaker polynomials have been named after Boubaker Boubaker (1897-1966)).
%D A172254 S. Fridjine, M. Amlouk, , A new parameter: An ABACUS for optimizig functional materials using the Boubaker polynomials expansion scheme, Modern Phys. Lett. B 23(2009)2179-2185
%D A172254 A. Belhadj, J. Bessrour, M. Bouhafs, L. Barrallier, Experimental and theoretical cooling velocity profile inside laser welded metals using keyhole approximation and Boubaker polynomials expansion, J. of Thermal Analysis and Calorimetry 97 (2009)911-916
%D A172254 D. H. Zhang, F.W. Li, A Boubaker Polynomials Expansion Scheme BPES-Related Analytical Solution to Williams-Brinkmann Stagnation Point Flow Equation at a Blunt Body, Ir. Journal of App. Phys. Lett. IJAPLett. 2 (2009) 25-31
%F A172254 for n>1:B_{4n}(X)=(X^4-4X^2+2)B_{4(n-1)}(X)-B_{4(n-2)}(X),
%Y A172254 Cf. A162180
%K A172254 nonn,new
%O A172254 1,3
%A A172254 L. Naing (lnaing.yangon_univ(AT)yahoo.in), Jan 29 2010

%I A172253
%S A172253 1,3,7,9,11,13,17,19,23,27,29,31,33,37,41,43,47,49,51,53,57,59,61,67,69,
%T A172253 71,73,77,79,81,83,87,89,91,93,97,99
%N A172253 a(n) = Successive numbers x such that value of function N(9^x-1,9^x) defined as product of different prime factors of product 9^x(9^x-1) is equal 3(9^x-1)/4
%C A172253 Maximal value of radical function N(a,b,9^x) for every number 9^x and everyone combination of partitions a,b such that a + b = 9^x and GCD[a,b,3]=1 is never less as 3(9^x-1)/4 and is exactly equal 3(9^x-1)/4 for exponents x in this sequence (*Artur Jasinski*).
%C A172253 Conjecture (*Artur Jasinski*): This sequence is infinite.
%K A172253 nonn,new
%O A172253 1,2
%A A172253 Artur Jasinski (grafix(AT)csl.pl), Jan 29 2010

%I A172252
%S A172252 1,7,23,55,119,247,503,1015,2039,4087,8183,16375,32759,65527,131063,
%T A172252 262135,524279,1048567,2097143,4194295,8388599,16777207,33554423,
%U A172252 67108855,134217719,268435447,536870903,1073741815,2147483639
%V A172252 -1,7,23,55,119,247,503,1015,2039,4087,8183,16375,32759,65527,131063,
%W A172252 262135,524279,1048567,2097143,4194295,8388599,16777207,33554423,
%X A172252 67108855,134217719,268435447,536870903,1073741815,2147483639
%N A172252 a(n) = 4 2^n-9
%C A172252 a(1)=-1, rest of terms are positive. Some functions N(9^n-1,9^n) defined as product different prime divisors of (9^n-1)9^n are equal 3(9^n+4*2^m-9)/2^(m+1) for some n (*Artur Jasinski*)
%t A172252 Table[4 2^n - 9, {n, 1, 100}]
%K A172252 sign,new
%O A172252 1,2
%A A172252 Artur Jasinski (grafix(AT)csl.pl), Jan 29 2010

%I A172251
%S A172251 1,2,3,4,5,6,8,9,10,11,12,13,14,16,17,19,20,23,24,25,26,29,32,33,34,35,
%T A172251 38,41,46,47,48,50,53,54,58,62,63,75,86,96,101,102,113,117,129,162,195,
%U A172251 204,233
%N A172251 Arises in the representability of integers as sums of triangular numbers.
%C A172251 Wieb Bosma, p.10: Following the bounds given in the proof of Theorem 1.6, computational evidence suggests that... a proof of the above identity using the techniques of Bhargava and Hanke developed in the proof of the 290-Theorem may require a careful analysis of a possible Siegel zero. The sequence given is thus conjectured to be complete as shown.
%D A172251 M. Bhargava, J. Hanke, Universal Quadratic Forms and the 290-Theorem, preprint.
%H A172251 Wieb Bosma, Ben Kane, <a href="http://arxiv.org/abs/0905.3594">The triangular theorem of eight and a certain non-finiteness theorem </a>, v.2, Jan 28, 2010.
%Y A172251 Cf. A030051.
%K A172251 fini,full,nonn,new
%O A172251 1,2
%A A172251 Jonathan Vos Post (jvospost3(AT)gmail.com), Jan 29 2010

%I A172248
%S A172248 2,3,4,5,6,8,9,10,12,14,16,18,20,22,24,26,27,28,30,32,33,34,36,38,42,44,
%T A172248 45,46,48,50,51,52,54,56,60,62,64,66,68,69,70,72,74,75,76,78,80,81,82,
%U A172248 84,86,87,88,90,92,94,96,98,100,102,104,105,106,108,110,112,114,116,118
%N A172248 a(n) = Numbers n for which doesn't exist partitions n on sum a + b such that a<=b and gcd(a,b,n)=1 having that same values of function N(a,b,n) defined as product different prime divisors of a*b*n.
%C A172248 Number of partitions n on sum a + b such that a<=b gcd(a,b,n)=1 see: A023022
%C A172248 Number of partitions having different value of function N(a,b,n) see: A172245
%C A172248 Numbers n for which existed cases with that same value of function N(a,b,n) see: A172247
%C A172248 Numbers n for which all partitions have different value of function N(a,b,n) see: A172248.
%e A172248 7 doesn't belong to this sequence because for 7 we have two partitions 7=1+6 and 7=3+4 with that same values of N(a,b,n) respectively 1*2*3*7=42 and 2*3*7=42.
%Y A172248 A023022, A172245, A172246, A172247.
%K A172248 nonn,uned,new
%O A172248 2,1
%A A172248 Artur Jasinski (grafix(AT)csl.pl), Jan 29 2010

%I A172247
%S A172247 7,11,13,15,17,19,21,23,25,29,31,35,37,39,40,41,43,47,49,53,55,57,58,59,
%T A172247 61,63,65,67,71,73,77,79,83,85,89,91,93,95,97,99,101,103,107,109,111,
%U A172247 113,115,117,119,121,123,125,127,129,131,133,136,137,139,143,145,147
%N A172247 a(n) = Numbers n for which existed such of partitions n on sum a + b such that a<=b and gcd(a,b,n)=1 having that same values of function N(a,b,n) defined as product different prime divisors of a*b*n.
%C A172247 Number of partitions n on sum a + b such that a<=b gcd(a,b,n)=1 see: A023022
%C A172247 Number of partitions having different value of function N(a,b,n) see: A172245
%C A172247 Numbers n for which existed cases with that same value of function N(a,b,n) see: A172247
%C A172247 Numbers n for which all partitions have different value of function N(a,b,n) see: A172248.
%e A172247 a(1)=7 because for 7 we have two partitions 7=1+6 and 7=3+4 with that same values of N(a,b,n) respectively 1*2*3*7=42 and 2*3*7=42.
%Y A172247 A023022, A172245, A172246, A172248.
%K A172247 nonn,new
%O A172247 2,1
%A A172247 Artur Jasinski (grafix(AT)csl.pl), Jan 29 2010

%I A172246
%S A172246 0,0,0,0,0,2,0,0,0,2,0,2,0,2,0,2,0,3,0,2,0,4,0,3,0,0,0,2,0,4,0,0,0,3,0,
%T A172246 3,0,2,2,3,0,3,0,0,0,5,0,4,0,0,0,4,0,4,0,2,2,3,0,5,0,2,0,4,0,4,0,0,0,7,
%U A172246 0,7,0,0,0,5,0,6,0,0,0,4,0,3,0,0,0,5,0,5,0,3,0,4,0,5,0,3,0,5,0,6,0,0,0
%N A172246 a(n) = Number of partitions n on sum a + b such that a<=b and gcd(a,b,n)=1 and having that same values of function N(a,b,n) defined as product different prime divisors of a*b*n.
%C A172246 Number of partitions n on sum a + b such that a<=b gcd(a,b,n)=1 see: A023022
%C A172246 Number of partitions having different value of function N(a,b,n) see: A172245
%C A172246 Numbers n for which existed cases with that same value of function N(a,b,n) see: A172247
%C A172246 Numbers n for which all partitions have different value of function N(a,b,n) see: A172248.
%e A172246 a(7)=2 because we have two partitions 7=1+6 and 7=3+4 with different values of N(a,b,n) respectively 1*2*3*7=42 and 2*3*7=42.
%Y A172246 A023022, A172245, A172247, A172248.
%K A172246 nonn,new
%O A172246 2,6
%A A172246 Artur Jasinski (grafix(AT)csl.pl), Jan 29 2010

%I A172250
%S A172250 1,0,1,0,1,0,0,0,2,1,0,0,1,1,1,0,0,0,3,2,0,0,0,0,1,3,4,1,0,0,0,0,4,
%T A172250 2,2,1,0,0,0,0,1,6,9,3,0,0,0,0,0,0,5,0,9,6,1,0,0,0,0,0,1,10,15,3,
%U A172250 3,1,0,0,0,0,0,0,6,5,24,18,4,0,0,0,0,0,0,0,1,15,20,6,18,8,1
%V A172250 1,0,1,0,1,0,0,0,2,-1,0,0,1,1,-1,0,0,0,3,-2,0,0,0,0,1,3,-4,1,0,0,0,0,4,
%W A172250 -2,-2,1,0,0,0,0,1,6,-9,3,0,0,0,0,0,0,5,0,-9,6,-1,0,0,0,0,0,1,10,-15,3,
%X A172250 3,-1,0,0,0,0,0,0,6,5,-24,18,-4,0,0,0,0,0,0,0,1,15,-20,-6,18,-8,1
%N A172250 Triangle, read by rows, given by [0,1,-1,0,0,0,0,0,0,0,...] DELTA [1,-1,1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938.
%F A172250 T(n,k)= T(n-1,k-1) + T(n-2,k-1) - T(n-2,k-2), T(0,0)=1, T(n,k) = 0 if k>n or if k<0. Sum_{k, 0<=k<=n} T(n,k)= 1^n = A000012(n).
%e A172250 Triangle begins : 1 ; 0,1 ; 0,1,0 ; 0,0,2,-1 ; 0,0,1,1,-1 ; 0,0,0,3,-2,0 ; ...
%Y A172250 Cf. A101950
%K A172250 sign,tabl,new
%O A172250 0,9
%A A172250 Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Jan 29 2010

%I A172249
%S A172249 1,0,3,0,1,8,0,0,6,21,0,0,1,25,55,0,0,0,9,90,144,0,0,0,1,51,300,377,0,0,
%T A172249 0,0,12,234,954,987,0,0,0,0,1,86,951,2939,2584,0,0,0,0,0,15,480,3573,
%U A172249 8850,6765,0,0,0,0,0,1,130,2305,12707,26195,17711,0,0,0,0,0,0,18,855
%N A172249 Triangle, read by rows, given by [0,1/3,-1/3,0,0,0,0,0,0,0,...] DELTA [3,-1/3,1/3,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938.
%C A172249 Diagonal sums : |A077897|. Column sums : A001353 .
%F A172249 T(n,k)= 3*T(n-1,k-1) + T(n-2,k-1) - T(n-2,k-2), T(0,0)=1, T(n,k) = 0 if k>n or if k<0. Sum_{k, 0<=k<=n} T(n,k)= 3^n = A000244(n).
%e A172249 Triangle begins : 1 ; 0,3 ; 0,1,8 ; 0,0,6,21 ; 0,0,1,25,55 ; 0,0,0,9,90,144 ; ...
%Y A172249 Cf. A001871, A001906, A125662
%K A172249 nonn,tabl,new
%O A172249 0,3
%A A172249 Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Jan 29 2010

%I A172245
%S A172245 1,1,1,2,1,2,2,3,2,4,2,5,3,3,4,7,3,7,4,5,5,8,4,8,6,9,6,13,4,12,8,10,8,
%T A172245 10,6,16,9,11,7,18,6,19,10,12,11,19,8,18,10,16,12,23,9,17,12,17,13,27,8,
%U A172245 26,15,17,16,21,10,30,16,22,12,29,12,30,18,20,18,26,12,34,16,27,20,38
%N A172245 a(n) = Number of partitions n on sum a + b such that a<=b and gcd(a,b,n)=1 and having different values of function N(a,b,n) defined as product different prime divisors of a*b*n.
%C A172245 Number of partitions n on sum a + b such that a<=b gcd(a,b,n)=1 see: A023022
%C A172245 Number of partitions having that same value of function N(a,b,n) see: A172246
%C A172245 Numbers n for which existed cases with that same value of function N(a,b,n) see: A172247
%C A172245 Numbers n for which all partitions have different value of function N(a,b,n) see: A172248.
%e A172245 a(5)=2 because we have two partitions 5=1+4 and 5=2+3 with different values of N(a,b,n) respectively 1*2*5=10 and 2*3*5=30.
%Y A172245 A023022, A172246, A172247, A172248.
%K A172245 nonn,new
%O A172245 2,4
%A A172245 Artur Jasinski (grafix(AT)csl.pl), Jan 29 2010

%I A172244
%S A172244 1,599,14951,9314449
%N A172244 (+/-)Integers for which equation 2x^2 + xy + 3y^2 + z^3 - z = a(n) have not integer solutions.
%H A172244 William C. Jagy, following Irving Kaplansky, 2009. <a href="http://zakuski.math.utsa.edu/~jagy/jagy_conjecture_23.pdf">Integers not represented by 2x^2 + xy + 3y^2 + z^3 - z</a>
%K A172244 nonn,new
%O A172244 1,2
%A A172244 Artur Jasinski (grafix(AT)csl.pl), Jan 29 2010

%I A172243
%S A172243 3,8,21,38,79,176,289,482,723,980,1333,1782,2359,3000,3673,4442,5371,
%T A172243 6524,7741,9150,10751,12864,15553,18306,21443,24772,28229,32710,37703,
%U A172243 44232
%N A172243 Partial sums of Proth primes A080076.
%C A172243 The subsequence of primes in this sequence begins 3, 79, 3673, 7741, 28229. The subsubsequence of Proth prime partial sum of Proth primes begins 3, and then what comes next?
%H A172243 Weisstein, Eric W., <a href="http://mathworld.wolfram.co/ProthPrime.html">Proth Prime</a>.
%F A172243 a(n) = SUM[i=1..n] A080076 (i) = SUM[i=1..n] {Proth numbers p such that p is prime, i.e., a number of the form N=k*2^n+1 for odd k, n a positive integer, and 2^n>k}.
%e A172243 a(5) = 3 + 5 + 13 + 17 + 41 = 79, which is prime but not a Proth prime. a(27) = 3 + 5 + 13 + 17 + 41 + 97 + 113 + 193 + 241 + 257 + 353 + 449 + 577 + 641 + 673 + 769 + 929 + 1153 + 1217 + 1409 + 1601 + 2113 + 2689 + 2753 + 3137 + 3329 + 3457 = 28229 is prime, but not Proth prime.
%Y A172243 Cf. A000040, A002253-A002256, A080076.
%K A172243 easy,more,nonn,new
%O A172243 1,1
%A A172243 Jonathan Vos Post (jvospost3(AT)gmail.com), Jan 29 2010

%I A172242
%S A172242 1,22,264,2288,16016,96096,512512,2489344,11202048,47297536,189190144,
%T A172242 722362368,2648662016,9372188672,32133218304,107110727680,348109864960,
%U A172242 1105760747520,3440144547840,10501493882880,31504481648640
%N A172242 Number of 10-D hypercubes in an n-dimensional hypercube.
%C A172242 With a different offset, number of n-permutations (n>=8) of 3 objects: u, v, z with repetition allowed, containing exactly ten (10) u's.
%F A172242 a(n) =C[n + 10, 10]*2^n, n>=0
%t A172242 Table[Binomial[n + 10, 10]*2^n, {n, 0, 22}]
%o A172242 (Other) sage:[lucas_number2(n, 2, 0)*binomial(n,10)/2^10for n in xrange(10, 31)]# [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Feb 05 2010]
%Y A172242 Cf. A001788, A001789, A003472, A054849, A002409, A140325, A140354
%K A172242 nonn,new
%O A172242 10,2
%A A172242 Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jan 29 2010

%I A172239
%S A172239 2,5,12,43,254,2565,200560492696,
%T A172239 1719620105458406433483340568317543019584575635895742560438771105058321655238562613083979651479555788009994557822024565226932906295208262756822476224186807
%N A172239 Partial sums of primorial primes A018239.
%C A172239 The subsequence of primes in this sequence begin: a(1) = 2, a(2) = 5, a(4) = 43. The subsubsequence of primorial prime partial sums of primorial primes begins 2, 5. Will someone extend these subsequences?
%F A172239 a(n) = SUM[i=1..n] A018239(i) = SUM[i=1..n] {p in A006862 and p prime} = SUM[i=1..n] (1 + Product[j=1..i] A000040(i)).
%e A172239 a(4) = 2 + 3 + 7 + 31 = 43, which is prime, but not primorial prime.
%Y A172239 Cf. A000040, A002110, A018239.
%K A172239 nonn,new
%O A172239 1,1
%A A172239 Jonathan Vos Post (jvospost3(AT)gmail.com), Jan 29 2010

%I A172238
%S A172238 2,3,7,13,19,37,43,67,79,97,103,109,127,163,193,223,229,277,307,313,349,
%T A172238 379,397,439,457,463,487,499,613,643,673,739,757,769,823,853,859,877,
%U A172238 883,907,937,967,1009,1087,1093,1213,1279,1297,1303,1423,1429,1447,1483
%N A172238 Primes p such that either p+5/2+-3/2 is prime.
%C A172238 Two together with primes p such that p and p+4 are both primes.
%e A172238 a(1)=2 because 2+5/2-3/2=3(prime) and 2+5/2+3/2=6(nonprime); a(2)=3 because 3+5/2-3/2=4(nonprime) and 3+5/2+3/2=7(prime).
%Y A172238 Cf. A023200.
%K A172238 nonn,new
%O A172238 1,1
%A A172238 Juri-Stepan Gerasimov (2stepan(AT)rambler.ru), Jan 29 2010

%I A172237
%S A172237 0,0,1,0,1,1,0,1,1,2,0,1,1,3,3,0,1,1,4,5,5,0,1,1,5,7,11,8,0,1,1,6,9,19,
%T A172237 21,13,0,1,1,7,11,29,40,43,21,0,1,1,8,13,41,65,97,85,34
%N A172237 Anti-diagonal triangle sequence of sequences of the type: t(n,k)=t(n-1,k)+k*t(n-2,k)
%C A172237 Row sums are:
%C A172237 {0, 1, 2, 4, 8, 16, 33, 70, 153, 345,...}.
%C A172237 Characteristic polynomials are:
%C A172237 x^2 - x - a == 0.
%F A172237 t(n,k)=t(n-1,k)+k*t(n-2,k);
%F A172237 a(n,k)=Antidiagonal(t(n,k))
%e A172237 {0},
%e A172237 {0, 1},
%e A172237 {0, 1, 1},
%e A172237 {0, 1, 1, 2},
%e A172237 {0, 1, 1, 3, 3},
%e A172237 {0, 1, 1, 4, 5, 5},
%e A172237 {0, 1, 1, 5, 7, 11, 8},
%e A172237 {0, 1, 1, 6, 9, 19, 21, 13},
%e A172237 {0, 1, 1, 7, 11, 29, 40, 43, 21},
%e A172237 {0, 1, 1, 8, 13, 41, 65, 97, 85, 34}
%t A172237 f[0, a_] := 0; f[1, a_] := 1;
%t A172237 f[n_, a_] := f[n, a] = f[n - 1, a] + a*f[n - 2, a];
%t A172237 m1 = Table[f[n, a], {n, 0, 10}, {a, 1, 11}];
%t A172237 Table[Table[m1[[m, n - m + 1]], {m, 1, n}], {n, 1, 10}];
%t A172237 Flatten[%]
%K A172237 nonn,tabl,uned,new
%O A172237 0,10
%A A172237 Roger L. Bagula and Gary W. Adamson (rlbagulatftn(AT)yahoo.com), Jan 29 2010

%I A172236
%S A172236 0,0,1,0,1,1,0,1,2,2,0,1,3,5,3,0,1,4,10,12,5,0,1,5,17,33,29,8,0,1,6,26,
%T A172236 72,109,70,13,0,1,7,37,135,305,360,169,21,0,1,8,50,228,701,1292,1189,
%U A172236 408,34
%N A172236 Anti-diagonal triangle sequence of sequences of the type: t(n,k)=k*t(n-1,k)+t(n-2,k)
%C A172236 Row sums are:
%C A172236 {0, 1, 2, 5, 12, 32, 93, 297, 1035, 3911,...}.
%C A172236 Characteristic polynomials are:
%C A172236 x^2 - a*x - 1 == 0
%C A172236 and they gives Binet root functions as: b0 = x /. Solve[x^2 - a*x - 1 == 0, x][[1]];
%C A172236 a0 = x /. Solve[x^2 - a*x - 1 == 0, x][[2]];
%C A172236 fb[n_, a_] := (a0^n - b0^n)/Sqrt[4 + a^2].
%F A172236 t(n,k)=k*t(n-1,k)+t(n-2,k);
%F A172236 a(n,k)=Antidiagonal(t(n,k))
%e A172236 {0},
%e A172236 {0, 1},
%e A172236 {0, 1, 1},
%e A172236 {0, 1, 2, 2},
%e A172236 {0, 1, 3, 5, 3},
%e A172236 {0, 1, 4, 10, 12, 5},
%e A172236 {0, 1, 5, 17, 33, 29, 8},
%e A172236 {0, 1, 6, 26, 72, 109, 70, 13},
%e A172236 {0, 1, 7, 37, 135, 305, 360, 169, 21},
%e A172236 {0, 1, 8, 50, 228, 701, 1292, 1189, 408, 34}
%t A172236 f[0, a_] := 0; f[1, a_] := 1;
%t A172236 f[n_, a_] := f[n, a] = a*f[n - 1, a] + f[n - 2, a];
%t A172236 m1 = Table[f[n, a], {n, 0, 10}, {a, 1, 11}];
%t A172236 Table[Table[m1[[m, n - m + 1]], {m, 1, n}], {n, 1, 10}];
%t A172236 Flatten[%]
%K A172236 nonn,tabl,uned,new
%O A172236 0,9
%A A172236 Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Jan 29 2010

%I A172234
%S A172234 0,2,1478,50726,573797,3581924,15516804,52550366,149162199,370817854,
%T A172234 831571604,1717417198,3316210152,6054985120,10545491888
%N A172234 Number of ways to place 7 nonattacking wazirs on an 7 X n board
%C A172234 Wazir is a (fairy chess) leaper [0,1]
%H A172234 V. Kotesovec, <a href="http://web.telecom.cz/vaclav.kotesovec/math.htm">Number of ways of placing non-attacking queens and kings on boards of various sizes</a>
%F A172234 Explicit formula (Vaclav Kotesovec, 29.1.2010): a(n) = (117649*n^7-1663893*n^6+10942729*n^5-43685355*n^4+114945646*n^3-199980312*n^2+213228096*n-107390880)/720, n>=6. For any fixed value of k > 1, a(n) = 1/k!*(kn)^k - (k-1)(5k-2)/2/k!*(kn)^(k-1) + ...
%Y A172234 A172229, A172230, A172231, A172232, A061993
%K A172234 nonn,new
%O A172234 1,2
%A A172234 Vaclav Kotesovec (kotesovec(AT)chello.cz), Jan 29 2010

%I A172232
%S A172232 0,2,504,10010,78052,368868,1280832,3612344,8774380,19049692,37898664,
%T A172232 70311824,123209012,205885204,330502992,512631720,771833276,1132294540,
%U A172232 1623506488,2280989952
%N A172232 Number of ways to place 6 nonattacking wazirs on an 6 X n board
%C A172232 Wazir is a (fairy chess) leaper [0,1]
%H A172232 V. Kotesovec, <a href="http://web.telecom.cz/vaclav.kotesovec/math.htm">Number of ways of placing non-attacking queens and kings on boards of various sizes</a>
%F A172232 Explicit formula (Vaclav Kotesovec, 29.1.2010): a(n) = 2*(486*n^6-5670*n^5+30240*n^4-95230*n^3+187899*n^2-220775*n+120540)/15, n>=5
%Y A172232 A172229, A172230, A172231, A061992
%K A172232 nonn,new
%O A172232 1,2
%A A172232 Vaclav Kotesovec (kotesovec(AT)chello.cz), Jan 29 2010

%I A172231
%S A172231 0,2,174,1998,10741,38438,107004,251354,522528,990816,1748883,2914894,
%T A172231 4635639,7089658,10490366,15089178,21178634,29095524,39224013,51998766
%N A172231 Number of ways to place 5 nonattacking wazirs on an 5 X n board
%C A172231 Wazir is a (fairy chess) leaper [0,1]
%H A172231 V. Kotesovec, <a href="http://web.telecom.cz/vaclav.kotesovec/math.htm">Number of ways of placing non-attacking queens and kings on boards of various sizes</a>
%F A172231 Explicit formula (Vaclav Kotesovec, 29.1.2010): a(n) = (625*n^5-5750*n^4+23535*n^3-54202*n^2+70640*n-41616)/24, n>=4
%Y A172231 A172228, A172229, A172230, A061991
%K A172231 nonn,new
%O A172231 1,2
%A A172231 Vaclav Kotesovec (kotesovec(AT)chello.cz), Jan 29 2010

%I A172230
%S A172230 0,2,61,405,1502,4072,9091,17791,31660,52442,82137,123001,177546,248540,
%T A172230 339007,452227,591736,761326,965045,1207197
%N A172230 Number of ways to place 4 nonattacking wazirs on an 4 X n board
%C A172230 Wazir is a (fairy chess) leaper [0,1]
%H A172230 V. Kotesovec, <a href="http://web.telecom.cz/vaclav.kotesovec/math.htm">Number of ways of placing non-attacking queens and kings on boards of various sizes</a>
%F A172230 Explicit formula (Vaclav Kotesovec, 29.1.2010): a(n) = (64*n^4-432*n^3+1235*n^2-1797*n+1122)/6, n>=3
%Y A172230 A172227, A172229, A061990
%K A172230 nonn,new
%O A172230 1,2
%A A172230 Vaclav Kotesovec (kotesovec(AT)chello.cz), Jan 29 2010

%I A172229
%S A172229 0,2,22,84,215,442,792,1292,1969,2850,3962,5332,6987,8954,11260,13932,
%T A172229 16997,20482,24414,28820
%N A172229 Number of ways to place 3 nonattacking wazirs on an 3 X n board
%C A172229 Wazir is a (fairy chess) leaper [0,1]
%H A172229 V. Kotesovec, <a href="http://web.telecom.cz/vaclav.kotesovec/math.htm">Number of ways of placing non-attacking queens and kings on boards of various sizes</a>
%F A172229 Explicit formula (Vaclav Kotesovec, 29.1.2010): a(n) = (3*n-5)*(3*n^2-8*n+8)/2, n>=2
%Y A172229 A172226, A061989
%K A172229 nonn,new
%O A172229 1,2
%A A172229 Vaclav Kotesovec (kotesovec(AT)chello.cz), Jan 29 2010

%I A172209
%S A172209 1,2,6,8,9,13,16,19,21,23,28,34,36,41,42,44,54,57,58,61,72,78,82,83,86,
%T A172209 89,96,99,104,111,113,132,133,149,152,154,167,173,177,187,201,211,218,
%U A172209 236,237,247,251,258,266,273,278,288,296,302,307,314,316,317,322,336
%N A172209 Numbers n of the form 18*n-+5.
%e A172209 a(1)=1 because 18*1-5=13=prime and 18*1+5=23=prime.
%Y A172209 Cf. A000027, A000040, A172147, A172154.
%K A172209 nonn,new
%O A172209 1,2
%A A172209 Juri-Step[an Gerasimov (2stepan(AT)rambler.ru), Jan 29 2010

%I A172228
%S A172228 0,0,1,359,10741,127960,870589,4197456,16005187,51439096,145085447,
%T A172228 369074128,863338777,1883786680,3875953561,7583888944
%N A172228 Number of ways to place 5 nonattacking wazirs on an n X n board
%C A172228 Wazir is a (fairy chess) leaper [0,1]
%H A172228 V. Kotesovec, <a href="http://web.telecom.cz/vaclav.kotesovec/math.htm">Number of ways of placing non-attacking queens and kings on boards of various sizes</a>
%F A172228 Explicit formula (Vaclav Kotesovec, 29.1.2010): a(n) = (n^10-50n^8+40n^7+995n^6-1560n^5-8890n^4+21080n^3+24264n^2-97440n+59520)/120, n>=5. For any fixed value of k > 1, a(n) = n^(2k)/k! - 5/2/(k-2)!*n^(2k-2) + ...
%Y A172228 A172225, A172226, A172227, A108792, A061998, A172129, A172136, A172140
%K A172228 nonn,new
%O A172228 1,4
%A A172228 Vaclav Kotesovec (kotesovec(AT)chello.cz), Jan 29 2010

%I A172227
%S A172227 0,0,9,405,5024,31320,133544,446421,1258590,3126724,7042930,14669709,
%T A172227 28658436,53069000,93909924,159819965,262913874,419816676,652912510,
%U A172227 991835749
%N A172227 Number of ways to place 4 nonattacking wazirs on an n X n board
%C A172227 Wazir is a (fairy chess) leaper [0,1]
%H A172227 V. Kotesovec, <a href="http://web.telecom.cz/vaclav.kotesovec/math.htm">Number of ways of placing non-attacking queens and kings on boards of various sizes</a>
%F A172227 Explicit formula (Vaclav Kotesovec, 29.1.2010): a(n) = (n^8-30n^6+24n^5+323n^4-504n^3-1110n^2+2760n-1224)/24, n>=4
%Y A172227 A172225, A172226, A061994, A061997, A172127, A172135, A172139
%K A172227 nonn,new
%O A172227 1,3
%A A172227 Vaclav Kotesovec (kotesovec(AT)chello.cz), Jan 29 2010

%I A172226
%S A172226 0,0,22,276,1474,5248,14690,35012,74326,144544,262398,450580,739002,
%T A172226 1166176,1780714,2642948,3826670,5420992,7532326,10286484
%N A172226 Number of ways to place 3 nonattacking wazirs on an n X n board
%C A172226 Wazir is a (fairy chess) leaper [0,1]
%H A172226 V. Kotesovec, <a href="http://web.telecom.cz/vaclav.kotesovec/math.htm">Number of ways of placing non-attacking queens and kings on boards of various sizes</a>
%F A172226 Explicit formula (Vaclav Kotesovec, 29.1.2010): a(n) = (n-2)(n^5+2n^4-11n^3-10n^2+42n-12)/6, n>=2
%Y A172226 A172225, A047659, A061996, A172124, A172134, A172138
%K A172226 nonn,new
%O A172226 1,3
%A A172226 Vaclav Kotesovec (kotesovec(AT)chello.cz), Jan 29 2010

%I A172225
%S A172225 0,2,24,96,260,570,1092,1904,3096,4770,7040,10032,13884,18746,24780,
%T A172225 32160,41072,51714,64296,79040
%N A172225 Number of ways to place 2 nonattacking wazirs on an n X n board
%C A172225 Wazir is a (fairy chess) leaper [0,1]
%D A172225 Christian Poisson, Echecs et mathematiques, Rex Multiplex 29/1990, p.829
%H A172225 V. Kotesovec, <a href="http://web.telecom.cz/vaclav.kotesovec/math.htm">Number of ways of placing non-attacking queens and kings on boards of various sizes</a>
%F A172225 Explicit formula (Christian Poisson, 1990): a(n) = n(n-1)(n^2+n-4)/2
%Y A172225 A036464, A061995, A172132, A172123, A172137
%K A172225 nonn,new
%O A172225 1,2
%A A172225 Vaclav Kotesovec (kotesovec(AT)chello.cz), Jan 29 2010

%I A172224
%S A172224 1,924,8989,37270,145233,525796,1605490,4136952,9435413,19632414,
%T A172224 37957424,69050898,119351315,197524064,314935542,486171662,729604121,
%U A172224 1068003424,1529198580,2146783422,2960869583,4018886128,5376425842
%N A172224 Number of ways to place 6 nonattacking zebras on an 6 X n board
%C A172224 Zebra is a (fairy chess) leaper [2,3]
%H A172224 V. Kotesovec, <a href="http://web.telecom.cz/vaclav.kotesovec/math.htm">Number of ways of placing non-attacking queens and kings on boards of various sizes</a>
%F A172224 Explicit formula (Vaclav Kotesovec, 26.1.2010): a(n) = (1944n^6-27540n^5+227070n^4-1222555n^3+4366071n^2-9580580n+9925860)/30, n>=15. For any fixed value of k > 1, a(n) = 1/k!*(kn)^k - (k-1)(9k-20)/2/k!*(kn)^(k-1) + ...
%Y A172224 A061992, A172221, A172222, A172223
%K A172224 nonn,new
%O A172224 1,2
%A A172224 Vaclav Kotesovec (kotesovec(AT)chello.cz), Jan 29 2010

%I A172223
%S A172223 1,252,1925,6534,20502,57710,142312,308254,606051,1105332,1897899,
%T A172223 3100250,4857000,7344010,10771530,15387310,21479725,29380900,39469835,
%U A172223 52175530,67980110,87421950,111098800,139670910,173864155
%N A172223 Number of ways to place 5 nonattacking zebras on an 5 X n board
%C A172223 Zebra is a (fairy chess) leaper [2,3]
%H A172223 V. Kotesovec, <a href="http://web.telecom.cz/vaclav.kotesovec/math.htm">Number of ways of placing non-attacking queens and kings on boards of various sizes</a>
%F A172223 Explicit formula (Vaclav Kotesovec, 26.1.2010): a(n) = 5*(125n^5-1250n^4+7575n^3-28426n^2+64000n-67056)/24, n>=12
%Y A172223 A172140, A061991, A172221, A172222
%K A172223 nonn,new
%O A172223 1,2
%A A172223 Vaclav Kotesovec (kotesovec(AT)chello.cz), Jan 29 2010

%I A172222
%S A172222 1,70,406,1168,2948,6576,13122,23808,40168,63996,97344,142516,202072,
%T A172222 278828,375856,496484,644296,823132,1037088,1290516,1588024,1934476,
%U A172222 2334992,2794948,3319976
%N A172222 Number of ways to place 4 nonattacking zebras on an 4 X n board
%C A172222 Zebra is a (fairy chess) leaper [2,3]
%H A172222 V. Kotesovec, <a href="http://web.telecom.cz/vaclav.kotesovec/math.htm">Number of ways of placing non-attacking queens and kings on boards of various sizes</a>
%F A172222 Explicit formula (Vaclav Kotesovec, 26.1.2010): a(n) = 4*(8n^4-48n^3+202n^2-471n+507)/3, n>=9
%Y A172222 A172139, A061990, A172221
%K A172222 nonn,new
%O A172222 1,2
%A A172222 Vaclav Kotesovec (kotesovec(AT)chello.cz), Jan 29 2010

%I A172221
%S A172221 1,20,84,200,403,720,1180,1808,2631,3676,4970,6540,8413,10616,13176,
%T A172221 16120,19475,23268,27526,32276,37545,43360,49748,56736,64351
%N A172221 Number of ways to place 3 nonattacking zebras on an 3 X n board
%C A172221 Zebra is a (fairy chess) leaper [2,3]
%H A172221 V. Kotesovec, <a href="http://web.telecom.cz/vaclav.kotesovec/math.htm">Number of ways of placing non-attacking queens and kings on boards of various sizes</a>
%F A172221 Explicit formula (Vaclav Kotesovec, 26.1.2010): a(n) = (9n^3-21n^2+50n-48)/2, n>=6
%Y A172221 A172138, A061989
%K A172221 nonn,new
%O A172221 1,2
%A A172221 Vaclav Kotesovec (kotesovec(AT)chello.cz), Jan 29 2010

%I A172220
%S A172220 1,28,157,1248,4650,15162,37988,86958,181423,351708,648441,1127392,
%T A172220 1874194,2988466,4602096,6870240,9983347,14163972,19672403,26812260,
%U A172220 35929480,47418482,61723238,79341720,100828175,126796852,157924785
%N A172220 Number of ways to place 5 nonattacking nightriders on an 5 X n board
%C A172220 A nightrider is a fairy chess piece that can move (proportionate to how a knight moves) in any direction.
%H A172220 V. Kotesovec, <a href="http://web.telecom.cz/vaclav.kotesovec/math.htm">Number of ways of placing non-attacking queens and kings on boards of various sizes</a>
%H A172220 Vaclav Kotesovec, <a href="http://www.research.att.com/~njas/sequences/b172220.txt">Table of n, a(n) for n=1..40</a>
%F A172220 Explicit formula (Vaclav Kotesovec, 28.1.2010): a(n) = (625n^5-15250n^4+197915n^3-1588634n^2+7645896n-17283552)/24, n>=32
%Y A172220 A061991, A172214, A172218, A172219
%K A172220 nonn,new
%O A172220 1,2
%A A172220 Vaclav Kotesovec (kotesovec(AT)chello.cz), Jan 29 2010

%I A172219
%S A172219 1,16,84,412,1126,2760,5739,10982,19695,33068,52801,80638,118731,169368,
%T A172219 235135,318890,423733,553028,710389,899690,1125059,1390880,1701793,
%U A172219 2062694,2478735,2955324,3498125,4113058,4806299,5584280,6453689
%N A172219 Number of ways to place 4 nonattacking nightriders on an 4 X n board
%C A172219 A nightrider is a fairy chess piece that can move (proportionate to how a knight moves) in any direction.
%H A172219 V. Kotesovec, <a href="http://web.telecom.cz/vaclav.kotesovec/math.htm">Number of ways of placing non-attacking queens and kings on boards of various sizes</a>
%F A172219 Explicit formula (Vaclav Kotesovec, 26.1.2010): a(n) = (32n^4-432n^3+3190n^2-13323n+25530)/3, n>=18
%Y A172219 A061990, A172213, A172218
%K A172219 nonn,new
%O A172219 1,2
%A A172219 Vaclav Kotesovec (kotesovec(AT)chello.cz), Jan 29 2010

%I A172218
%S A172218 1,12,36,100,213,408,712,1148,1745,2528,3524,4760,6263,8060,10178,12644,
%T A172218 15485,18728,22400,26528,31139,36260,41918,48140,54953,62384,70460,
%U A172218 79208,88655,98828,109754,121460,133973,147320,161528,176624,192635
%N A172218 Number of ways to place 3 nonattacking nightriders on an 3 X n board
%C A172218 A nightrider is a fairy chess piece that can move (proportionate to how a knight moves) in any direction.
%H A172218 V. Kotesovec, <a href="http://web.telecom.cz/vaclav.kotesovec/math.htm">Number of ways of placing non-attacking queens and kings on boards of various sizes</a>
%F A172218 Explicit formula (Vaclav Kotesovec, 26.1.2010): a(n) = (9n^3-57n^2+210n-344)/2, n>=8
%Y A172218 A172141, A061989, A172212
%K A172218 nonn,new
%O A172218 1,2
%A A172218 Vaclav Kotesovec (kotesovec(AT)chello.cz), Jan 29 2010

%I A172217
%S A172217 1,78,1758,38588,383246,2135344,8891854,30108310,86669806,219845764,
%T A172217 504261973,1065642840,2104251027,3924818982,6973786593,11884673662,
%U A172217 19532410762,31097451768,48140491605,72688612756,107333684073
%N A172217 Number of ways to place 7 nonattacking knights on an 7 X n board
%H A172217 V. Kotesovec, <a href="http://web.telecom.cz/vaclav.kotesovec/math.htm">Number of ways of placing non-attacking queens and kings on boards of various sizes</a>
%F A172217 Explicit formula (Vaclav Kotesovec, 29.1.2010): a(n) = (117649n^7-2571471n^6+29223943n^5-216954465n^4+1114503256n^3-3907492824n^2+8562799512n-8962924320)/720,n>=12. For any fixed value of k > 1, a(n) = 1/k!*(kn)^k - 3(k-1)(3k-4)/2/k!*(kn)^(k-1) + ...
%Y A172217 A061993, A172212, A172213, A172214, A172215
%K A172217 nonn,new
%O A172217 1,2
%A A172217 Vaclav Kotesovec (kotesovec(AT)chello.cz), Jan 29 2010

%I A172216
%S A172216 1,1,1,1,1,3,2,5,1,1,7,2,1,1,1,2,5,1,1,6,2,2,1,1,4,1,4,2,2,1,2,1,1,1,2,
%T A172216 1,1,4,6,1,1,2,1,1,1,1,3,1,1,1,5,6,1,4,4,1,1,2,2,1,1,4,3,1,1,1,1,1,8,2,
%U A172216 1,1,2,1,1,5,2,1,1,1,8,1,4,2,3,1,1,2,1,1,1,4,1,8,3,2,6,2,3,6,2,1,10,8,1
%N A172216 Smallest k such that sum of digits of prime(n)^k is prime.
%C A172216 For all n, prime(n)^0 = 1 has non-prime sum of digits 1.
%C A172216 a(n) = 1 iff prime(n) is in A046704, an additive prime. a(n) = 1 iff n is in A075177.
%e A172216 prime(1) = 2; 2^1 = 2 has prime sum of digits 2. Hence a(1) = 1.
%e A172216 prime(6) = 13; 13^1 = 13 has non-prime sum of digits 4; 13^2 = 169 has non-prime sum of digits 16; 13^3 = 2197 has prime sum of digits 19. Hence a(6) = 3.
%o A172216 (MAGMA) S:=[]; for n in [1..105] do j:=1; while not IsPrime(&+Intseq(NthPrime(n)^j)) do j+:=1; end while; Append(~S, j); end for; S;
%Y A172216 Cf. A046704, A075177, A172035.
%K A172216 base,nonn,new
%O A172216 1,6
%A A172216 Klaus Brockhaus (klaus-brockhaus(AT)t-online.de), Jan 29 2010

%I A172215
%S A172215 1,58,729,8830,60285,257318,858262,2404448,5879329,12927182,26115008,
%T A172215 49238436,87675623,148787822,242366502,381127124,581249573,862965246,
%U A172215 1251190796,1776208532,2474393475,3388987070,4570917554,6079666980
%N A172215 Number of ways to place 6 nonattacking knights on an 6 X n board
%H A172215 V. Kotesovec, <a href="http://web.telecom.cz/vaclav.kotesovec/math.htm">Number of ways of placing non-attacking queens and kings on boards of various sizes</a>
%F A172215 Explicit formula (Vaclav Kotesovec, 26.1.2010): a(n) = (648n^6-11340n^5+103770n^4-606645n^3+2328317n^2-5466660n+6051720)/10, n>=10
%Y A172215 A061992, A172212, A172213, A172214
%K A172215 nonn,new
%O A172215 1,2
%A A172215 Vaclav Kotesovec (kotesovec(AT)chello.cz), Jan 29 2010

%I A172214
%S A172214 1,28,259,1968,9386,30842,82738,192336,400277,763984,1360797,2291056,
%T A172214 3681226,5687022,8496534,12333352,17459691,24179516,32841667,43842984,
%U A172214 57631432,74709226,95635956,121031712,151580209
%N A172214 Number of ways to place 5 nonattacking knights on an 5 X n board
%H A172214 V. Kotesovec, <a href="http://web.telecom.cz/vaclav.kotesovec/math.htm">Number of ways of placing non-attacking queens and kings on boards of various sizes</a>
%F A172214 Explicit formula (Vaclav Kotesovec, 26.1.2010): a(n) = (625n^5-8250n^4+57235n^3-242778n^2+608440n-705984)/24, n>=8
%Y A172214 A172136, A061991, A172212, A172213
%K A172214 nonn,new
%O A172214 1,2
%A A172214 Vaclav Kotesovec (kotesovec(AT)chello.cz), Jan 29 2010

%I A172213
%S A172213 1,16,84,412,1416,3640,7928,15384,27352,45432,71480,107608,156184,
%T A172213 219832,301432,404120,531288,686584,873912,1097432,1361560,1670968,
%U A172213 2030584,2445592,2921432
%N A172213 Number of ways to place 4 nonattacking knights on an 4 X n board
%H A172213 V. Kotesovec, <a href="http://web.telecom.cz/vaclav.kotesovec/math.htm">Number of ways of placing non-attacking queens and kings on boards of various sizes</a>
%F A172213 Explicit formula (Vaclav Kotesovec, 26.1.2010): a(n) = 8*(4n^4-36n^3+170n^2-450n+537)/3, n>=6
%Y A172213 A172135, A061990, A172212
%K A172213 nonn,new
%O A172213 1,2
%A A172213 Vaclav Kotesovec (kotesovec(AT)chello.cz), Jan 29 2010

%I A172212
%S A172212 1,12,36,100,233,456,796,1280,1935,2788,3866,5196,6805,8720,10968,13576,
%T A172212 16571,19980,23830,28148,32961,38296,44180,50640,57703
%N A172212 Number of ways to place 3 nonattacking knights on an 3 X n board
%H A172212 V. Kotesovec, <a href="http://web.telecom.cz/vaclav.kotesovec/math.htm">Number of ways of placing non-attacking queens and kings on boards of various sizes</a>
%F A172212 Explicit formula (Vaclav Kotesovec, 26.1.2010): a(n) = (9n^3-45n^2+122n-144)/2, n>=4
%Y A172212 A172134, A061989, A047659, A172202
%K A172212 nonn,new
%O A172212 1,2
%A A172212 Vaclav Kotesovec (kotesovec(AT)chello.cz), Jan 29 2010

%I A172211
%S A172211 1,16,313,2320,12160,53744,209428,683524,1905625,4664384,10297579,
%T A172211 20907590,39664250,71114916,121559433,199459466,315906248,485124352,
%U A172211 725031335,1057839684,1510706686,2116429956,2914190277,3950340692
%N A172211 Number of ways to place 6 nonattacking bishops on an 6 X n board
%H A172211 V. Kotesovec, <a href="http://web.telecom.cz/vaclav.kotesovec/math.htm">Number of ways of placing non-attacking queens and kings on boards of various sizes</a>
%H A172211 Vaclav Kotesovec, <a href="http://www.research.att.com/~njas/sequences/b172211.txt">Table of n, a(n) for n=1..32</a>
%F A172211 Explicit formula (Vaclav Kotesovec, 28.1.2010): a(n) = (648n^6-17820n^5+240930n^4-2011545n^3+10806047n^2-35094560n+53430940)/10, n>=25. For any fixed value of k > 1, a(n) = 1/k!*(kn)^k - (2k-1)/2/(k-2)!*(kn)^(k-1) + ...
%Y A172211 A061992, A172207, A172208, A172210
%K A172211 nonn,new
%O A172211 1,2
%A A172211 Vaclav Kotesovec (kotesovec(AT)chello.cz), Jan 29 2010

%I A172210
%S A172210 1,12,143,770,3368,12632,38566,98968,222351,450682,843169,1479116,
%T A172210 2460912,3917228,6006056,8917888,12878847,18153806,25049515,33917724,
%U A172210 45158308,59222392,76615476,97900560,123701269,154704978,191665937
%N A172210 Number of ways to place 5 nonattacking bishops on an 5 X n board
%H A172210 V. Kotesovec, <a href="http://web.telecom.cz/vaclav.kotesovec/math.htm">Number of ways of placing non-attacking queens and kings on boards of various sizes</a>
%F A172210 Explicit formula (Vaclav Kotesovec, 26.1.2010): a(n) = (625n^5-11250n^4+98875n^3-515250n^2+1566016n-2194944)/24, n>=16
%Y A172210 A172129, A061991, A172207, A172208
%K A172210 nonn,new
%O A172210 1,2
%A A172210 Vaclav Kotesovec (kotesovec(AT)chello.cz), Jan 29 2010

%I A172208
%S A172208 1,9,61,260,927,2578,5965,12066,22135,37678,60457,92488,136043,193650,
%T A172208 268093,362412,479903,624118,798865,1008208,1256467,1548218,1888293,
%U A172208 2281780,2734023,3250622,3837433,4500568,5246395,6081538,7012877
%N A172208 Number of ways to place 4 nonattacking bishops on an 4 X n board
%H A172208 V. Kotesovec, <a href="http://web.telecom.cz/vaclav.kotesovec/math.htm">Number of ways of placing non-attacking queens and kings on boards of various sizes</a>
%F A172208 Explicit formula (Vaclav Kotesovec, 26.1.2010): a(n) = (32n^4-336n^3+1702n^2-4701n+5844)/3, n>=9
%Y A172208 A172127, A061990, A172207
%K A172208 nonn,new
%O A172208 1,2
%A A172208 Vaclav Kotesovec (kotesovec(AT)chello.cz), Jan 29 2010

%I A172207
%S A172207 1,6,26,86,211,426,758,1234,1881,2726,3796,5118,6719,8626,10866,13466,
%T A172207 16453,19854,23696,28006,32811,38138,44014,50466,57521,65206,73548,
%U A172207 82574,92311,102786,114026,126058
%N A172207 Number of ways to place 3 nonattacking bishops on an 3 X n board
%H A172207 V. Kotesovec, <a href="http://web.telecom.cz/vaclav.kotesovec/math.htm">Number of ways of placing non-attacking queens and kings on boards of various sizes</a>
%F A172207 Explicit formula (Vaclav Kotesovec, 26.1.2010): a(n) = (9n^3-45n^2+106n-108)/2, n>=4
%Y A172207 A172124, A061989, A047659, A061996
%K A172207 nonn,new
%O A172207 1,2
%A A172207 Vaclav Kotesovec (kotesovec(AT)chello.cz), Jan 29 2010

%I A172206
%S A172206 0,0,24,926,37282,394202,2484382,10999618,38168864,110899878,281638602,
%T A172206 643766432,1352358921,2651129458,4906381466,8648792662,14623854922,
%U A172206 23851793294
%N A172206 Number of ways to place 7 nonattacking kings on an 7 X n board
%H A172206 V. Kotesovec, <a href="http://web.telecom.cz/vaclav.kotesovec/math.htm">Number of ways of placing non-attacking queens and kings on boards of various sizes</a>
%F A172206 Explicit formula (Vaclav Kotesovec, 28.1.2010): a(n) = (117649n^7-2873997n^6+32197753n^5-215350695n^4+932130286n^3-2618213868n^2+4424623272n-3468569760)/720, n>=6. For any fixed value of k > 1, a(n) = 1/k!*(kn)^k - 3(k-1)(3k-2)/2/k!*(kn)^(k-1) + ...
%Y A172206 A061993, A172202, A172203, A172204, A172205
%K A172206 nonn,new
%O A172206 1,3
%A A172206 Vaclav Kotesovec (kotesovec(AT)chello.cz), Jan 29 2010

%I A172205
%S A172205 0,0,16,408,8544,62266,291908,1021254,2916232,7179314,15790572,31795390,
%T A172205 59638832,105546666,177953044,287974838,449932632,681918370,1006409660,
%U A172205 1450930734,2048760064,2839684634,3870800868,5197362214,6883673384
%N A172205 Number of ways to place 6 nonattacking kings on an 6 X n board
%H A172205 V. Kotesovec, <a href="http://web.telecom.cz/vaclav.kotesovec/math.htm">Number of ways of placing non-attacking queens and kings on boards of various sizes</a>
%F A172205 Explicit formula (Vaclav Kotesovec, 27.1.2010): a(n) = 2*(162n^6-3240n^5+29160n^4-151830n^3+483798n^2-895085n+749335)/5, n>=5
%Y A172205 A172158, A061992, A172202, A172203, A172204
%K A172205 nonn,new
%O A172205 1,3
%A A172205 Vaclav Kotesovec (kotesovec(AT)chello.cz), Jan 29 2010

%I A172204
%S A172204 0,0,15,194,1974,9856,34475,95466,224589,468854,893646,1585850,2656976,
%T A172204 4246284,6523909,9693986,13997775,19716786,27175904,36746514,48849626,
%U A172204 63959000,82604271,105374074,132919169
%N A172204 Number of ways to place 5 nonattacking kings on an 5 X n board
%H A172204 V. Kotesovec, <a href="http://web.telecom.cz/vaclav.kotesovec/math.htm">Number of ways of placing non-attacking queens and kings on boards of various sizes</a>
%F A172204 Explicit formula (Vaclav Kotesovec, 27.1.2010): a(n) = (625n^5-9750n^4+66415n^3-247626n^2+504664n-446544)/24, n>=4
%Y A172204 A061998, A061991, A172202, A172203
%K A172204 nonn,new
%O A172204 1,3
%A A172204 Vaclav Kotesovec (kotesovec(AT)chello.cz), Jan 29 2010

%I A172203
%S A172203 0,0,9,79,454,1566,4103,9009,17484,30984,51221,80163,120034,173314,
%T A172203 242739,331301,442248,579084,745569,945719,1183806,1464358,1792159,
%U A172203 2172249,2609924
%N A172203 Number of ways to place 4 nonattacking kings on an 4 X n board
%H A172203 V. Kotesovec, <a href="http://web.telecom.cz/vaclav.kotesovec/math.htm">Number of ways of placing non-attacking queens and kings on boards of various sizes</a>
%F A172203 Explicit formula (Vaclav Kotesovec, 27.1.2010): a(n) = (64n^4-720n^3+3347n^2-7569n+6894)/6, n>=3
%Y A172203 A061997, A061990, A172202
%K A172203 nonn,new
%O A172203 1,3
%A A172203 Vaclav Kotesovec (kotesovec(AT)chello.cz), Jan 29 2010

%I A172202
%S A172202 0,0,8,34,105,248,490,858,1379,2080,2988,4130,5533,7224,9230,11578,
%T A172202 14295,17408,20944,24930,29393,34360,39858,45914,52555
%N A172202 Number of ways to place 3 nonattacking kings on an 3 X n board
%H A172202 V. Kotesovec, <a href="http://web.telecom.cz/vaclav.kotesovec/math.htm">Number of ways of placing non-attacking queens and kings on boards of various sizes</a>
%F A172202 Explicit formula (Vaclav Kotesovec, 27.1.2010): a(n) = (n-2)(9n^2-45n+70)/2, n>=2
%Y A172202 A061996, A061989, A047659
%K A172202 nonn,new
%O A172202 1,3
%A A172202 Vaclav Kotesovec (kotesovec(AT)chello.cz), Jan 29 2010

%I A172201
%S A172201 0,0,0,0,48,424,1976,6616,17852,41544,86660,166288,298616,508200,827168,
%T A172201 1296744,1968676,2907016,4189772,5910944,8182400,11136168,14926536,
%U A172201 19732600,25760588,33246664,42459476,53703216,67320392,83695144
%N A172201 Number of ways to place 3 nonattacking amazons (superqueens) on an n X n board
%C A172201 A amazon (superqueen) moves like a queen and a knight.
%D A172201 Panos Louridas, idee & form 93/2007, p. 2936-2938
%H A172201 V. Kotesovec, <a href="http://web.telecom.cz/vaclav.kotesovec/math.htm">Number of ways of placing non-attacking queens and kings on boards of various sizes</a>
%F A172201 Explicit formula (Panos Louridas, 2007): a(n) = (2n^6 - 20n^5 + 31n^4 + 314n^3 - 1452n^2 + 2040n - 672)/12 if n is even (n >= 4) and a(n) = (2n^6 - 20n^5 + 31n^4 + 314n^3 - 1452n^2 + 2034n - 669)/12 if n is odd (n >= 5)
%Y A172201 A051223, A051224, A047659, A061989, A172200
%K A172201 nonn,new
%O A172201 1,5
%A A172201 Vaclav Kotesovec (kotesovec(AT)chello.cz), Jan 29 2010

%I A172200
%S A172200 0,0,0,20,92,260,580,1120,1960,3192,4920,7260,10340,14300,19292,25480,
%T A172200 33040,42160,53040,65892,80940,98420,118580,141680,167992,197800,231400,
%U A172200 269100,311220,358092,410060,467480,530720,600160,676192,759220,849660
%N A172200 Number of ways to place 2 nonattacking amazons (superqueens) on an n X n board
%C A172200 A amazon (superqueen) moves like a queen and a knight.
%D A172200 Christian Poisson, Echecs et mathematiques, Rex Multiplex 29/1990, p.829
%H A172200 V. Kotesovec, <a href="http://web.telecom.cz/vaclav.kotesovec/math.htm">Number of ways of placing non-attacking queens and kings on boards of various sizes</a>
%F A172200 Explicit formula (Christian Poisson, 1990): a(n) = (n - 1)(n - 2)(n - 3)(3n + 8)/6
%Y A172200 A051223, A051224, A036464
%K A172200 nonn,new
%O A172200 1,4
%A A172200 Vaclav Kotesovec (kotesovec(AT)chello.cz), Jan 29 2010

%I A172199
%S A172199 0,1,1,5,4,79,71,128,117,971,919
%V A172199 0,1,-1,5,-4,79,-71,128,-117,971,-919
%N A172199 From Bernoulli numbers to b(n)=0,0,1,-1,2,-2,3,-3,4,-4,=mix A001477,-A001477=A004526 signed;(see A172030,A172031,A072032). Inverse binomial transform of A172032/A172031 is 0,1,-1/2,5/3,-4/3,79/30,-71/30,128/35,-117/35,971/210,-919/210, . Numerators are a(n). ((Submitted A172194)/A172031)=0,1,1/2,2/3,2/3,19/30,19/30,) - (0,1,-1/2,5/3,-4/3 companion) = b(n).
%C A172199 Bernoulli numbers are based on A000127 or A001477. Note (5+4)/3=9/3=3 , (79+71)/30=150/30=5, (128+117)/35=245/35=7 , (971+919)/210=1890/210=9 ;see A144396.
%Y A172199 A166687.
%K A172199 nonn,uned,new
%O A172199 0,4
%A A172199 Paul Curtz (bpcrtz(AT)free.fr), Jan 29 2010

%I A172198
%S A172198 1,1,1,1,325,1,1,178849,178849,1,1,1121470273,1106493697,1121470273,1,1,
%T A172198 65131063096321,64859828626945,64859828626945,65131063096321,1,1,
%U A172198 34423599076368353281,34376183545107456001,34376383642256188417
%N A172198 A q-form based product triangular sequence:q=3;c(n,q)=Product[1 - q^i, {i, 1, n}]; t(n,m,q)=1 + Abs[c(n, q) - c(m, q)]*Abs[c(n, q) - c(n - m, q)]
%C A172198 Row sums are:
%C A172198 {1, 2, 327, 357700, 3349434245, 259981783446534, 171975948885207806983,
%C A172198 985775044264004540477767688, 49486309390027311684794921366323209,
%C A172198 21907966958329449995754873283658380770017290,
%C A172198 85933223490421173788627731935098309659305222246236171,...}
%F A172198 q=3;
%F A172198 c(n,q)=Product[1 - q^i, {i, 1, n}];
%F A172198 t(n,m,q)=1 + Abs[c(n, q) - c(m, q)]*Abs[c(n, q) - c(n - m, q)]
%e A172198 {1},
%e A172198 {1, 1},
%e A172198 {1, 325, 1},
%e A172198 {1, 178849, 178849, 1},
%e A172198 {1, 1121470273, 1106493697, 1121470273, 1},
%e A172198 {1, 65131063096321, 64859828626945, 64859828626945, 65131063096321, 1},
%e A172198 {1, 34423599076368353281, 34376183545107456001, 34376383642256188417, 34376183545107456001, 34423599076368353281, 1},
%e A172198 {1, 164345972820689217363886081, 164270722833364659313950721, 164270826477948393561047041, 164270826477948393561047041, 164270722833364659313950721, 164345972820689217363886081, 1},
%e A172198 {1, 7070242636814107505803921053941761, 7069164527232402102869015830855681, 7069165020872356787292825709608961, 7069165020189578892863396177510401, 7069165020872356787292825709608961, 7069164527232402102869015830855681, 7070242636814107505803921053941761, 1},
%e A172198 {1, 2738600226570617128979968109300172796723201, 2738461070055039353749136141519872827064321, 2738461091274392140130202872128129546321921, 2738461091264676375018129518881015214899201, 2738461091264676375018129518881015214899201, 2738461091274392140130202872128129546321921, 2738461070055039353749136141519872827064321, 2738600226570617128979968109300172796723201, 1},
%e A172198 {1, 9548261712378179136510752747630538293064212466892801, 9548100003512000594710518784172787524135789002752001, 9548100011728914526547034585819628526290815961006081, 9548100011727661566200369560238909114222701864550401, 9548100011727662140690380579374582743878183655833601, 9548100011727661566200369560238909114222701864550401, 9548100011728914526547034585819628526290815961006081, 9548100003512000594710518784172787524135789002752001, 9548261712378179136510752747630538293064212466892801, 1}
%t A172198 Clear[t, n, m, c, q];
%t A172198 c[n_, q_] = Product[1 - q^i, {i, 1, n}];
%t A172198 t[n_, m_, q_] = 1 + Abs[c[n, q] - c[m, q]]*Abs[c[n, q] - c[n - m, q]];
%t A172198 Table[Table[Table[t[n, m, q], {m, 0, n}], {n, 0, 10}], {q, 2, 12}];
%t A172198 Table[Flatten[Table[Table[t[n, m, q], {m, 0, n}], {n, 0, 10}]], {q, 2, 12}]
%K A172198 nonn,tabl,uned,new
%O A172198 0,5
%A A172198 Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Jan 29 2010

%I A172197
%S A172197 1,0,9,4,5,7,6,1,0,5,2,3,1,6,4,5,6,7,0,1,0,8,8,3,0,5,4,7,9,8,5,2,9,9,4,
%T A172197 6,3,0,0,9,9,4,3,5,9,8,4,9,5,9,9,6,9,2,0,7,3,3,3,1,7,4,5,0,9,7,8,7,4,1,
%U A172197 0,6,7,3,9,7,7,5,8,0,4,6,9,5,1,1,2,9,6,4,7,3,6,8,6,0,3,3,2,4,2,9,0,0,8
%N A172197 Decimal expansion of the abscissa x of a local maximum of the Fibonacci function F(x).
%C A172197 Define the Fibonacci Function F(x) and its derivative dF/dx as in A172081.
%C A172197 At the local maximum, dF(x)/dx=0.
%C A172197 This constant x=1.0945... here satisfies this condition of vanishing first derivative.
%H A172197 Gerd Lamprecht, <a href="http://freenet-homepage.de/gerdlamprecht/Roemisch_JAVA.htm">Iterationsrechner mit Algorithmus</a>
%H A172197 Gerd Lamprecht, <a href="http://freenet-homepage.de/gerdlamprecht/Zahlenfolgen.html">Zahlenfolgen (sequences)</a>
%H A172197 E. Weisstein, <a href="http://mathworld.wolfram.com/FibonacciNumber.html">Fibonacci Number</a>, Mathworld.
%e A172197 F(1.0945761052316...) = 1.0098243...
%p A172197 p := (1+sqrt(5))/2 ; F := (p^x - cos(Pi*x)/p^x )/sqrt(5);
%p A172197 Fpr := diff(F,x) ; Fpr2 := diff(Fpr,x) ;
%p A172197 Digits := 80 ; x0 := 1.0 ;
%p A172197 for n from 1 to 10 do
%p A172197 x0 := evalf(x0-subs(x=x0,Fpr)/subs(x=x0,Fpr2)) ;
%p A172197 end do ; # R. J. Mathar, Feb 02 2010
%o A172197 (Other) Gerd Lamprecht online Iterationsrechner: #(@P@C1],x+x)*@C2]+cos(x*PI)*@C2]+sin(x*PI)*PI)*@P@C1], -x)/@C0]@N@C0]=@Q5); @C1]=@C0]/2+0.5; @C2]=log(@C1]); @B1]=1.09; @B2]=1.1; @B3]=Fx(@B1]); @B4]=Fx(@B2]); d=4e-16; IM=2; @N@B4]=Fx(@B2]); @B0]=(@B4]-@B3])/ (@B2]-@B1]); a=@B1]-@B3]/@B0]; b=Fx(a); if(b*@B4]%3C0){@B1]=@B2]; @B2]=a; @B3]=@B4]; }@F@B2]=a; @B3]*=@H2,@B4],b); }@N(@A@B4])%3Cd)@O(@A@B4])%3Cd)@O@A@B2]-@B1])%3Cd@N0@N1@Nif(@A@B4]) %3Cd)c=@B2]; @Eif(@A@B3])%3C1e-16)c=@B1]; @Ec=(@B1]+@B2])/2;
%Y A172197 Cf. A171909, A172081
%K A172197 cons,nonn,new
%O A172197 1,3
%A A172197 Gerd Lamprecht (gerdlamprecht(AT)googlemail.com), Jan 29 2010
%E A172197 Edited, embedded JavaScript source code of URL removed - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Feb 02 2010

%I A172196
%S A172196 1,1,1,1,17,1,1,481,481,1,1,106177,97345,106177,1,1,98421121,95179393,
%T A172196 95179393,98421121,1,1,384472892161,378269256961,378490726657,
%U A172196 378269256961,384472892161,1,1,6152325140989441,6103497476160001
%N A172196 A q-form based product triangular sequence:q=2;c(n,q)=Product[1 - q^i, {i, 1, n}]; t(n,m,q)=1 + Abs[c(n, q) - c(m, q)]*Abs[c(n, q) - c(n - m, q)]
%C A172196 Row sums are:
%C A172196 {1, 2, 19, 964, 309701, 387201030, 1903975024903, 36720211532497928,
%C A172196 2781595663780054625289, 829577145112419282515435530,
%C A172196 976434509539483869421893849108491}
%F A172196 q=2;
%F A172196 c(n,q)=Product[1 - q^i, {i, 1, n}];
%F A172196 t(n,m,q)=1 + Abs[c(n, q) - c(m, q)]*Abs[c(n, q) - c(n - m, q)]
%e A172196 {1},
%e A172196 {1, 1},
%e A172196 {1, 17, 1},
%e A172196 {1, 481, 481, 1},
%e A172196 {1, 106177, 97345, 106177, 1},
%e A172196 {1, 98421121, 95179393, 95179393, 98421121, 1},
%e A172196 {1, 384472892161, 378269256961, 378490726657, 378269256961, 384472892161, 1},
%e A172196 {1, 6152325140989441, 6103497476160001, 6104283149099521, 6104283149099521, 6103497476160001, 6152325140989441, 1},
%e A172196 {1, 398486104502198307841, 396917261491397391361, 396929713103923184641, 396929505585016857601, 396929713103923184641, 396917261491397391361, 398486104502198307841, 1},
%e A172196 {1, 103849463689095182481561601, 103645837289745522254407681, 103646638968238943190190081, 103646632609129993331558401, 103646632609129993331558401, 103646638968238943190190081, 103645837289745522254407681, 103849463689095182481561601, 1},
%e A172196 {1, 108575237383741823092231867084801, 108468999382387826615775448780801, 108469207692194652954214206935041, 108469206872073804836983867084801, 108469206878687654423483069337601, 108469206872073804836983867084801, 108469207692194652954214206935041, 108468999382387826615775448780801, 108575237383741823092231867084801, 1}
%t A172196 Clear[t, n, m, c, q];
%t A172196 c[n_, q_] = Product[1 - q^i, {i, 1, n}];
%t A172196 t[n_, m_, q_] = 1 + Abs[c[n, q] - c[m, q]]*Abs[c[n, q] - c[n - m, q]];
%t A172196 Table[Table[Table[t[n, m, q], {m, 0, n}], {n, 0, 10}], {q, 2, 12}];
%t A172196 Table[Flatten[Table[Table[t[n, m, q], {m, 0, n}], {n, 0, 10}]], {q, 2, 12}]
%K A172196 nonn,tabl,uned,new
%O A172196 0,5
%A A172196 Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Jan 29 2010

%I A175091
%S A175091 4,7,13,15,21,22,23,24,40,49,50,51,52,53,54,57,64,65,66,67,68,73,74,79,
%T A175091 80,94,95,96,99,100,101,102,111,112,122,123,124,127,134,135,136,137,138,
%U A175091 145,146,147,148,149,150,159,160,172,173,174,177,178,179,180,181,182
%N A175091 Numbers n such that A074753(n) = number congruent (2,4) mod 6
%p A175091 A074753 := proc(n) option remember ; local a,k ; a := 0 ; for k from 1 to n do if numtheory[sigma](k) < n then a := a+1 ; end if; end do ; a ; end proc: isA175091 := proc(n) return ( (A074753(n) mod 6 )in {2,4}) ; end proc ; for n from 1 to 300 do if isA175091(n) then printf("%d,",n) ; end if; end do ; [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Feb 08 2010]
%Y A175091 Cf. A074753
%K A175091 nonn,new
%O A175091 1,1
%A A175091 Giovanni Teofilatto (g.teofilatto(AT)tiscalinet.it), Jan 29 2010
%E A175091 Corrected by R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Feb 08 2010

%I A172195
%S A172195 29,37,43,47,53,59,67,73,79,83,97,101,113,131,151,181,191,211,223,227,
%T A172195 229,251,281,311,313,331,401,409,443,449,461,463,467,521,601,607,641,
%U A172195 643,647,661,683,809,811,821,863,881,883,911,1013,1019,1031
%N A172195 Prime numbers for which the absolute difference between the summation of its digits & the product of its digits is a prime.
%Y A172195 Cf. A007605 (Sum of digits of n-th prime), A053666 (Product of digits of n-th prime). For the sequence terms, abs(A007605(n) - A053666(n)) is prime.
%K A172195 base,nonn,new
%O A172195 1,1
%A A172195 Umut Uludag (uludagum(AT)yahoo.com), Jan 29 2010

%I A172193
%S A172193 1,37,83,139,205,281,367,463,569,685,811,947,1093,1249,1415,1591,1777,
%T A172193 1973,2179,2395,2621,2857,3103,3359,3625,3901,4187,4483,4789,5105,5431,
%U A172193 5767,6113,6469,6835,7211,7597,7993,8399,8815,9241,9677,10123,10579
%N A172193 Numbers n such that n^2+1=0(mod.p), (with p=2 or p=4k+1)
%C A172193 See sequence of prime:(37, 83, 139, 281, 367, 463, 569, 811, 947, 1093, 1249, 1777, 1973, 2179, 2621, 2857, 3359, 4483, 4789, 5431, 6113, 6469, 7211, 7993, 9241, 9677, 12007, 12503, 13009, 14051 and so on)
%F A172193 a(n)=5*n^2+31n+1
%e A172193 For n=0, a(0)=1; n=1, a(1)=37; n=2, a(2)=83; n=3, a(3)=139
%K A172193 nonn,new
%O A172193 0,2
%A A172193 Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Jan 29 2010

%I A172194
%S A172194 0,1,1,2,2,19,19,23,23,131,131,808,808
%N A172194 Numerators of inverse binomial transform of (A172030/A172031)=0,1,5/2,31/6,31/3,..
%C A172194 Inverse binomial transform:0,1,1/2,2/3,2/3,19/30,19/30,23/35,23/35,. Denominators:1,1,2,3,3,30,30,35,35,210,210,1155,1155,=1,A100650. See A165142.
%K A172194 nonn,uned,new
%O A172194 0,4
%A A172194 Paul Curtz (bpcrtz(AT)free.fr), Jan 29 2010

%I A172044
%S A172044 1,17,43,79,125,181,247,323,409,505,611,727,853,989,1135,1291,1457,1633,
%T A172044 1819,2015,2221,2437,2663,2899,3145,3401,3667,3943,4229,4525,4831,5147,
%U A172044 5473,5809,6155,6511,6877,7253,7639,8035,8441,8857,9283,9719,10165
%N A172044 Numbers n such that n^2+1=0(mod.p), (with p=2 or p=4k+1)
%F A172044 a(n)=5*n^2+11n+1
%e A172044 For n=0, a(0)=1; n=1, a(1)=17; n=2, a(2)=43; n=3, a(3)=79
%K A172044 nonn,new
%O A172044 0,2
%A A172044 Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Jan 29 2010

%I A172043
%S A172043 1,5,19,43,77,121,175,239,313,397,491,595,709,833,967,1111,1265,1429,
%T A172043 1603,1787,1981,2185,2399,2623,2857,3101,3355,3619,3893,4177,4471,4775,
%U A172043 5089,5413,5747,6091,6445,6809,7183,7567,7961,8365,8779,9203,9637,10081
%N A172043 Numbers n such that n^2+1=0(mod.p), (with p=2 or p=4k+1)
%F A172043 a(n)=5*n^2-n+1
%e A172043 For n=0, a(0)=1; n=1, a(1)=5; n=2, a(2)=19, n=3, a(3)=43
%K A172043 nonn,new
%O A172043 0,2
%A A172043 Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Jan 29 2010

%I A172192
%S A172192 4,9,10,12,14,19,29,46,57,59,66,71,72,84,85,90,95,96,97,114,119,122,155,
%T A172192 157,190,191,204,207,212,221
%N A172192 Naturals n with n^6-(n+1)^5 prime (A171771)
%D A172192 Leonard E. Dickson: History of the Theory of numbers, vol. I, Dover Publications 2005
%D A172192 Derrick H. Lehmer, Guide to Tables in the Theory of Numbers Washington, D.C. 1941
%e A172192 4^6 - (4+1)^5 = 971 = prime(164) => a(1) = 4
%e A172192 9^6 - (9+1)^5 = 431441 = prime(36274) => a(2) = 9
%Y A172192 Cf. A171771, A002327, A140719, A087191
%K A172192 nonn,new
%O A172192 1,1
%A A172192 Ulrich Krug (leuchtfeuer37(AT)gmx.de), Jan 29 2010

%I A172191
%S A172191 32,162,1250,4802,29282,57122,167042,260642,559682,1414562,1847042,
%T A172191 3748322,5651522,6837602,9759362,15780962,24234722,27691682,40302242,
%U A172191 50823362,56796482,77900162,94916642,125484482,177058562,208120802
%N A172191 2*p^4
%Y A172191 Cf. A030514
%K A172191 nonn,new
%O A172191 1,1
%A A172191 Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Jan 29 2010

%I A172190
%S A172190 16,54,250,686,2662,4394,9826,13718,24334,48778,59582,101306,137842,
%T A172190 159014,207646,297754,410758,453962,601526,715822,778034,986078,1143574,
%U A172190 1409938,1825346,2060602,2185454,2450086,2590058,2885794,4096766
%N A172190 2*p^3
%Y A172190 Cf. A030078
%K A172190 nonn,new
%O A172190 1,1
%A A172190 Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Jan 29 2010

%I A172189
%S A172189 3,10,23,42,73,110,153,214,281,354,433,530,633,742,869,1003,1159,1316,
%T A172189 1479,1660,1853,2052,2263,2486,2715,2956,3227,3504,3787,4094,4407,4738,
%U A172189 5075,5424,5791,6164,6543,6940,7349,7770,8203,8642,9099,9562
%N A172189 Partial sums of primes of the form 3*m+1/2-+1/2.
%e A172189 a(1)=3*1+1/2-1/2=3, a(2)=3+3*2+1/2+1/2=10.
%Y A172189 Cf. A007645, A038349.
%K A172189 nonn,new
%O A172189 1,1
%A A172189 Juri-Stepan Gerasimov (2stepan(AT)rambler.ru), Jan 29 2010

%I A172188
%S A172188 2,7,18,35,58,87,128,175,228,287,358,441,530,631,738,851,982,1119,1268,
%T A172188 1435,1608,1787,1978,2175,2402,2635,2874,3125,3382,3645,3914,4195,4488,
%U A172188 4799,5116,5463,5816,6175,6558,6947,7348,7767,8198,8641,9090
%N A172188 Partial sums of primes of the form 3*k-1.
%e A172188 a(1)=3*1-1=2, a(2)=2+3*2-1=7.
%Y A172188 Cf. A038361.
%K A172188 nonn,new
%O A172188 1,1
%A A172188 Juri-Stepan-Gerasimov (2stepan(AT)rambler.ru), Jan 29 2010, Feb 01 2010

%I A172187
%S A172187 13,22,37,58,73,85,94,122,130,137,157,166,181,193,202,229,237,238,253,
%T A172187 262,265,301,302,310,318,346,373,382,409,418,433,437,445,454,462,465,
%U A172187 481,514,517,526,537,541,553,562,589,598,634,661,662,670,697,706,733
%N A172187 Numbers n such that n and n+1 are square-free but 2n+1 isn't
%F A172187 Complement of A007674 and A172186.
%t A172187 ff = {}; gg = {}; Do[kk = FactorInteger[n]; tak = False; Do[If[kk[[m]][[2]] > 1, tak = True], {m, 1, Length[kk]}];If[tak == False, jj = FactorInteger[n + 1]; tak1 = False; Do[If[jj[[m]][[2]] > 1, tak1 = True], {m, 1, Length[jj]}];If[tak1 == False, hh = FactorInteger[2 n + 1]; tak2 = False; Do[If[hh[[m]][[2]] > 1, tak2 = True], {m, 1, Length[hh]}]; If[tak2 == False, AppendTo[ff, n], AppendTo[gg, n]]]], {n, 1, 1000}]; gg (*Artur Jasinski*)
%Y A172187 A007674, A172186
%K A172187 nonn,new
%O A172187 1,1
%A A172187 Artur Jasinski (grafix(AT)csl.pl), Jan 28 2010

%I A172186
%S A172186 1,2,5,6,10,14,21,29,30,33,34,38,41,42,46,57,61,65,66,69,70,77,78,82,86,
%T A172186 93,101,102,105,106,109,110,113,114,118,129,133,138,141,142,145,154,158,
%U A172186 165,173,177,178,182,185,186,190,194,201,205,209,210,213,214,217,218
%N A172186 Numbers n such that n and n+1 and 2n+1 are square-free
%C A172186 This sequence is similar to A007674. For terms in A007674 which lack in this sequence see A172187.
%t A172186 ff = {}; gg = {}; Do[kk = FactorInteger[n]; tak = False; Do[If[kk[[m]][[2]] > 1, tak = True], {m, 1, Length[kk]}];If[tak == False, jj = FactorInteger[n + 1]; tak1 = False; Do[If[jj[[m]][[2]] > 1, tak1 = True], {m, 1, Length[jj]}];If[tak1 == False, hh = FactorInteger[2 n + 1]; tak2 = False; Do[If[hh[[m]][[2]] > 1, tak2 = True], {m, 1, Length[hh]}]; If[tak2 == False, AppendTo[ff, n], AppendTo[gg, n]]]], {n, 1, 500}]; ff (*Artur Jasinski*)
%Y A172186 A007674, A172187
%K A172186 nonn,new
%O A172186 1,2
%A A172186 Artur Jasinski (grafix(AT)csl.pl), Jan 28 2010

%I A175090
%S A175090 9,10,15,16,21,22,25,26,28,33,34,36,39,40,45,46,49,50,52,56,58,63,64,66,
%T A175090 69,70,75,76,78,81,82,85,86,88,91,92,94,96,99,100
%N A175090 Composites c with result 0 under iterations of {r mod (max prime p <= r)} starting at r = c.
%C A175090 Subsequence of A175089. Union of a(n) and A000040 is A175089.
%e A175090 Iteration procedure for a(3) = 15: 15 mod 13 = 2, 2 mod 2 = 0.
%K A175090 nonn,new
%O A175090 1,1
%A A175090 Jaroslav Krizek (jaroslav.krizek(AT)atlas.cz), Jan 28 2010

%I A175089
%S A175089 2,3,5,7,9,10,11,13,15,16,17,19,21,22,23,25,26,28,29,31,33,34,36,37,39,
%T A175089 40,41,43,45,46,47,49,50,52,53,55,56,58,59,61
%N A175089 Natural numbers m with result 0 under iterations of {r mod (max prime p <= r)} starting at r = m.
%C A175089 Complement of A175088.
%C A175089 Union of A000040 (primes) and A175090. [From Jaroslav Krizek (jaroslav.krizek(AT)atlas.cz), Feb 05 2010]
%e A175089 Iteration procedure for a(3) = 5: 5 mod 5 = 0. Iteration procedure for a(5) = 9: 9 mod 7 = 2, 2 mod 2 = 0.
%K A175089 nonn,new
%O A175089 1,1
%A A175089 Jaroslav Krizek (jaroslav.krizek(AT)atlas.cz), Jan 28 2010

%I A175088
%S A175088 1,4,6,8,12,14,18,20,24,27,30,32,35,38,42,44,48,51,54,57,60,62,65,68,72,
%T A175088 74,77,80,84,87,90,93,95,98,102
%N A175088 Natural numbers m with result 1 under iterations of {r mod (max prime p <= r)} starting at r = m.
%C A175088 a(1) = 1, a(n) = composites for all n >= 2.
%C A175088 Complement of A175089. [From Jaroslav Krizek (jaroslav.krizek(AT)atlas.cz), Feb 05 2010]
%e A175088 Iteration procedure for a(6) = 14: 14 mod 13 = 1. Iteration procedure for a(10) = 27: 27 mod 23 = 4, 4 mod 3 = 1.
%K A175088 nonn,new
%O A175088 1,2
%A A175088 Jaroslav Krizek (jaroslav.krizek(AT)atlas.cz), Jan 28 2010

%I A172185
%S A172185 10,9,11,9,20,11,9,29,31,11,9,38,60,42,11,9,47,98,102,53,11,9,56,145,
%T A172185 200,155,64,11,9,65,201,345,355,219,75,11
%N A172185 (9,11)Pascal triangle
%C A172185 Sums of NW-SE diagonals give A022114 (apart from first two terms)
%e A172185 Triangle begins:
%e A172185 .....10
%e A172185 ....9,11
%e A172185 ...9,20,11
%e A172185 .9,29,31,11
%e A172185 9,38,60,42,11
%Y A172185 Cf. A093644, A172179, A022114
%K A172185 nonn,tabl,new
%O A172185 1,1
%A A172185 M. Dols (markdols99(AT)yahoo.com), Jan 28 2010

%I A172182
%S A172182 1,8,14,20,25,26,32,38,44,49,50,55,56,62,68,74,80,85,86,91,92,98,104,
%T A172182 110,115,116,121,122,128,133,134,140,145,146,152,158,164,169,170,175,
%U A172182 176,182,187,188,194,200,205,206,212,217,218,224,230,235,236,242,247
%N A172182 Nonprimes of the form 6*k-3/2-+5/2.
%e A172182 a(1)=6*0-3/2+5/2=1.
%Y A172182 Cf. A000027, A045375, A141468.
%K A172182 nonn,new
%O A172182 1,2
%A A172182 Juri-Stepan Gerasimov (2stepan(AT)rambler.ru), Jan 28 2010

%I A172181
%S A172181 9,15,21,27,33,35,39,45,51,57,63,65,69,75,77,81,87,93,95,99,105,111,117,
%T A172181 119,123,125,129,135,141,143,147,153,155,159,161,165,171,177,183,185,
%U A172181 189,195,201,203,207,209,213,215,219,221,225,231,237,243,245,249,255
%N A172181 Odd nonprimes of the form 6*k-2-+1.
%e A172181 a(1)=6*2-2-1=9.
%Y A172181 Cf. A005408, A045375, A045410.
%K A172181 nonn,new
%O A172181 1,1
%A A172181 Juri-Stepan Gerasimov (2stepan(AT)rambler.ru), Jan 28 2010

%I A172180
%S A172180 1,1,1,1,1,1,1,1,4,5,9,9,9,12,15,19,17,21,22,23,27,33,39,39,42,41,43,53,
%T A172180 54,60,59,67,66,71,75,76,81,85,86,93,93,93,93,101,111,112,113,115,120,
%U A172180 120,129,133,138,141,141,144,147,147,153,165,168,168,171,183,185,193
%N A172180 The absolute value of nth prime of the form 3*n-+1 minus nth nonprime of the form 3*n-+1.
%F A172180 a(n)=abs(A045344(n)-A171993(n)).
%p A172180 a(1)=abs(2-1)=1, a(2)=abs(5-4)=1, a(3)=abs(7-8)=1, a(4)=abs(11-10)=1, a(5)=abs(13-14)=1, a(6)=abs(17-16)=1, a(7)=abs(19-20)=1, a(8)=abs(23-22)=1, a(9)=abs(29-25)=4.
%Y A172180 Cf. A045344, A171993.
%K A172180 nonn,new
%O A172180 1,9
%A A172180 Juri-Stepan Gerasimov (2stepan(AT)rambler.ru), Jan 28 2010

%I A172183
%S A172183 5,11,7,13,13,31,11,17,13,19,19,37,17,23,19,41,8209,43,23,29,29,31,31,
%T A172183 73,29,53,31,37,37,79,0,41,37,43,43,61,41,47,43,67,73,67,47,53,53,71,79,
%U A172183 73,53,59,59,61,61,79,59,83,61,67,67,109,0,71,67,73,73,191,71,193,73,79
%N A172183 a(n) is the smallest prime of the form p^q+n, where p and q are prime, or zero if no such prime exists.
%C A172183 If n mod 6 = 1, both p and q must be 2, and a(n)=0 if n+4 is not a prime. Conjecture: a(n)=0 for n=87,257,297,353,383,557,717,773,927,...
%e A172183 a(1)=5 because 5=2^2+1 is the smallest prime of the form p^q+1. a(2)=11 because 11=3^2+2. a(3)=7, because 7=2^2+3. a(17)=8209, because 8209=2^13+17. a(31)=0, because p^q+31 cannot be a prime.
%t A172183 For[l={};n=1,n<100,n++, For[found=False;rm=Infinity;i=1,i<100,i++, For[j=1,j<100,j++, p=Prime[i];q=Prime[j];r=p^q+n; If[r>=rm,Break[],If[PrimeQ[r],rm=r;found=True]]]] If[ !found,rm=0];l=Append[l,rm] ] l
%Y A172183 Cf. A123318, A056208, A056206, A057733, A123250, A104066, A156940, A104067, A156973
%K A172183 nonn,new
%O A172183 1,1
%A A172183 Cheng Zhang (cz1(AT)rice.edu), Jan 28 2010

%I A172310
%S A172310 0,1,3,7,13,21,33,47,61,79,97,117,141,165,203,237,279,313,339,367,399,
%T A172310 437,487,541,599,657,723,783,839,889,955,1021,1087,1157,1199,1251,1307,
%U A172310 1361,1423,1489,1567,1655
%N A172310 L-toothpick sequence (see Comment lines for definition).
%C A172310 We define an "L-toothpick" to consist of two line segments forming an "L".
%C A172310 There are two size for L-toothpicks: Small and large. Each component of small L-toothpick has length 1. Each component of large L- toothpick has length sqrt(2).
%C A172310 The rule for the nth stage:
%C A172310 If n is odd then we add the large L-toothpicks to the structure, otherwise we add the small L-toothpicks to the structure.
%C A172310 Note that, on the infinite square grid, every large L-toothpick is placed with angle = 45 degrees and every small L-toothpick is placed with angle = 90 degrees.
%C A172310 The special rule: L-toothpicks are not added if this would lead to overlap with another L-toothpick branch in the same generation. In these structure, "exposed" ends persist in generations.
%C A172310 We start at stage 0 with no L-toothpicks.
%C A172310 At stage 1 we place a large L-toothpick in the horizontal direction, as a "V", anywhere in the plane (Note that there are two exposed endpoints).
%C A172310 At stage 2 we place two small L-toothpicks.
%C A172310 At stage 3 we place four large L-toothpicks.
%C A172310 At stage 4 we place six small L-toothpicks.
%C A172310 And so on...
%C A172310 The sequence gives the number of L-toothpick after n stages. A172311 (the first differences) gives the number of L-toothpicks added at the n-th stage.
%C A172310 For more information see A139250, the toothpick sequence.
%C A172310 In calculating the extension, the "special rule" was strengthened to prohibit intersections as well as overlappings. [From John W. Layman (layman(AT)math.vt.edu), Feb 04 2010]
%H A172310 David Applegate, Omar E. Pol and N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/tooth.pdf">The Toothpick Sequence and Other Sequences from Cellular Automata</a>
%H A172310 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/toothlist.html">Catalog of Toothpick and Cellular Automata Sequences in the OEIS</a>
%H A172310 O. E. Pol, <a href="http://www.polprimos.com/imagenespub/poltp061.jpg">Illustration of initial terms</a>
%H A172310 <a href="Sindx_To.html#toothpick">Index entries for sequences related to toothpick sequences</a>
%H A172310 <a href="Sindx_Ce.html#cell">Index entries for sequences related to cellular automata</a>
%Y A172310 Cf. A139250, A160120, A160170, A160172, A161206, A161328, A172311, A172312.
%K A172310 more,nonn,new
%O A172310 0,3
%A A172310 Omar E. Pol (info(AT)polprimos.com), Jan 31 2010
%E A172310 Terms a(9) - a(41) from John W. Layman (layman(AT)math.vt.edu), Feb 04 2010

%I A170919
%S A170919 1,1,3,3,5,45,105,315,2835,14175,5775,467775,6081075,2837835,212837625,
%T A170919 70945875,3618239625,97692469875,206239658625,9280784638125,1031198293125,
%U A170919 142924083427125,322279795963125,101111706320625,136968913284328125,161872352063296875
%N A170919 Write tan x = Prod_{n>=1} (1 + g_n x^n); a(n) = denominator(g_n).
%e A170919 1, -1, 7/3, -14/3, 54/5, -1112/45, 6574/105, -48488/315, 1143731/2835, ...
%Y A170919 Cf. A170918, A170910-A170917.
%K A170919 nonn,new
%O A170919 1,3
%A A170919 N. J. A. Sloane (njas(AT)research.att.com), Jan 30 2010

%I A170918
%S A170918 1,1,7,14,54,1112,6574,48488,1143731,14813072,16252211,3500388967,125127865048,
%T A170918 158589803803,33133618166566,30512906279732,4378989933312913,330336346477870319,
%U A170918 1981395373839282068,251479418962683770473,79893293800974935213,31493610597939643431532
%V A170918 1,-1,7,-14,54,-1112,6574,-48488,1143731,-14813072,16252211,-3500388967,125127865048,
%W A170918 -158589803803,33133618166566,-30512906279732,4378989933312913,-330336346477870319,
%X A170918 1981395373839282068,-251479418962683770473,79893293800974935213,-31493610597939643431532
%N A170918 Write tan x = Prod_{n>=1} (1 + g_n x^n); a(n) = numerator(g_n).
%e A170918 1, -1, 7/3, -14/3, 54/5, -1112/45, 6574/105, -48488/315, 1143731/2835, ...
%p A170918 t1:=tan(x);
%p A170918 L:=100;
%p A170918 t0:=series(t1,x,L):
%p A170918 g:=[]; M:=40; t2:=t0:
%p A170918 for n from 1 to M do
%p A170918 t3:=coeff(t2,x,n); t2:=series(t2/(1+t3*x^n),x,L); g:=[op(g),t3];
%p A170918 od:
%p A170918 g;
%p A170918 g1:=map(numer,g);
%p A170918 g2:=map(denom,g);
%Y A170918 Cf. A170919, A170910-A170917.
%K A170918 sign,frac,new
%O A170918 1,3
%A A170918 N. J. A. Sloane (njas(AT)research.att.com), Jan 30 2010

%I A170917
%S A170917 6,120,840,362880,14968800,9340531200,49037788800,3201186852864000,8485288812000,
%T A170917 182467650613248000,908859963476424960000,1424498881530396672000000,10633661572674172032000000,
%U A170917 8289151869130970582384640000000,1720739115690134518218240000000,97858575719142221963014963200000000
%N A170917 Write sin(x)/x = Prod_{n>=1} (1 + g_n x^2n); a(n) = denominator(g_n).
%D A170917 H. Gingold, H. W. Gould and M. E. Mays, Power product expansions, Until. Math., 34 (1988), 143-161.
%e A170917 -1/6, 1/120, 1/840, 73/362880, 353/14968800, 36499/9340531200, 24257/49037788800, ...
%Y A170917 Cf. A170916, A170912-A170915.
%K A170917 nonn,frac,new
%O A170917 1,1
%A A170917 N. J. A. Sloane (njas(AT)research.att.com), Jan 30 2010

%I A170916
%S A170916 1,1,1,73,353,36499,24257,302426881,87721,348958703,226786069421,62199570679633,
%T A170916 62531659610839,8559230855533306387,235495453816743509,2644298730170939345197,
%U A170916 281737789368631676609,39043444996461526437828311,6203284926188598376335167
%V A170916 -1,1,1,73,353,36499,24257,302426881,87721,348958703,226786069421,62199570679633,
%W A170916 62531659610839,8559230855533306387,235495453816743509,2644298730170939345197,
%X A170916 281737789368631676609,39043444996461526437828311,6203284926188598376335167
%N A170916 Write sin(x)/x = Prod_{n>=1} (1 + g_n x^2n); a(n) = numerator(g_n).
%D A170916 H. Gingold, H. W. Gould and M. E. Mays, Power product expansions, Until. Math., 34 (1988), 143-161.
%e A170916 -1/6, 1/120, 1/840, 73/362880, 353/14968800, 36499/9340531200, 24257/49037788800, ...
%Y A170916 Cf. A170917, A170912-A170915.
%K A170916 sign,frac,new
%O A170916 1,4
%A A170916 N. J. A. Sloane (njas(AT)research.att.com), Jan 30 2010

%I A170915
%S A170915 1,1,6,6,120,120,5040,280,72576,362880,39916800,11975040,1245404160,88957440,
%T A170915 1307674368000,11675664000,71137485619200,1067062284288000,121645100408832000,
%U A170915 101370917007360000,10218188434341888000,5109094217170944000,25852016738884976640000
%N A170915 Write 1 + sin x = Prod_{n>=1} (1 + g_n x^n); a(n) = denominator(g_n).
%D A170915 H. Gingold, H. W. Gould and M. E. Mays, Power product expansions, Until. Math., 34 (1988), 143-161.
%e A170915 1, 0, -1/6, 1/6, -19/120, 19/120, -659/5040, 37/280, -7675/72576, ...
%Y A170915 Cf. A170914, A170910, A170911,
%K A170915 nonn,frac,new
%O A170915 1,3
%A A170915 N. J. A. Sloane (njas(AT)research.att.com), Jan 30 2010

%I A170914
%S A170914 1,0,1,1,19,19,659,37,7675,40043,3578279,1123009,95259767,7091713,85215100151,
%T A170914 832857559,4180679675171,63804880881241,6399968826052559,5697831990097981,
%U A170914 478887035449041839,252737248941887573,1123931378903214542099,35703551772944759
%V A170914 1,0,-1,1,-19,19,-659,37,-7675,40043,-3578279,1123009,-95259767,7091713,-85215100151,
%W A170914 832857559,-4180679675171,63804880881241,-6399968826052559,5697831990097981,
%X A170914 -478887035449041839,252737248941887573,-1123931378903214542099,35703551772944759
%N A170914 Write 1 + sin x = Prod_{n>=1} (1 + g_n x^n); a(n) = numerator(g_n).
%D A170914 H. Gingold, H. W. Gould and M. E. Mays, Power product expansions, Until. Math., 34 (1988), 143-161.
%e A170914 1, 0, -1/6, 1/6, -19/120, 19/120, -659/5040, 37/280, -7675/72576, ...
%Y A170914 Cf. A170915, A170910, A170911,
%K A170914 sign,frac,new
%O A170914 1,5
%A A170914 N. J. A. Sloane (njas(AT)research.att.com), Jan 30 2010

%I A170913
%S A170913 2,24,360,13440,453600,47900160,5448643200,2988969984000,3126159036000,
%T A170913 101370917007360000,4390627842881280000,552984315270266880000,393839317506450816000000,
%U A170913 1465809349094778175488000000,129517997955171415349760000000,263130836933693530167218012160000000
%N A170913 Write cos(x) = Prod_{n>=1} (1 + g_n x^2n); a(n) = denominator(g_n).
%D A170913 H. Gingold, H. W. Gould and M. E. Mays, Power product expansions, Until. Math., 34 (1988), 143-161.
%e A170913 -1/2, 1/24, 7/360, 131/13440, 1843/453600, 97261/47900160, ...
%Y A170913 Cf. A170912.
%K A170913 nonn,frac,new
%O A170913 1,1
%A A170913 N. J. A. Sloane (njas(AT)research.att.com), Jan 30 2010

%I A170912
%S A170912 1,1,7,131,1843,97261,4683059,1331727679,568285777,9521655609199,175554688130609,
%T A170912 11334988388673161,3457026400678609391,6594042537777612027841,249248595232521829462213,
%U A170912 268938575250382935485761673113,3929672369519648081411955883,4719016202742955262333630268611
%V A170912 -1,1,7,131,1843,97261,4683059,1331727679,568285777,9521655609199,175554688130609,
%W A170912 11334988388673161,3457026400678609391,6594042537777612027841,249248595232521829462213,
%X A170912 268938575250382935485761673113,3929672369519648081411955883,4719016202742955262333630268611
%N A170912 Write cos(x) = Prod_{n>=1} (1 + g_n x^2n); a(n) = numerator(g_n).
%D A170912 H. Gingold, H. W. Gould and M. E. Mays, Power product expansions, Until. Math., 34 (1988), 143-161.
%e A170912 -1/2, 1/24, 7/360, 131/13440, 1843/453600, 97261/47900160, ...
%p A170912 t1:=cos(x);
%p A170912 L:=100;
%p A170912 t0:=series(t1,x,L):
%p A170912 g:=[]; M:=40; t2:=t0:
%p A170912 for n from 1 to M do
%p A170912 t3:=coeff(t2,x,n); t2:=series(t2/(1+t3*x^n),x,L); g:=[op(g),t3];
%p A170912 od:
%p A170912 g;
%p A170912 h:=[seq(g[2*n],n=1..nops(g)/2)];
%p A170912 h1:=map(numer,h);
%p A170912 h2:=map(denom,h);
%Y A170912 Cf. A170913.
%K A170912 sign,frac,new
%O A170912 1,3
%A A170912 N. J. A. Sloane (njas(AT)research.att.com), Jan 30 2010

%I A170911
%S A170911 1,2,3,8,5,72,7,128,81,800,11,13824,13,6272,30375,32768,17,419904,19,20480000,
%T A170911 750141,247808,23,1528823808,15625,1384448,1594323,5035261952,29,30233088000000,
%U A170911 31,2147483648,235782657,37879808,1313046875,240734712102912,37,189267968
%N A170911 Write exp(-x) = Prod_{n>=1} (1 + g_n x^n); a(n) = denominator(g_n).
%D A170911 H. Gingold, H. W. Gould and M. E. Mays, Power product expansions, Until. Math., 34 (1988), 143-161.
%e A170911 -1, 1/2, 1/3, 3/8, 1/5, 13/72, 1/7, 27/128, 8/81, 91/800, 1/11, ...
%p A170911 L:=100; t1:=exp(-x); t0:=series(t1,x,L): g:=[]; M:=40; t2:=t0:
%p A170911 for n from 1 to M do t3:=coeff(t2,x,n); t2:=series(t2/(1+t3*x^n),x,L); g:=[op(g),t3]; od: g;
%Y A170911 Cf. A170910.
%K A170911 nonn,frac,new
%O A170911 1,2
%A A170911 N. J. A. Sloane (njas(AT)research.att.com), Jan 30 2010

%I A170910
%S A170910 1,1,1,3,1,13,1,27,8,91,1,1213,1,505,1919,2955,1,24557,1,1136313,34943,
%T A170910 12277,1,65978519,624,57331,58528,195948483,1,1052424027703,1,77010795,
%U A170910 7085759,1179631,37497599,7047825380633,1,5242861,89281919,355723139681937
%V A170910 -1,1,1,3,1,13,1,27,8,91,1,1213,1,505,1919,2955,1,24557,1,1136313,34943,
%W A170910 12277,1,65978519,624,57331,58528,195948483,1,1052424027703,1,77010795,
%X A170910 7085759,1179631,37497599,7047825380633,1,5242861,89281919,355723139681937
%N A170910 Write exp(-x) = Prod_{n>=1} (1 + g_n x^n); a(n) = numerator(g_n).
%D A170910 H. Gingold, H. W. Gould and M. E. Mays, Power product expansions, Until. Math., 34 (1988), 143-161.
%e A170910 -1, 1/2, 1/3, 3/8, 1/5, 13/72, 1/7, 27/128, 8/81, 91/800, 1/11, ...
%p A170910 L:=100; t1:=exp(-x); t0:=series(t1,x,L): g:=[]; M:=40; t2:=t0:
%p A170910 for n from 1 to M do t3:=coeff(t2,x,n); t2:=series(t2/(1+t3*x^n),x,L); g:=[op(g),t3]; od: g;
%Y A170910 Cf. A170911.
%K A170910 sign,frac,new
%O A170910 1,4
%A A170910 N. J. A. Sloane (njas(AT)research.att.com), Jan 30 2010

%I A170909
%S A170909 1,1,2,3,24,15,360,5040,40320,11340,1814400,39916800,47900160,6227020800,
%T A170909 7925299200,1307674368000,20922789888000,11115232128000,3201186852864000,
%U A170909 24329020081766400,48658040163532800,51090942171709440000,1124000727777607680000
%N A170909 Denominators in Taylor series expansion of Product_{n >= 1} (1+x^n/n!).
%D A170909 H. Gingold, H. W. Gould and M. E. Mays, Power product expansions, Until. Math., 34 (1988), 143-161.
%e A170909 1+x+(1/2)*x^2+(2/3)*x^3+(5/24)*x^4+(2/15)*x^5+(41/360)*x^6+(169/5040)*x^7+...
%Y A170909 Cf. A170908.
%K A170909 nonn,frac,new
%O A170909 0,4
%A A170909 N. J. A. Sloane (njas(AT)research.att.com), Jan 30 2010

%I A170908
%S A170908 1,1,1,2,5,2,41,169,541,71,8983,44419,20183,802751,445223,52275409,166257661,
%T A170908 26261353,2160586067,4871649347,3667033133,2762567051857,10112898715063,
%U A170908 12453960597367,24546527305109,48002125894859,5216471357244949,159144839200310539
%N A170908 Numerators in Taylor series expansion of Product_{n >= 1} (1+x^n/n!).
%D A170908 H. Gingold, H. W. Gould and M. E. Mays, Power product expansions, Until. Math., 34 (1988), 143-161.
%e A170908 1+x+(1/2)*x^2+(2/3)*x^3+(5/24)*x^4+(2/15)*x^5+(41/360)*x^6+(169/5040)*x^7+...
%Y A170908 Cf. A170909.
%K A170908 nonn,frac,new
%O A170908 0,4
%A A170908 N. J. A. Sloane (njas(AT)research.att.com), Jan 30 2010

%I A054634
%S A054634 0,1,2,3,4,5,6,7,1,0,1,1,1,2,1,3,1,4,1,5,1,6,1,7,2,0,2,1,2,2,2,3,2,
%T A054634 4,2,5,2,6,2,7,3,0,3,1,3,2,3,3,3,4,3,5,3,6,3,7,4,0,4,1,4,2,4,3,4,4,
%U A054634 4,5,4,6,4,7,5,0,5,1,5,2,5,3,5,4,5,5,5,6,5,7,6,0,6,1,6,2,6,3,6,4,6
%N A054634 Champernowne sequence: write n in base 8 and juxtapose.
%C A054634 Apart from the initial term, identical to A031035.
%C A054634 Should not be merged with A031035 because there are many sequences which depend on the latter starting with a 1. - N. J. A. Sloane, Jan 30 2010
%Y A054634 Cf. A007376, A030190.
%K A054634 nonn,base,easy,new
%O A054634 0,3
%A A054634 N. J. A. Sloane (njas(AT)research.att.com), Apr 16 2000

%I A031035
%S A031035 1,2,3,4,5,6,7,1,0,1,1,1,2,1,3,1,4,1,5,1,6,1,7,2,0,2,1,2,2,2,
%T A031035 3,2,4,2,5,2,6,2,7,3,0,3,1,3,2,3,3,3,4,3,5,3,6,3,7,4,0,4,1,4,
%U A031035 2,4,3,4,4,4,5,4,6,4,7,5,0,5,1,5,2,5,3,5,4,5,5,5,6,5,7,6,0,6
%N A031035 Write n in base 8 and juxtapose.
%C A031035 Apart from the initial term, identical to A054634.
%C A031035 Should not be merged with A054634 because there are many sequences which depend on this sequence starting with a 1. - N. J. A. Sloane, Jan 30 2010
%K A031035 nonn,new
%O A031035 1,2
%A A031035 Clark Kimberling (ck6(AT)evansville.edu)

%I A160753
%S A160753 0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,1,0,0,1,1,1,0,1,1,1,0,0,0,0,1,
%T A160753 1,1,1,1,0,1,0
%N A160753 Binary expansion of the Chaitin halting probability Omega_L for a certain programming language L.
%C A160753 If this sequence were extended to 5000 terms, it would settle the Riemann hypothesis.
%D A160753 C. S. Calude, E. Calude and M. J. Dinneen, A new measure of the difficulty of problems, J. Mult.-Valued Logic Soft. Comput., 12 (2006), 285-307.
%D A160753 C. S. Calude and G. J. Chaitin, What is a Halting Probability?, Notices Amer. Math. Soc., 57 (No. 2, 2010), 236-237.
%D A160753 C. S. Calude and M. J. Dinneen, Exact approximations of omega numbers, Internat. J. Bifur. Chaos, 17 (6) (2007), 1937-1954.
%Y A160753 Cf. A079365.
%K A160753 nonn,hard,more,new
%O A160753 0,1
%A A160753 N. J. A. Sloane (njas(AT)research.att.com), Jan 29 2010

%I A172179
%S A172179 1,1,100,1,101,199,1,102,300,298,1,103,402,598,397,1,104,505,1000,995,
%T A172179 496,1,105,609,1505,1995,1491,595,1,106,714,2114,3500,3486,2086,694,1,
%U A172179 107,820,2828,5614,6986,5572,2780,793,1,108,927,3648,8442,12600,12558
%N A172179 (1,[99n+1])Pascal Triangle
%e A172179 Triangle starts:
%e A172179 1
%e A172179 1,100
%e A172179 1,101,199
%e A172179 1,102,300,298
%Y A172179 Cf. A172178, A172171, A007318
%K A172179 nonn,tabl,new
%O A172179 1,3
%A A172179 M. Dols (markdols99(AT)yahoo.com), Jan 28 2010

%I A172178
%S A172178 1,100,199,298,397,496,595,694,793,892,991,1090,1189,1288,1387,1486,
%T A172178 1585,1684,1783,1882,1981,2080,2179,2278,2377,2476,2575,2674,2773,2872,
%U A172178 2971,3070,3169,3268,3367,3466,3565,3664,3763,3862,3961,4060,4159,4258
%N A172178 99n+1
%F A172178 a(n)= a(n-1)+99 with a(1)=1
%Y A172178 Cf. A017173
%K A172178 nonn,new
%O A172178 1,2
%A A172178 M. Dols (markdols99(AT)yahoo.com), Jan 28 2010
%E A172178 More terms a(15)-a(47) from Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Jan 29 2010

%I A172177
%S A172177 1,1,1,1,2,1,1,21,21,1,1,415,485,415,1,1,11425,13453,13453,11425,1,1,
%T A172177 431401,499729,509797,499729,431401,1,1,21768481,24786961,25250545,
%U A172177 25250545,24786961,21768481,1,1,1422454321,1596592801,1620622801
%N A172177 Symmetrical triangle sequence: t(n,m)=1 + Abs[n! - m! ]*Abs[n! - (n - m)! ].
%C A172177 Row sums are:
%C A172177 {1, 2, 4, 44, 1317, 49758, 2372059, 143611976, 10903107465, 1019901208330,
%C A172177 115544527447691...}.
%F A172177 t(n,m)=1 + Abs[n! - m! ]*Abs[n! - (n - m)! ].
%e A172177 {1},
%e A172177 {1, 1},
%e A172177 {1, 2, 1},
%e A172177 {1, 21, 21, 1},
%e A172177 {1, 415, 485, 415, 1},
%e A172177 {1, 11425, 13453, 13453, 11425, 1},
%e A172177 {1, 431401, 499729, 509797, 499729, 431401, 1},
%e A172177 {1, 21768481, 24786961, 25250545, 25250545, 24786961, 21768481, 1},
%e A172177 {1, 1422454321, 1596592801, 1620622801, 1623767617, 1620622801, 1596592801, 1422454321, 1},
%e A172177 {1, 117050250241, 129852263521, 131418447841, 131629642561, 131629642561, 131418447841, 129852263521, 117050250241, 1},
%e A172177 {1, 11851367230081, 13021869047041, 13149878545441, 13165489630081, 13167318542401, 13165489630081, 13149878545441, 13021869047041, 11851367230081, 1}
%t A172177 Clear[t, n, m];
%t A172177 t[n_, m_] = 1 + Abs[n! - m! ]*Abs[n! - (n - m)! ];
%t A172177 Table[Table[t[n, m], {m, 0, n}], {n, 0, 10}];
%t A172177 Flatten[%]
%K A172177 nonn,tabl,uned,new
%O A172177 0,5
%A A172177 Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Jan 28 2010

%I A172176
%S A172176 1,2,2,1,2,1,8,0,0,8,31,4,5,4,31,74,10,22,22,10,74,143,18,
%T A172176 57,82,57,18,143,244,28,116,188,188,116,28,244,383,40,205,352,
%U A172176 401,352,205,40,383,566,54,330,586,714,714,586,330,54,566,799
%V A172176 1,2,2,1,2,1,-8,0,0,-8,-31,-4,5,-4,-31,-74,-10,22,22,-10,-74,-143,-18,
%W A172176 57,82,57,-18,-143,-244,-28,116,188,188,116,-28,-244,-383,-40,205,352,
%X A172176 401,352,205,-40,-383,-566,-54,330,586,714,714,586,330,-54,-566,-799
%N A172176 Symmetrical triangle sequence: t(n,m)=1 + (n + m - n*m)*(n + (n - m) - n(n - m)).
%C A172176 Row sums are:
%C A172176 {1, 4, 4, -16, -65, -124, -126, 64, 669, 2020, 4576...}.
%e A172176 {1},
%e A172176 {2, 2},
%e A172176 {1, 2, 1},
%e A172176 {-8, 0, 0, -8},
%e A172176 {-31, -4, 5, -4, -31},
%e A172176 {-74, -10, 22, 22, -10, -74},
%e A172176 {-143, -18, 57, 82, 57, -18, -143},
%e A172176 {-244, -28, 116, 188, 188, 116, -28, -244},
%e A172176 {-383, -40, 205, 352, 401, 352, 205, -40, -383},
%e A172176 {-566, -54, 330, 586, 714, 714, 586, 330, -54, -566}, {-799, -70, 497, 902, 1145, 1226, 1145, 902, 497, -70, -799}
%t A172176 Clear[t, n, m];
%t A172176 t[n_, m_] = 1 + (n + m - n*m)*(n + (n - m) - n(n - m));
%t A172176 Table[Table[t[n, m], {m, 0, n}], {n, 0, 10}];
%t A172176 Flatten[%]
%K A172176 sign,tabl,uned,new
%O A172176 0,2
%A A172176 Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Jan 28 2010

%I A172175
%S A172175 1,111,12211,1343211,147753211,16252853211,1787813853211,
%T A172175 196659523853211,21632547623853211,2379580238623853211,
%U A172175 261753826248623853211,28792920887348623853211
%N A172175 a(n)= a(n-1)*110+1
%C A172175 Sum of pairs of integers given in A162849.Sum of digits give A000225
%Y A172175 Cf. A165155, A162849, A000225.
%K A172175 nonn,new
%O A172175 1,2
%A A172175 M. Dols (markdols99(AT)yahoo.com), Jan 28 2010
%E A172175 More terms from Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Jan 29 2010

%I A172174
%S A172174 1,91,8191,737191,66347191,5971247191,537412247191,48367102247191
%N A172174 a(n)= a(n-1)*90+1
%C A172174 Difference of pairs of integers given in A162849.Sum of digits give A017173.
%Y A172174 Cf. A165154, A162849, A017173.
%K A172174 nonn,new
%O A172174 1,2
%A A172174 M. Dols (markdols99(AT)yahoo.com), Jan 28 2010

%I A172173
%S A172173 0,1,1,11,12,32,44,85,129,223,352,584,936,1529,2465,4003,6468
%N A172173 Sums of NE-SW diagonals of triangle A172171
%F A172173 For n=even: a(n)=a(n-1)+a(n-2); for n=odd: a(n)=a(n-1)+a(n-2)+9 ; with a(0)=0 and a(1)=1
%Y A172173 Cf. A172171, A000045, A172172
%K A172173 nonn,new
%O A172173 1,4
%A A172173 M. Dols (markdols99(AT)yahoo.com), Jan 28 2010

%I A172172
%S A172172 0,1,10,20,39,68,116,193,318,520,847,1376,2232,3617,5858,9484
%N A172172 Sums of NW-SE diagonals of triangle A172171
%F A172172 a(n)=a(n-1)+a(n-2)+9 with a(0)=0 and a(1)=1
%Y A172172 Cf. A172171, A000045
%K A172172 nonn,new
%O A172172 1,3
%A A172172 M. Dols (markdols99(AT)yahoo.com), Jan 28 2010

%I A172171
%S A172171 1,1,10,1,11,19,1,12,30,28,1,13,42,58,37,1,14,55,100,95,46,1,15,69,155,
%T A172171 195,141,55
%N A172171 (1,9) Pascal Triangle read by horizontal rows.Same as A093644, but mirrored and without the additional row/column (1,9,9,9,9,..).
%C A172171 Binomial transform of A017173.Row sums give A139634.Central axis gives A050489
%e A172171 Triangle begins:
%e A172171 .....1
%e A172171 ....1,10
%e A172171 ...1,11,19
%e A172171 ..1,12,30,28
%e A172171 .1,13,42,58,37
%e A172171 1,14,55,100,95,46
%Y A172171 Cf. A139634, A007318, A017173, A093644, A050489
%K A172171 nonn,tabl,new
%O A172171 1,3
%A A172171 M. Dols (markdols99(AT)yahoo.com), Jan 28 2010

%I A172170
%S A172170 1,1,1,3,3,5,5,7,7,3,3,11,11,13,13,3,3,17,17,19,19,3,3,23,23,5,5,3,3,29,
%T A172170 29,31,31,3,3,5,5,37,37,3,3,41,41,43,43,3,3,47,47,7,7,3,3,53,53,5,5,3,3,
%U A172170 59,59,61,61,3,3,5,5,67,67,3,3,71,71,73,73,3,3,7,7,79,79,3,3,83,83,5,5
%N A172170 A090368=1,3,5,7,3,11,13,3,. a(n)= 1, doubled A090368.
%C A172170 For -differences of ,from Euler A000111 and Bernoulli tan x + sec x , ( (A099612/A099617)=1,1,1/2,1/3,5/24,2/15, ) =0,1/2,1/6,3/24,9/120,35/720,=((0,A034428)/A000142(n+1))=b(n).Numerators and denominators of b(n) will be divided by a(n).
%K A172170 nonn,uned,new
%O A172170 0,4
%A A172170 Paul Curtz (bpcrtz(AT)]free.fr), Jan 28 2010

%I A172169
%S A172169 3,3,0,1,1,4,2,1,4,8,5,2,8,7,0,2,0,2,8,8,9,3,2,9,5,8,8,7,7,2,2,8,2,6,
%T A172169 8,2,5,7,3,6,9,8,5,0,0,8,3,2,6,3,7,6,3,8,7,8,1,9,6,0,0,2,4,5,1,9,3,5,9,
%U A172169 1,5,2,7,5,6,1,6,5,6,9,8,3,7,2,6,6,8,5,0,4,2,4,0,4,4,2,0,6,3,6,7,6,4,6
%N A172169 Decimal expansion of solution to x=Fibonacci(x);0<x<1
%C A172169 x=Fibonacci(x);x<1; expliz. Fibonacci = pow(sqrt(5)*0.5+0.5,x)/sqrt(5)+pow(sqrt(5)*0.5-0.5,x)*sin(PI*(x-0.5))/sqrt(5)
%H A172169 Gerd Lamprecht, <a href="http://freenet-homepage.de/gerdlamprecht/Roemisch_JAVA.htm#@P@Q5)*0.5+0.5,x)/@Q5)+@P@Q5)*0.5-0.5,x)*sin(PI*(x-0.5))/@Q5)-x@Na=0.33;b=0.331;c=(a+b)/2;@Nd=(Fx(c)*Fx(a)%3C0);a=d?a:c;b=d?c:b;c=(c+(d?a:b))/2;@N@AFx(c))%3C%205e-17@N0@N1@Nc=c;@B0]=GetKoDezi(-11,0,56);">Iterationsrechner mit Algorithmus</a>
%H A172169 Gerd Lamprecht, <a href="http://freenet-homepage.de/gerdlamprecht/Zahlenfolgen.html">Zahlenfolgen (sequence)</a>
%F A172169 Gerd Lamprecht online Iterationsrechner Beispiel 59
%e A172169 0.3301142148528702028...=Fibonacci(0.3301142148528702028...)
%o A172169 (Other) Gerd Lamprecht online Iterationsrechner: #@P@Q5)*0.5+0.5,x)/@Q5)+@P@Q5)*0.5-0.5,x)*sin(PI*(x-0.5))/@Q5)-x@Na=0.33;b=0.331;c=(a+b)/2;@Nd=(Fx(c)*Fx(a)%3C0);a=d?a:c;b=d?c:b;c=(c+(d?a:b))/2;@N@AFx(c))%3C%205e-17@N0@N1@Nc=c;@B0]=GetKoDezi(-11,0,56);
%Y A172169 A171909, A172081
%K A172169 cons,nonn,new
%O A172169 0,2
%A A172169 Gerd Lamprecht (gerdlamprecht(AT)googlemail.com), Jan 28 2010
%E A172169 Adjusted offset and leading zero - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jan 30 2010

%I A172168
%S A172168 8,1,4,5,9,6,5,7,1,7,0,2,9,9
%N A172168 Decimal expansion of Marek Wolf's analog of the Brun constant for primes of the form q = m^2 + 1.
%C A172168 The sum given in the formula below is trivially convergent because each term is less than the corresponding term of SUM[i=i..infinity] 1/(i^2) = (pi^2)/6.
%C A172168 Eight significant digits of this constant are mentioned in A083844, which counts the number of primes of the form m^2 + 1 < 10^n. [From T. D. Noe (noe(AT)sspectra.com), Jan 28 2010]
%H A172168 Marek Wolf, <a href="http://arxiv.org/abs/0803.1456">Search for primes of the form m^2+1</a>, version 3, pp.6-8, Jan 27, 2010.
%H A172168 G. L. Honaker Jr. and C. Caldwell, <a href="http://primes.utm.edu/curios/page.php?number_id=3508">Prime Curios!: 0.81459657</a> [From T. D. Noe (noe(AT)sspectra.com), Jan 28 2010]
%F A172168 SUM[q in {primes of form m^2 + 1] 1/q = SUM[i=1..infinity] 1/A002496(i) = 1/2 + 1/5 + 1/17 + 1/37 + 1/101 + ...
%e A172168 0.81459657170299.
%Y A172168 Cf. A002496, A005597.
%K A172168 nonn,new
%O A172168 0,2
%A A172168 Jonathan Vos Post (jvospost3(AT)gmail.com), Jan 28 2010
%E A172168 Adjusted offset and leading zero - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jan 30 2010

%I A172167
%S A172167 2,5,10,17,30,47,66,103,176,273,382,545,738,995,1428,1915,2492,3261,
%T A172167 4414,5711,7170,9763,12680,16137,20026,30395,42684,60181,78614,117981,
%U A172167 170470,236007,375976,523433,733386,1065163,1537556,2167413,2913910
%N A172167 Partial sums of Class 1- (or Pierpont) primes A005109.
%C A172167 The subsequence of primes in this sequence begins a(1) = 2, a(2) = 5, a(4) = 17, a(6) = 47, a(8) = 103, a(20) = 5711. The subsubsequence which are Pierpont prime partial sums of Pierpont primes begins 2, 5, 17, and then which occur next?
%F A172167 a(n) = SUM[i=1..n] A005109(i) = SUM[i=1..n] (primes of the form 2^t*3^u + 1}.
%e A172167 a(20) = 2 + 3 + 5 + 7 + 13 + 17 + 19 + 37 + 73 + 97 + 109 + 163 + 193 + 257 + 433 + 487 + 577 + 769 + 1153 + 1297 = 5711 is prime.
%Y A172167 Cf. A000040, A048135, A048136, A056637, A005105, A005110, A005111, A005112, A081424, A081425, A081426, A081427, A081428, A081429, A081430, A122259.
%K A172167 easy,nonn,new
%O A172167 1,1
%A A172167 Jonathan Vos Post (jvospost3(AT)gmail.com), Jan 28 2010
%E A172167 More terms from R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Feb 07 2010

%I A172166
%S A172166 5,16,33,62,99,140,193,252,319,390,487,588,715,864,1043,1234,1457,1684,
%T A172166 1935,2192,2461,2768,3079,3410,3757,4176,4607,5148,5705,6268,6837,7424,
%U A172166 8017,8616,9257,9984,10717,11456,12265,13086,13939,14868,15805,16772
%N A172166 Partial sums of A028388 good primes (version 2).
%C A172166 Prime partial sums of good primes begin a(1) = 5, a(7) = 193, a(11) = 487, a(23) = 3079. Good prime partial sums of good primes begin a(1) = 5, and what are the next of that subsequence?
%F A172166 a(n) = SIM[i=1..n] {p_n such that (p_n)^2 > p_{n-i}p_{n+i} for all 1 <= i <= n-1}.
%e A172166 a(20) = 5 + 11 + 17 + 29 + 37 + 41 + 53 + 59 + 67 + 71 + 97 + 101 + 127 + 149 + 179 + 191 + 223 + 227 + 251 + 257 = 2192.
%Y A172166 Cf. A000040, A028388.
%K A172166 easy,nonn,new
%O A172166 1,1
%A A172166 Jonathan Vos Post (jvospost3(AT)gmail.com), Jan 27 2010
%E A172166 More terms from R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jan 30 2010

%I A172165
%S A172165 2,4,12,68,630,7782,117656,2097160,43046730,1000000010,25937424612,
%T A172165 743008370700,23298085122494,793714773254158,29192926025390640,
%U A172165 1152921504606846992,48661191875666868498,2185911559738696531986
%N A172165 A simple sequence a(n) = n + n ^(n-1)
%F A172165 a(n) = n + n ^(n-1)
%F A172165 a(n) = n+A000169(n). [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jan 30 2010]
%e A172165 a(10) = 000000010
%K A172165 easy,nonn,new
%O A172165 1,2
%A A172165 Lorenzo Cococcia (lorenzo.cococcia(AT)yahoo.com), Jan 27 2010
%E A172165 Every second term removed by R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jan 30 2010

%I A172164
%S A172164 20,19,20,19,20,21,18,21,19,20,20,20,19,20,20,19,20,20,20,19,21,18,21,
%T A172164 19,21,18,21,19,21,18,21,19,20,20,20,19,20,20,19,20,20,20,19,20,20,20,
%U A172164 19,21,18,21,19,20,20,20,19,20,20,20,19,20,20,19,20,20,20,19,21,18,21
%N A172164 Differences between numbers of triangles entierly contained in two consecutive turns of Pythagore's snail (Theodorus spiral).
%C A172164 Conjecture : The terms are only 18,19,20,21 (From the first thousand turns, there are 2,3% of 18, 36,5% of 19, 46,2% of 20 and 15% of 21). No period found. Probably due to Pi transcendence.
%e A172164 Exemple : in the first turn, 16 triangles are complete. In the 2nd turn, there are 36 triangles completly included. The difference is 20.
%o A172164 (Python) See A137515 for Python code, and then OooCalc for more.
%Y A172164 Cf. A072895, A137515
%K A172164 nonn,new
%O A172164 2,1
%A A172164 Sebastien Dumortier (sdumortier(AT)ac-limoges.fr), Jan 27 2010

%I A172163
%S A172163 0,10,1020,103030,10307040,1030814050,103082025060,10308214641070,
%T A172163 1030821549763080
%N A172163 (A165155 - A165154)/2
%Y A172163 Cf. A165155, A165154, A162741, A162849
%K A172163 nonn,new
%O A172163 1,2
%A A172163 M. Dols (markdols(AT)yahoo.com), Jan 27 2010

%I A172162
%S A172162 1,101,10201,1020401,102050701,10205121101,1020513261601,
%T A172162 102051333512201
%N A172162 (A165155 + A165154)/2
%Y A172162 Cf. A165155, A165154, A162741, A162849
%K A172162 nonn,new
%O A172162 1,2
%A A172162 M. Dols (markdols(AT)yahoo.com), Jan 27 2010

%I A172161
%S A172161 0,1,2,3,4,6,9,13,20,30,45,67,101,151,227,340,510,765,1148
%N A172161 Greedy Coppersmith-Winograd sequence.
%C A172161 Coppersmith & Winograd asked for dense sets S of integers such that if A,B,C are three disjoint subsets of S, their sums are cannot all be equal. Such sets yield new matrix multiplication algorithms. This is the "greedy sequence" obeying this property, that is, we start with S={0, 1} and adjoin new integers one at a time, always adjoining the least new integer such that the Coppersmith-Winograd property remains valid. It looks as though each term is approximately 1.5 times the preceding term. The sequence is clearly infinite because each term is no greater than the sum of all previous terms.
%C A172161 Amazingly, this sequence appears to agree with McRae's sequence A120134 after the "3". (This probably can be proved, but I haven't as yet.) [From Warren D. Smith (warren.wds(AT)gmail.com), Jan 29 2010]
%D A172161 Don Coppersmith and Shmuel Winograd. Matrix multiplication via arithmetic progressions, Journal of Symbolic Computation, 9 (1990) 251-280.
%Y A172161 A120134 [From Warren D. Smith (warren.wds(AT)gmail.com), Jan 29 2010]
%K A172161 hard,nonn,new
%O A172161 1,3
%A A172161 Warren D. Smith (warren.wds(AT)gmail.com), Jan 27 2010
%E A172161 Added two more terms. Warren D. Smith (warren.wds(AT)gmail.com), Jan 29 2010

%I A172160
%S A172160 1,2,3,4,4,0,16,64,192,512,1280,3072,7168,16384,36864,81920
%V A172160 1,2,3,4,4,0,-16,-64,-192,-512,-1280,-3072,-7168,-16384,-36864,-81920
%N A172160 a(n+1)-2a(n)=-A131577.
%C A172160 A059165=16*A001787. b(n)=0,A001787;b(n+1)-2b(n)=A131577. a(n)+b(n)=A000079. Inverse binomial transform:1,1,0,0,-1,-1,-2,-2,-3,-3,=doubled -A023443.
%F A172160 a(n)=1,2,3,4,4,-A059165=1,2,3,4,4*A159964.
%K A172160 nonn,uned,new
%O A172160 0,2
%A A172160 Paul Curtz (bpcrtz(AT)free.fr), Jan 27 2010

%I A172158
%S A172158 0,0,0,0,978,62266,1220298,12033330,77784658,377818258,1492665418,
%T A172158 5042436754,15062292834,40736208186,101489568538
%N A172158 Number of ways to place 6 nonattacking kings on an n X n board
%H A172158 V. Kotesovec, <a href="http://web.telecom.cz/vaclav.kotesovec/math.htm">Number of ways of placing non - attacking queens and kings on boards of various sizes</a>
%F A172158 Explicit formula (Vaclav Kotesovec, 27.1.2010): a(n) = (n^12 - 135n^10 + 180n^9 + 7465n^8 - 18840n^7 - 202665n^6 + 751860n^5 + 2442334n^4 - 13441200n^3 - 3643800n^2 + 89860320n - 108217440)/720, n>=5. For any fixed value of k > 1, a(n) = n^(2k)/k! - 9n^(2k-2)/2/(k-2)! + 6n^(2k-3)/(k-2)! ...
%Y A172158 A061995, A061996, A061997, A061998
%K A172158 hard,nonn,new
%O A172158 1,5
%A A172158 Vaclav Kotesovec (kotesovec(AT)chello.cz), Jan 27 2010

%I A172157
%S A172157 1,1,3,1,8,5,1,15,3,7,1,24,21,16,9,1,35,2,1,5,11,
%T A172157 1,48,45,40,33,24,13,1,63,15,55,3,39,7,15,1,80,77,8,
%U A172157 65,56,5,32,17,1,99,6,91,21,3,4,51,9,19,1,120,117,112
%V A172157 -1,-1,-3,-1,-8,-5,-1,-15,-3,-7,-1,-24,-21,-16,-9,-1,-35,-2,-1,-5,-11,
%W A172157 -1,-48,-45,-40,-33,-24,-13,-1,-63,-15,-55,-3,-39,-7,-15,-1,-80,-77,-8,
%X A172157 -65,-56,-5,-32,-17,-1,-99,-6,-91,21,-3,4,-51,-9,-19,-1,-120,-117,-112
%N A172157 Triangle read by rows.Respectively rows before A005563 (Lyman), A061037 (Balmer), A061039 (Paschen), A061041, A061043, A061045, A061047, A061049, . .See A067998, A144477, A171709, submitted A171825=0, -7, -3, -15, -1, -15, -3, -7, A061041.Also A165795 (in array, -1, -3, 0, 5, 3, 21, 2, is the good omitted third row);1's not necessary. Half extended Rydberg-Ritz spectrum of hydrogen.
%F A172157 a(n)= -1, (mix -1, -n-th row of A120072) .
%K A172157 nonn,uned,new
%O A172157 1,3
%A A172157 Paul Curtz (bpcrtz(AT)free.fr), Jan 27 2010

%I A172156
%S A172156 7,17,37,157,317,1277,2557,20477,655357,5242877,671088637,2684354557,
%T A172156 5368709117,343597383677,23058430092136939517,23611832414348226068477,
%U A172156 48357032784585166988247037
%N A172156 Primes in the chain of repeated application of x->2*x+3, starting at x=2.
%K A172156 nonn,new
%O A172156 1,1
%A A172156 Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Jan 27 2010

%I A172153
%S A172153 0,4,8,10,13,16,20,24,25,29,32,34,40,44,45,48,50,52,56,58,61,64,65,68,
%T A172153 72,73,74,78,80,85,89,90,94,96,97,100,104,106,109,112,113,116,120,122,
%U A172153 125,130,136,140,142,144,145,148,152,153,156,157,160,164,168,169,170
%N A172153 Where records occur for number of partitions of n into two non-squares.
%C A172153 A172151(a(n))=A172152(n) and A172151(m)<A172152(n) for m<a(n).
%K A172153 nonn,new
%O A172153 1,2
%A A172153 Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Jan 26 2010

%I A172152
%S A172152 0,1,2,3,4,5,7,8,9,10,11,13,15,16,17,18,19,20,21,23,24,25,26,27,28,29,
%T A172152 30,31,33,35,36,37,38,39,40,42,43,44,45,46,47,49,50,51,53,56,58,59,60,
%U A172152 61,62,63,64,65,66,67,69,71,72,73,74,75,77,78,79,81,82,83,84,85,87,88
%N A172152 Records for number of partitions of n into two non-squares.
%C A172152 a(n)=A172151(A172153(n)) and A172151(m)<a(n) for m<A172153(n).
%K A172152 nonn,new
%O A172152 0,3
%A A172152 Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Jan 26 2010

%I A172151
%S A172151 0,0,0,0,1,1,1,1,2,2,3,2,3,4,4,4,5,5,5,5,7,6,7,7,8,9,9,8,9,10,10,10,11,
%T A172151 11,13,12,13,13,13,13,15,15,15,15,16,17,17,17,18,18,19,18,20,20,20,20,
%U A172151 21,21,23,22,23,24,24,24,25,26,25,25,27,26,27,27,28,29,30,29,30,30,31
%N A172151 Number of partitions of n into two non-squares.
%C A172151 A172152 and A172153 give record values and where they occur: a(A172153(n))=A172152(n) and a(m)<A172152(n) for m<A172153(n).
%H A172151 R. Zumkeller, <a href="A172151.txt">Table of n, a(n) for n = 0..10000</a>
%e A172151 a(8) = #{6+2, 5+3} = 2;
%e A172151 a(9) = #{7+2, 6+3} = 2;
%e A172151 a(10) = #{8+2, 7+3, 5+5} = 3;
%e A172151 a(11) = #{8+3, 6+5} = 2;
%e A172151 a(12) = #{10+2, 7+5, 6+6} = 3;
%e A172151 a(13) = #{11+2, 10+3, 8+5, 7+6} = 4.
%Y A172151 A000037, A004526, A087153.
%K A172151 nonn,new
%O A172151 0,9
%A A172151 Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Jan 26 2010

%I A172155
%S A172155 1,1,1,2,1,2,1,2,2,1,2,1,2,2,2,2,1,1,2,4,3,2,1,4,1,2,2,2,3,1,4,1,1,4,2,
%T A172155 4,2,1,2,1,6,2,2,2,1,3,1,4,2,4,2,1,2,6,2,2,6,1,2,2,2,2,2,3,2,2,2,2,4,1,
%U A172155 6,1,2,2,2,3,1,2,6,2,2,2,2,2,2,4,1,4,4,3,2,4,2,2,1,3,4,2,3,1,3,2,1,8,2
%N A172155 Omega(6*n-1)*Omega(6*n+1).
%e A172155 For n=1, Omega(6*n-1)*Omega(6*n+1)=Omega(5)*Omega(7)=1, so a(1)=1.
%p A172155 A172155 := proc(n) numtheory[bigomega](6*n-1)*numtheory[bigomega](6*n+1) ; end proc: seq(A172155(n),n=1..100) ; [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jan 30 2010]
%Y A172155 Cf. A145193.
%K A172155 nonn,new
%O A172155 1,4
%A A172155 Juri-Stepan Gerasimov (2stepan(AT)rambler.ru), Jan 26 2010
%E A172155 Corrected by R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jan 30 2010

%I A172154
%S A172154 1,2,3,6,7,13,16,23,27,29,33,34,44,49,54,62,68,71,72,78,83,89,92,98,99,
%T A172154 103,112,114,119,148,149,154,163,167,176,177,181,182,187,188,197,216,
%U A172154 218,222,232,236,237,244,252,254,257,266,274,279,288,301,302,313,328
%N A172154 Numbers n such that 24*n+-5 are both prime.
%e A172154 a(1)=1 because 24*1-5=19(prime) and 24*1+5=29(prime).
%Y A172154 Cf. A000027, A000040, A172147.
%K A172154 nonn,new
%O A172154 1,2
%A A172154 Juri-Stepan Gerasimov (2stepan(AT)rambler.ru), Jan 26 2010

%I A172148
%S A172148 1,2,2,4,4,8,8,14,14,28,28,56,56,112,112,192,192,384,384,768,768,1536,
%T A172148 1536,2688,2688
%N A172148 Number of subsets S of {1,2,3,...,n} with the property that if x is a member of S then at least one of x/2 and 2x is also a member of S
%C A172148 For the terms a(1) through a(25) is it seen that if n is odd then a(n)=a(n-1); also that if n is not a multiple of 4 then a(2n)=2a(2n-2). Does this behavior continue?
%Y A172148 Cf. A172020.
%K A172148 nonn,new
%O A172148 1,2
%A A172148 John W. Layman (layman(AT)math.vt.edu), Jan 26 2010

%I A172147
%S A172147 1,2,4,9,11,12,17,18,21,22,26,28,29,31,34,37,43,49,56,57,64,66,68,79,86,
%T A172147 88,104,114,117,119,121,133,138,144,148,152,166,171,172,182,183,192,199,
%U A172147 204,207,208,213,218,219,221,224,229,233
%N A172147 Numbers n such that 42*n+-5 are both prime.
%e A172147 a(1)=1 because 42*1-5=37(prime) and 42*1+5=47(prime).
%Y A172147 Cf. A000027, A000040.
%K A172147 nonn,new
%O A172147 1,2
%A A172147 Juri-Stepan Gerasimov (2stepan(AT)rambler.ru), Jan 26 2010

%I A172146
%S A172146 3,5,7,11,13,17,19,23,29,31,37,41,43,47,101,2531,4241,5393,94933,262469,
%T A172146 16797953,48989177,78371693633,2552470327703,4747732369319,
%U A172146 17832200896513,131621703955647041,4052555153019035587
%N A172146 Primes of form x^y + y^x + 1
%e A172146 a(1)=1^1+1^1+1=3 a(2)=1^3+3^1+1=5 a(3)=1^5+5^1+1=7 ... a(15)=2^6+6^2+1=101
%t A172146 a[n_] := Block[{}, For[l = {}; i = 1, i < n, i++, For[j = i, j < n, j++, x = i^j + j^i + 1; If[PrimeQ[x], l = Append[l, x]]]]; Print[Sort[Union[l]]]]; a[50]
%Y A172146 Cf. A094133, A172143
%K A172146 nonn,new
%O A172146 1,1
%A A172146 Cheng Zhang (cz1(AT)rice.edu), Jan 26 2010

%I A172142
%S A172142 2531,94933,48989177,19088056323407827075424486287615602692671561637,
%T A172142 10027860709531471276608129899567499096303854889970269316268113271,
%U A172142 88537996291958256446260440678593208943077817551131498658191653913030830300434060998128240895267
%N A172142 Primes of form p^q+q^p+1, where p, q are also primes.
%e A172142 a(1)=3^7+7^3+1=2531 a(2)=5^7+7^5+1=94933 a(3)=5^11+11^5+1=48989177 a(4)=3^97+97^3+1=19088056323407827075424486287615602692671561637 a(5)=23^47+47^23+1
%t A172142 a[n_] := Block[{}, For[l = {}; i = 1, i < n, i++, For[j = i, j < n, j++, p = Prime[i]; q = Prime[j]; x = p^q + q^p + 1; If[PrimeQ[x], l = Append[l, x]]]]; Print[Sort[Union[l]]]]; a[50]
%Y A172142 Cf. A118097
%K A172142 nonn,new
%O A172142 1,1
%A A172142 Cheng Zhang (cz1(AT)rice.edu), Jan 26 2010

%I A172141
%S A172141 0,6,28,96,240,518,980,1712,2784,4310,6380,9136,12688,17206,22820,29728,
%T A172141 38080,48102,59964,73920,90160,108966,130548,155216,183200,214838,
%U A172141 250380,290192,334544,383830,438340,498496,564608,637126
%N A172141 Number of ways to place 2 nonattacking nightriders on an n X n board
%C A172141 A nightrider is a fairy chess piece that can move (proportionate to how a knight moves) in any direction.
%D A172141 Christian Poisson, Echecs et mathematiques, Rex Multiplex 29/1990, p.829
%H A172141 V. Kotesovec, <a href="http://web.telecom.cz/vaclav.kotesovec/math.htm">Number of ways of placing non-attacking queens and kings on boards of various sizes</a>
%F A172141 Explicit formula (Christian Poisson, 1990): a(n) = n(3n^3 - 5n^2 + 9n - 4)/6 if n is even and a(n) = n(n - 1)(3n^2 - 2n + 7)/6 if n is odd.
%Y A172141 A036464, A172123, A172132, A172137
%K A172141 easy,nonn,new
%O A172141 1,2
%A A172141 Vaclav Kotesovec (kotesovec(AT)chello.cz), Jan 26 2010

%I A172140
%S A172140 0,0,126,2032,20502,160696,929880,4117520,15037036,47368960,132577826,
%T A172140 336828368,789558314,1729320120,3574328936,7027309888,13226773092,
%U A172140 23959787480,41954706558,71276149776,117848892710,190142197976
%N A172140 Number of ways to place 5 nonattacking zebras on an n X n board
%C A172140 Zebra is a (fairy chess) leaper [2,3]
%H A172140 V. Kotesovec, <a href="http://web.telecom.cz/vaclav.kotesovec/math.htm">Number of ways of placing non-attacking queens and kings on boards of various sizes</a>
%F A172140 Explicit formula (Vaclav Kotesovec, 26.1.2010): a(n) = (n^10 - 90n^8 + 400n^7 + 2915n^6 - 26880n^5 + 2430n^4 + 609920n^3 - 1517496n^2 - 4188480n + 16581120)/120, n >= 12. For any fixed value of k > 1, a(n) = n^(2k) /k! - 9n^(2k - 2) /2/(k - 2)! + 20n^(2k - 3) /(k - 2)! + ...
%Y A172140 A108792, A172129, A172136, A172137, A172138, A172139
%K A172140 hard,nonn,new
%O A172140 1,3
%A A172140 Vaclav Kotesovec (kotesovec(AT)chello.cz), Jan 26 2010

%I A172139
%S A172139 0,1,126,1168,7334,35749,137970,438984,1208246,2969389,6662480,13873100,
%T A172139 27144408,50389581,89424014,152638280,251834530,403250693,628798516,
%U A172139 957543164,1427453780,2087456085,2999819778,4242915176
%N A172139 Number of ways to place 4 nonattacking zebras on an n X n board
%C A172139 Zebra is a (fairy chess) leaper [2,3]
%H A172139 V. Kotesovec, <a href="http://web.telecom.cz/vaclav.kotesovec/math.htm">Number of ways of placing non-attacking queens and kings on boards of various sizes</a>
%F A172139 Explicit formula (Vaclav Kotesovec, 24.1.2010): a(n) = (n^8 - 54n^6 + 240n^5 + 827n^4 - 8592n^3 + 10362n^2 + 75600n - 204864)/24, n >= 9
%Y A172139 A061994, A172127, A172135, A172137, A172138
%K A172139 nonn,new
%O A172139 1,3
%A A172139 Vaclav Kotesovec (kotesovec(AT)chello.cz), Jan 26 2010

%I A172138
%S A172138 0,4,84,452,1772,5596,14888,34640,72712,140716,255036,437968,718980,
%T A172138 1136092,1737376,2582576,3744848,5312620,7391572,10106736,13604716,
%U A172138 18056028,23657560,30635152,39246296,49782956,62574508,77990800
%N A172138 Number of ways to place 3 nonattacking zebras on an n X n board
%C A172138 Zebra is a (fairy chess) leaper [2,3]
%H A172138 V. Kotesovec, <a href="http://web.telecom.cz/vaclav.kotesovec/math.htm">Number of ways of placing non-attacking queens and kings on boards of various sizes</a>
%F A172138 Explicit formula (Vaclav Kotesovec, 24.1.2010): a(n) = (n^6 - 27n^4 + 120n^3 + 74n^2 - 1608n + 2976)/6, n > =6
%Y A172138 A047659, A172124, A172134, A172137
%K A172138 nonn,new
%O A172138 1,2
%A A172138 Vaclav Kotesovec (kotesovec(AT)chello.cz), Jan 26 2010

%I A172137
%S A172137 0,6,36,112,276,582,1096,1896,3072,4726,6972,9936,13756,18582,24576,
%T A172137 31912,40776,51366,63892,78576,95652,115366,137976,163752,192976,225942,
%U A172137 262956,304336,350412,401526,458032,520296,588696,663622
%N A172137 Number of ways to place 2 nonattacking zebras on an n X n board
%C A172137 Zebra is a (fairy chess) leaper [2,3]
%D A172137 Christian Poisson, Echecs et mathematiques, Rex Multiplex 29/1990, p.829
%H A172137 V. Kotesovec, <a href="http://web.telecom.cz/vaclav.kotesovec/math.htm">Number of ways of placing non-attacking queens and kings on boards of various sizes</a>
%F A172137 Explicit formula (Christian Poisson, 1990): a(n) = (n^4 - 9n^2 + 40n - 48)/2, n >= 2
%Y A172137 A036464, A172123, A172132
%K A172137 easy,nonn,new
%O A172137 1,2
%A A172137 Vaclav Kotesovec (kotesovec(AT)chello.cz), Jan 26 2010

%I A172136
%S A172136 0,0,2,340,9386,97580,649476,3184708,12472084,41199404,119171110,
%T A172136 309957412,739123094,1639655452,3422020324,6778432292,12833460256,
%U A172136 23356032940,41051290730,69954580804
%N A172136 Number of ways to place 5 nonattacking knights on an n X n board
%C A172136 For any fixed value of k>1, a(n) = n^(2k) /k! - 9n^(2k - 2) /2/(k - 2)! + 12n^(2k - 3) /(k - 2)! + ...
%H A172136 V. Kotesovec, <a href="http://web.telecom.cz/vaclav.kotesovec/math.htm">Number of ways of placing non-attacking queens and kings on boards of various sizes</a>
%F A172136 Explicit formula (Vaclav Kotesovec, 25.1.2010): a(n) = (n^10 - 90n^8 + 240n^7 + 3235n^6 - 16320n^5 - 40530n^4 + 396480n^3 - 231656n^2 - 3359520n + 6509280)/120, n >= 8
%Y A172136 A108792, A172129, A172132, A172134, A172135
%K A172136 hard,nonn,new
%O A172136 1,3
%A A172136 Vaclav Kotesovec (kotesovec(AT)chello.cz), Jan 26 2010

%I A172135
%S A172135 0,1,18,412,4436,26133,111066,376560,1080942,2732909,6253408,13204356,
%T A172135 26100160,48819677,87137934,149398608,247349946,397168485,620696612,
%U A172135 946921684,1413726108,2069939461,2977725410,4215337872
%N A172135 Number of ways to place 4 nonattacking knights on an n X n board
%D A172135 E. Bonsdorff, K. Fabel, O. Riihimaa, Schach und Zahl, 1966, p. 51-63
%H A172135 V. Kotesovec, <a href="http://web.telecom.cz/vaclav.kotesovec/math.htm">Number of ways of placing non-attacking queens and kings on boards of various sizes</a>
%F A172135 Explicit formula (Karl Fabel, 1966): a(n) = (n^8 - 54n^6 + 144n^5 + 1019n^4 - 5232n^3 - 2022n^2 + 51120n - 77184)/24, n >= 6
%Y A172135 A061994, A172127, A172132, A172134
%K A172135 nonn,new
%O A172135 1,3
%A A172135 Vaclav Kotesovec (kotesovec(AT)chello.cz), Jan 26 2010

%I A172134
%S A172134 0,4,36,276,1360,4752,13340,32084,68796,135040,247152,427380,705144,
%T A172134 1118416,1715220,2555252,3711620,5272704,7344136,10050900,13539552,
%U A172134 17980560,23570764,30535956,39133580,49655552,62431200,77830324
%N A172134 Number of ways to place 3 nonattacking knights on an n X n board
%D A172134 E. Bonsdorff, K. Fabel, O. Riihimaa, Schach und Zahl, 1966, p. 51-63
%H A172134 V. Kotesovec, <a href="http://web.telecom.cz/vaclav.kotesovec/math.htm">Number of ways of placing non-attacking queens and kings on boards of various sizes</a>
%F A172134 Explicit formula (Karl Fabel, 1966): a(n) = (n - 2)(n + 5)(n^4 - 3n^3 - 8n^2 + 66n - 108)/6, n >= 4
%Y A172134 A047659, A172124, A172132
%K A172134 nonn,new
%O A172134 1,2
%A A172134 Vaclav Kotesovec (kotesovec(AT)chello.cz), Jan 26 2010

%I A172133
%S A172133 1,1,2,1,2,1,5,2,3,1,10,1,3,2,2,1,2,1,5,2,2,1,18,2,3,6,2,1,9,1,12,7,2,3,
%T A172133 8,1,74,2,5,1,2,1,14,3,6,1,9,18,3,2,3,1,3,2,19,3,2,1,2,1,3,2,115,5,6,1,
%U A172133 16,5,2,1,2,1,5,3,2,3,2,1,6,2,6,1,3,3,10,3,5,1,2,2,3,24,8,3,13,1,13,3,2
%N A172133 a(n) = the smallest k such that n(n+1)(n+2)...(n+k-1)+1 is a prime
%e A172133 a(2)=1 because 2+1=3 is a prime. a(3)=2 because 3+1=4 is not a prime, but 3*4+1=13 is a prime. a(9)=3 because neither 9+1=10 nor 9*10+1=91=13*7 is a prime, but 9*10*11+1=991 is.
%p A172133 a[n_] := Block[{}, For[s = n; k = 1, ! PrimeQ[s + 1], s *= n + k; k++ ]; k] Table[a[n], {n, 1, 100}]
%Y A172133 A087564
%K A172133 nonn,new
%O A172133 1,3
%A A172133 Cheng Zhang (cz1(AT)rice.edu), Jan 26 2010

%I A172132
%S A172132 0,6,28,96,252,550,1056,1848,3016,4662,6900,9856,13668,18486,24472,
%T A172132 31800,40656,51238,63756,78432,95500,115206,137808,163576,192792,225750,
%U A172132 262756,304128,350196,401302,457800,520056,588448,663366
%N A172132 Number of ways to place 2 nonattacking knights on an n X n board
%D A172132 E. Bonsdorff, K. Fabel, O. Riihimaa, Schach und Zahl, 1966, p. 51-63
%H A172132 V. Kotesovec, <a href="http://web.telecom.cz/vaclav.kotesovec/math.htm">Number of ways of placing non-attacking queens and kings on boards of various sizes</a>
%F A172132 a(n) = (n - 1)(n + 4)(n^2 - 3n + 4)/2
%Y A172132 A036464, A172123
%K A172132 easy,nonn,new
%O A172132 1,2
%A A172132 Vaclav Kotesovec (kotesovec(AT)chello.cz), Jan 26 2010

%I A172130
%S A172130 3,2,1,1,2,1,3,1,2,2,2,1,2,1,2,2,3,1,2,1,2,3,3,1,17,2,4,2,6,1,2,1,10,3,
%T A172130 2,9,3,1,2,3,2,1,4,1,2,2,3,1,4,2,8,7,2,1,2,2,3,3,2,1,11,1,3,2,2,2,5,1,3,
%U A172130 2,11,1,4,1,3,2,7,7,5,1,4,38,2,1,2,2,7,9,2,1,4,4,2,2,4,2,3,1,14,2,2,1,2
%N A172130 a(n) = the smallest k such that n(n+1)(n+2)...(n+k-1)-1 is a prime
%C A172130 a(534) = 624 is particularly large.
%t A172130 a[n_] := Block[{}, For[s = n; k = 1, ! PrimeQ[s - 1], s *= n + k; k++ ]; k] Table[a[n], {n, 1, 300}]
%Y A172130 Cf. A087564, A172126
%K A172130 nonn,new
%O A172130 1,1
%A A172130 Cheng Zhang (cz1(AT)rice.edu), Jan 26 2010

%I A172129
%S A172129 0,0,0,112,3368,39680,282248,1444928,5865552,20014112,59673360,
%T A172129 159698416,391202680,890095584,1902427800,3853570560,7450556064,
%U A172129 13829016768,24759442464,42930138864,72328779720,118747638592
%N A172129 Number of ways to place 5 nonattacking bishops on an n X n board
%C A172129 For any fixed value of k>1, a(n) = n^(2k) /k! - 2n^(2k - 1) /3/(k - 2)! + ...
%H A172129 V. Kotesovec, <a href="http://web.telecom.cz/vaclav.kotesovec/math.htm">Number of ways of placing non-attacking queens and kings on boards of various sizes</a>
%F A172129 Explicit formula (Vaclav Kotesovec, 26.1.2010): a(n) = n(n - 2)(3n^8 - 34n^7 + 177n^6 - 590n^5 + 1435n^4 - 2592n^3 + 3326n^2 - 2844n + 1344)/360 if n is even and a(n) = (n - 1)(n - 2)(n - 3)(3n^7 - 22n^6 + 80n^5 - 204n^4 + 379n^3 - 464n^2 + 378n - 270)/360 if n is odd.
%Y A172129 A108792, A172123, A172124, A172127
%K A172129 hard,nonn,new
%O A172129 1,4
%A A172129 Vaclav Kotesovec (kotesovec(AT)chello.cz), Jan 26 2010

%I A171139
%S A171139 2,3,17,19,23,29,47,67,73,97,113,151,163,173,223,227,229,239,251,257,
%T A171139 263,311,331,359,367,373,389,409,419,433,439,457,463,491,503,523,541,
%U A171139 563,569,607,701
%N A171139 Primes p such that p and 7*p^2+7*p-1 are both prime.
%Y A171139 Cf. A171138
%K A171139 nonn,new
%O A171139 1,1
%A A171139 Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Jan 26 2010

%I A172127
%S A172127 0,0,8,260,2728,16428,70792,242856,706048,1809464,4199064,8992684,
%T A172127 18024072,34170724,61784632,107243472,179645376,291667440,460615272,
%U A172127 709686228,1069477928,1579767068,2291594536,3269684088,4595235136
%N A172127 Number of ways to place 4 nonattacking bishops on an n X n board
%D A172127 E. Bonsdorff, K. Fabel, O. Riihimaa, Schach und Zahl, 1966, p. 51-63
%H A172127 V. Kotesovec, <a href="http://web.telecom.cz/vaclav.kotesovec/math.htm">Number of ways of placing non-attacking queens and kings on boards of various sizes</a>
%F A172127 Explicit formula (Karl Fabel, 1966): a(n) = n(n - 2)(15n^6 - 90n^5 + 260n^4 - 524n^3 + 727n^2 - 646n + 348)/360 if n is even and a(n) = (n - 1)(n - 2)(15n^6 - 75n^5 + 185n^4 - 339n^3 + 388n^2 - 258n + 180)/360 if n is odd.
%Y A172127 A061994, A172123, A172124
%K A172127 nonn,new
%O A172127 1,3
%A A172127 Vaclav Kotesovec (kotesovec(AT)chello.cz), Jan 26 2010

%I A172125
%S A172125 5,5,2,3,29,5,503,7,89,109,131,11,181,13,239,271,5813,17,379,19,461,
%T A172125 12143,13799,23,53916693990528892723199999,701,657719,811,968330879,29,
%U A172125 991,31,5339572260422399,42839,1259,257256702743039,54833,37,1559,68879
%N A172125 a(n) = smallest prime of the form n*(n+1)*(n+2)...(n+k-1) - 1, or zero if no such a prime exists.
%C A172125 A conjecture is that no term is zero.
%e A172125 a(1)=5 because 1-1=0 and 1*2-1=1 are not primes, but 1*2*3-1=5 is.
%t A172125 a[n_] := Block[{}, For[s = n; k = 1, ! PrimeQ[s - 1], s *= n + k; k++ ]; s - 1] Table[a[n], {n, 1, 100}]
%Y A172125 A087564
%K A172125 nonn,new
%O A172125 1,1
%A A172125 Cheng Zhang (cz1(AT)rice.edu), Jan 26 2010

%I A172124
%S A172124 0,0,26,232,1124,3896,10894,26192,56296,110960,204130,355000,589196,
%T A172124 940072,1450134,2172576,3172944,4530912,6342186,8720520,11799860,
%U A172124 15736600,20711966,26934512,34642744,44107856,55636594,69574232
%N A172124 Number of ways to place 3 nonattacking bishops on an n X n board
%D A172124 E. Bonsdorff, K. Fabel, O. Riihimaa, Schach und Zahl, 1966, p. 51-63
%H A172124 V. Kotesovec, <a href="http://web.telecom.cz/vaclav.kotesovec/math.htm">Number of ways of placing non-attacking queens and kings on boards of various sizes</a>
%F A172124 Explicit formula (Karl Fabel, 1966): a(n) = n(n - 2)(2n^4 - 4n^3 + 7n^2 - 6n + 4)/12 if n is even and a(n) = (n - 1)(2n^5 - 6n^4 + 9n^3 - 11n^2 + 5n - 3)/12 if n is odd.
%Y A172124 A047659, A172123
%K A172124 nonn,new
%O A172124 1,3
%A A172124 Vaclav Kotesovec (kotesovec(AT)chello.cz), Jan 26 2010

%I A171138
%S A171138 41,83,2141,2659,3863,6089,15791,31891,37813,66541,90173,160663,187123,
%T A171138 210713,349663,362291,368689,401519,442763,464141,486023,679223,769243,
%U A171138 904679,945391,976513,1061969,1173829,1231859,1315453,1352119,1465141
%N A171138 Primes of the form 7*p^2+7*p-1 (with p=prime)
%Y A171138 Cf. A171139
%K A171138 nonn,new
%O A171138 1,1
%A A171138 Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Jan 26 2010

%I A172123
%S A172123 0,4,26,92,240,520,994,1736,2832,4380,6490,9284,12896,17472,23170,30160,
%T A172123 38624,48756,60762,74860,91280,110264,132066,156952,185200,217100,
%U A172123 252954,293076,337792,387440,442370,502944,569536,642532
%N A172123 Number of ways to place 2 nonattacking bishops on an n X n board
%D A172123 E. Bonsdorff, K. Fabel, O. Riihimaa, Schach und Zahl, 1966, p. 51-63
%H A172123 V. Kotesovec, <a href="http://web.telecom.cz/vaclav.kotesovec/math.htm">Number of ways of placing non-attacking queens and kings on boards of various sizes</a>
%F A172123 a(n) = n(n - 1)(3n^2 - n + 2)/6
%Y A172123 A036464
%K A172123 nonn,new
%O A172123 1,2
%A A172123 Vaclav Kotesovec (kotesovec(AT)chello.cz), Jan 26 2010

%I A172122
%S A172122 2,5,17,29,59,197,227,257,317,359,467,509,569,587,797,929,1097,1187,
%T A172122 1259,1307,1439,1637,1697,1847,1877,1997,2027,2069,2099,2237,2297,2399,
%U A172122 2459,2477,2657,2687,2729,2939,3167,3359,3407
%N A172122 Primes p such that p and 7*p^2+7*p+1 are both prime.
%K A172122 nonn,new
%O A172122 1,1
%A A172122 Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Jan 26 2010

%I A172119
%S A172119 1,1,1,1,2,1,1,3,2,1,1,4,4,2,1,1,5,7,4,2,1,1,6,12,8,4,2,1,1,7,20,15,8,4,
%T A172119 2,1,1,8,33,28,16,8,4,2,1,1,9,54,52,31,16,8,4,2,1,1,10,88,96,60,32,16,8,
%U A172119 4,2,1,1,11,143,177,116,63,32,16,8,4,2,1,1,12,232,326,224,124,64,32,16
%N A172119 Sum the k preceeding elements in the same column and add 1 every time.
%C A172119 Columns are related to Fibonacci n-step numbers. Are there closed forms for the sequences in the columns?
%C A172119 We denote by a(n,k) the number which is in the (n+1)-th row and (k+1)-th-column. With help of the definition, we have also the recurrence relation: a(n+k+1,k)=2*a(n+k,k)-a(n,k). We see on the main diagonal the numbers 1,2,4, 8, ..., which is clear from the formula for the general term d(n)=2^n. [From Richard Choulet (richardchoulet(AT)yahoo.fr), Jan 31 2010]
%H A172119 Noe, Tony; Piezas, Tito III; and Weisstein, Eric W., <a href="http://mathworld.wolfram.com/Fibonaccin-StepNumber.html">Fibonacci n-Step Number</a>
%H A172119 Wikipedia, <a href="http://en.wikipedia.org/wiki/Fibonacci_number">Fibonacci number</a>
%F A172119 The general term in the n-th row and k-th column is given by: a(n,k)=sum((-1)^j binomial(n-k*j,n-(k+1)*j)*2^(n-(k+1)*j),j=0..floor(n/(k+1))).For example: a(5,3)=binomial(5,5)*2^5-binomial(2,1)*2^1= 28. The generating fonction of the (k+1)-th column satisfies: psi(k)(z)=1/(1-2*z+z^(k+1)) (for k=0 we have the known result psi(0)(z)=1/(1-z)). [From Richard Choulet (richardchoulet(AT)yahoo.fr), Jan 31 2010]
%e A172119 Triangle begins:
%e A172119 n\k|....0....1....2....3....4....5....6....7....8....9...10
%e A172119 ---|-------------------------------------------------------
%e A172119 0..|....1
%e A172119 1..|....1....1
%e A172119 2..|....1....2....1
%e A172119 3..|....1....3....2....1
%e A172119 4..|....1....4....4....2....1
%e A172119 5..|....1....5....7....4....2....1
%e A172119 6..|....1....6...12....8....4....2....1
%e A172119 7..|....1....7...20...15....8....4....2....1
%e A172119 8..|....1....8...33...28...16....8....4....2....1
%e A172119 9..|....1....9...54...52...31...16....8....4....2....1
%e A172119 10.|....1...10...88...96...60...32...16....8....4....2....1
%p A172119 for k from 0 to 20 do for n from 0 to 20 do b(n):=sum((-1)^j*binomial(n-k*j,n-(k+1)*j)*2^(n-(k+1)*j),j=0..floor(n/(k+1))):od: seq(b(n),n=0..20):od; [From Richard Choulet (richardchoulet(AT)yahoo.fr), Jan 31 2010]
%Y A172119 Cf. k=0 A000012, k=1 A000027, k=2 A000071, k=3 A008937.
%Y A172119 Cf. (1-((-1)^T(n, k)))/2 = A051731, see formula by Hieronymus Fischer in A022003.
%K A172119 nonn,tabl,new
%O A172119 0,5
%A A172119 Mats Granvik (mats.granvik(AT)abo.fi), Jan 26 2010

%I A172120
%S A172120 7,11,13,15,19,21,25,35,40,47,61,63,73,79,95,97,107,115,121,133,143,145,
%T A172120 149,151,156,166,167,169,181,184,187,191,203,205,207,211,215,221,223,
%U A172120 227,235,241,255,259,271,273,293,295,301,302,323,329,331,333,355,364
%N A172120 a(n) = numbers n for which maxima of function Log[n]/Log[N[a,n-a,n]] occured for two or more different values of a, (a < n-a , function N[a,n-a,n] is product of square free part of a*(n-a)*n and GCD[a,n-a,n]=1)
%C A172120 This sequence is related to ABC conjencture.
%e A172120 a(1)=7 because maxima of Log[7]/Log[N[a,7-a,7]] occured for two different values a=1 and a=3 (Log[c]/Log[N[a,b,c]] is equal in both cases Log[7]/Log[42]).
%t A172120 cc = {}; Do[k = x; w = Floor[(k - 1)/2]; logmax = 0; nmax = 0; nmax1 = 0; radmax = 0; logequal = 0; Do[If[(GCD[n, k] == 1) && (GCD[n, k - n] == 1) && (GCD[k, k - n] == 1), m = FactorInteger[k n (k - n)]; rad = 1; Do[rad = rad m[[s]][[1]], {s, 1, Length[m]}]; log = Log[k]/Log[rad]; If[log == logmax, logequal = log; nmax1 = n];If[log > logmax, nmax = n; logmax = log]], {n, 1, w}]; If[logequal == logmax, AppendTo[cc, k]], {x, 3, 100}]; cc (*Artur Jasinski*)
%Y A172120 A147638, A147306, A147639, A147640, A172121
%K A172120 nonn,new
%O A172120 3,1
%A A172120 Artur Jasinski (grafix(AT)csl.pl), Jan 26 2010

%I A172121
%S A172121 3,4,5,6,8,9,10,12,14,16,17,18,20,22,23,24,26,27,28,29,30,31,32,33,34,
%T A172121 36,37,38,39,41,42,43,44,45,46,48,49,50,51,52,53,54,55,56,57,58,59,60,
%U A172121 62,64,65,66,67,68,69,70,71,72,74,75,76,77,78,80,81,82,83,84,85,86,87
%N A172121 a(n) = numbers n for which maximum of function Log[n]/Log[N[a,n-a,n]] occured only for single value of a, (a < n-a , function N[a,n-a,n] is product of square free part of a*(n-a)*n and GCD[a,n-a,n]=1)
%C A172121 This sequence is related to ABC conjencture. Complement sequence to A172120 .
%Y A172121 A147638, A147306, A147639, A147640, A172120
%K A172121 nonn,new
%O A172121 3,1
%A A172121 Artur Jasinski (grafix(AT)csl.pl), Jan 26 2010

%I A172118
%S A172118 0,1,45,234,730,1755,3591,6580,11124,17685,26785,39006,54990,75439,
%T A172118 101115,132840,171496,218025,273429,338770,415170,503811,605935,722844,
%U A172118 855900,1006525,1176201,1366470,1578934,1815255,2077155,2366416,2684880
%N A172118 a(n)=(5*n^4+4*n^3-4*n^2-3*n)/2
%C A172118 Numbers: (0,1,2,3,4,5,6,7,8,9,10,..,) A172117: (0,1,23,86,21, .., ) 45 is in the sequence because 45=23*2-(1+0); 234=86*3-(23+1+0); 730=210*4-(86+23+1+0)
%F A172118 a(n)=n*(n+1)*(30*n^2-6*n-18)/12
%e A172118 For n=0, a(0)=0; n=1, a(1)=1; n=2, a(2)=45; n=3, a(3)=234
%Y A172118 Cf. A172117
%K A172118 nonn,new
%O A172118 0,3
%A A172118 Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Jan 26 2010

%I A172117
%S A172117 0,1,23,86,210,415,721,1148,1716,2445,3355,4466,5798,7371,9205,11320,
%T A172117 13736,16473,19551,22990,26810,31031,35673,40756,46300,52325,58851,
%U A172117 65898,73486,81635,90365,99696,109648,120241,131495,143430,156066
%N A172117 a(n)=(20*n^3+3*n^2-17*n)/6
%C A172117 Generated by formula: n*(n+1)*[2*d*n-(2*d-3)]/6, [with d=10]
%F A172117 a(n)=n*(n+1)*(20*n-17)/6
%e A172117 For n=0, a(0)=0; n=1, a(1)=1; n=2, a(2)=23; n=3, a(3)=86;
%K A172117 nonn,new
%O A172117 0,3
%A A172117 Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Jan 26 2010

%I A172116
%S A172116 103,4985,641687,326552783
%N A172116 Arises in lower bounds on the lengths of double-base representations.
%C A172116 Dimitrov: A double-base representation of an integer n is an expression n = n_1 + ... + n_r, where the n_i are (positive or negative) integers that are divisible by no primes other than 2 or 3; the length of the representation is the number r of terms. It is known that there is a constant a > 0 such that every integer n has a double-base representation of length at most a log n / log log n. We show that there is a constant c > 0 such that there are infinitely many integers n whose shortest double-base representations have length greater than c log n / (log log n log log log n). Our methods allow us to find the smallest positive integers with no double-base representations of several lengths.
%C A172116 In particular, we show that 103 is the smallest positive integer with no double-base representation of length 2, that 4985 is the smallest positive integer with no double-base representation of length 3, that 641687 is the smallest positive integer with no double-base representation of length 4, and that 326552783 is the smallest positive integer with no double-base representation of length 5.
%D A172116 Vassil Dimitrov, Laurent Imbert, and Pradeep Kumar Mishra, Efficient and secure elliptic curve point multiplication using double-base chains, Advances in cryptology, ASIACRYPT 2005, Lecture Notes in Comput. Sci., vol. 3788, Springer, Berlin, 2005, pp. 59-78.
%D A172116 Vassil Dimitrov, Laurent Imbert, and Pradeep K. Mishra, The double-base number system and its application to elliptic curve cryptography, Math. Comp. 77 (2008), no. 262, 1075-1104.
%D A172116 Pradeep Kumar Mishra and Vassil Dimitrov, A combinatorial interpretation of double base number system and some consequences, Adv. Math. Commun. 2 (2008), no. 2, 159-173.
%H A172116 Vassil S. Dimitrov, Everett W. Howe, <a href="http://arxiv.org/abs/1001.4133">Lower bounds on the lengths of double-base representations</a>, January 23, 2010.
%K A172116 hard,nonn,uned,new
%O A172116 2,1
%A A172116 Jonathan Vos Post (jvospost3(AT)gmail.com), Jan 26 2010

%I A171792
%S A171792 1,1,2,7,34,214,1652,15121,160110,1925442,25924260,386354366,6314171932,
%T A171792 112286067892,2158562109096,44605949528355,986049177712850,
%U A171792 23218586050641090,580198948211652348,15334750335623526670
%N A171792 G.f. satisfies: A(x) = [x + A(x+x^2)]/2 with A(0)=0.
%F A171792 G.f.: A(x) = Sum_{n>=0} G_{n}(x)/2^(n+1) where G_{n}(x) is the n-th iteration of (x+x^2) defined by G_{n}(x) = G_{n-1}(x+x^2) with G_0(x)=x.
%F A171792 a(k) = Sum_{n>=0} A122888(n,k)/2^(n+1).
%F A171792 a(k) is odd iff k is a power of 2: a(2^n) == 1 (mod 2) for n>=0.
%e A171792 G.f.: A(x) = x + x^2 + 2*x^3 + 7*x^4 + 34*x^5 + 214*x^6 +...
%e A171792 A(x+x^2) = x + 2*x^2 + 4*x^3 + 14*x^4 + 68*x^5 + 428*x^6 +...
%o A171792 (PARI) {a(n)=local(A=x+x^2);for(i=1,n*(n+1)/2,A=(x+subst(A,x,x+x^2+x*O(x^n)))/2);ceil(polcoeff(A,n))}
%o A171792 (PARI) {a(n)=if(n==1,1,polcoeff(sum(m=1,n-1,a(m)*(x+x^2+x*O(x^n))^m),n))} [From Paul D. Hanna (pauldhanna(AT)juno.com), Jan 30 2010]
%Y A171792 Cf. A122888.
%K A171792 nonn,new
%O A171792 1,3
%A A171792 Paul D. Hanna (pauldhanna(AT)juno.com), Jan 25 2010

%I A172115
%S A172115 3,7,12,19,30,43,60,83,112,155,202,285,416,553,912,1343,1776,2225,2734,
%T A172115 3303,3874,6845,11568,16955,26266,35943,50374,75935,106692,142691,
%U A172115 180202,231035,312874,417785,547806,695897,897004,1294383,1728164
%N A172115 Partial sums of A001605 indices of prime Fibonacci numbers.
%C A172115 Primes in this sequence include a(1) = 3, a(2) = 7, a(4) = 19, a(6) = 43, a(8) = 83. Because the underlying sequence is hard, so is this, and so it is hard to determine if there is ever another such prime sum of indices of prime Fibonacci numbers.
%e A172115 a(20) = 3 + 4 + 5 + 7 + 11 + 13 + 17 + 23 + 29 + 43 + 47 + 83 + 131 + 137 + 359 + 431 + 433 + 449 + 509 + 569 = 3303.
%Y A172115 Cf. A001605, A001578, A005478, A086597, A080345.
%K A172115 hard,nonn,new
%O A172115 1,1
%A A172115 Jonathan Vos Post (jvospost3(AT)gmail.com), Jan 25 2010

%I A172114
%S A172114 2,5,10,17,40,759,5798,39922599,518924198,87697215397,
%T A172114 10888869450418352248465215398,265263748681641476988556945215397,
%U A172114 263396100682375171644206569105215396
%N A172114 Partial sums of factorial primes A088054.
%C A172114 The primes in this sequence begin 2, 5, 17; where 5 is itself a factorial prime 3!-1. What is the next prime in the sequence?
%F A172114 a(n) = SUM[i=1..n] A088054(i) = SUM[i=1..n] {primes which are within 1 of a factorial number}.
%Y A172114 Cf. A000040, A000142, A002981, A002982.
%K A172114 more,nonn,new
%O A172114 1,1
%A A172114 Jonathan Vos Post (jvospost3(AT)gmail.com), Jan 25 2010

%I A172111
%S A172111 0,0,0,8,48,368,3408,36848,454608,6294128,96556368,1624775408,
%T A172111 29744591568,588384837488,12503968334928,284065406275568,
%U A172111 6869235761650128,176150548586638448,4774198652678411088
%N A172111 T_4(n) gives the number of ordered partitions of {1,1,1,1,2,3,...,n-3}
%F A172111 For n>=4, T_4(n)=Sum{m=1...n}Sum_{l=0..m}binomial(m,l)*binomial(l+3,4)*(-1)^(m-l)*l^(n-4)
%t A172111 f[r_, n_] := Sum[Sum[Binomial[m, l] Binomial[l + r - 1, r] (-1)^(m - l) l^(n - r), {l, 1, m}], {m, 1, n}]; Table[f[4, n], {n, 4, 21}]
%Y A172111 This gives the row sums of A172108.
%K A172111 nonn,new
%O A172111 1,4
%A A172111 Martin Griffiths (martin.griffiths(AT)manchester.ac.uk), Jan 25 2010

%I A172110
%S A172110 0,0,4,20,132,1076,10404,116180,1469892,20766836,323924964,5527326740,
%T A172110 102396386052,2046350191796,43876822764324,1004631156809300,
%U A172110 24463049576172612,631213045618035956,17203155473859536484
%N A172110 T_3(n) gives the number of ordered partitions of {1,1,1,2,3,...,n-2}
%F A172110 For n>=3, T_3(n)=Sum{m=1...n}Sum_{l=0..m}binomial(m,l)*binomial(l+2,3)*(-1)^(m-l)*l^(n-3)
%t A172110 f[r_, n_] := Sum[Sum[Binomial[m, l] Binomial[l + r - 1, r] (-1)^(m - l) l^(n - r), {l, 1, m}], {m, 1, n}]; Table[f[3, n], {n, 3, 21}]
%Y A172110 This gives the row sums of A172107.
%K A172110 nonn,new
%O A172110 1,3
%A A172110 Martin Griffiths (martin.griffiths(AT)manchester.ac.uk), Jan 25 2010

%I A172109
%S A172109 0,2,8,44,308,2612,25988,296564,3816548,54667412,862440068,14857100084,
%T A172109 277474957988,5584100659412,120462266974148,2772968936479604,
%U A172109 67843210855558628,1757952715142990612,48093560991292628228
%N A172109 T_2(n) gives the number of ordered partitions of {1,1,2,3,...,n-1}
%F A172109 For n>=2, T_2(n)=Sum{m=1...n}Sum_{l=0..m}binomial(m,l)*binomial(l+1,2)*(-1)^(m-l)*l^(n-2)
%t A172109 f[r_, n_] := Sum[Sum[Binomial[m, l] Binomial[l + r - 1, r] (-1)^(m - l) l^(n - r), {l, 1, m}], {m, 1, n}]; Table[f[2, n], {n, 2, 21}]
%Y A172109 This gives the row sums of A172106.
%Y A172109 Cf. A005649. [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jan 28 2010]
%K A172109 nonn,new
%O A172109 1,2
%A A172109 Martin Griffiths (martin.griffiths(AT)manchester.ac.uk), Jan 25 2010

%I A172113
%S A172113 3,10,23,42,73,110,153,214,281,354,433,530,633,742,869,1008,1159,1316,
%T A172113 1479,1660,1853,2052,2263,2486,2715,2956,3227,3504,3787,4094,4407,4738,
%U A172113 5075,5424,5791,6164,6543,6940,7349,7770,8203,8642,9099,9562,10049
%N A172113 Partial sums of Cuban primes A007645.
%C A172113 The primes in this sequence begin: a(1) = 3, a(3) = 23, a(5) = 73, a(9) = 281, a(11) = 433. Of these, the subset of cuban primes which are partial sums of cuban primes begins: 3, 73, 433.
%F A172113 a(n) = SUM[i=1..n] A007645(i) = SUM[i=1..n] {primes of the form x^2 + xy + y^2} = SUM[i=1..n] {primes of form x^2 + 3*y^2} = SUM[i=1..n] {primes == 0 or 1 mod 3}.
%e A172113 a(30) = 3 + 7 + 13 + 19 + 31 + 37 + 43 + 61 + 67 + 73 + 79 + 97 + 103 + 109 + 127 + 139 + 151 + 157 + 163 + 181 + 193 + 199 + 211 + 223 + 229 + 241 + 271 + 277 + 283 + 307 = 4094.
%Y A172113 Cf. A000040, A007645.
%K A172113 easy,nonn,new
%O A172113 1,1
%A A172113 Jonathan Vos Post (jvospost3(AT)gmail.com), Jan 25 2010
%E A172113 a(5) corrected and more terms appended by R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Feb 07 2010

%I A172112
%S A172112 3,10,23,42,79,122,189,268,365,468,577,704,867,1060,1283,1512,1789,2096,
%T A172112 2409,2758,3137,3534,3973,4430,4893,5380,5879,6492,7135,7808,8547,9304,
%U A172112 10073,10896,11749,12608,13485,14368,15275,16212,17179,18188,19275
%N A172112 Partial sums of A023200.
%C A172112 Primes in the partial sum begin: a(1) = 3, a(3) = 23, a(5) = 79, a(11) = 577, a(15) = 1283, a(17) = 1789, a(21) = 3137, a(27) = 5879. Of these, the smaller members of cousin prime pairs which appear among the partial sums of smaller member p of cousin prime pairs begin: 3, 79; which are the next in this subset?
%F A172112 a(n) = SUM[i=i..n] A023200(i) = SUM[i=i..n] {Primes p such that p and p + 4 are both primes}.
%e A172112 a(30) = 3 + 7 + 13 + 19 + 37 + 43 + 67 + 79 + 97 + 103 + 109 + 127 + 163 + 193 + 223 + 229 + 277 + 307 + 313 + 349 + 379 + 397 + 439 + 457 + 463 + 487 + 499 + 613 + 643 + 673 = 7808.
%Y A172112 Cf. A000040, A023200, A046132.
%K A172112 easy,nonn,new
%O A172112 1,1
%A A172112 Jonathan Vos Post (jvospost3(AT)gmail.com), Jan 25 2010
%E A172112 More terms from Max Alekseyev (maxale(AT)gmail.com), Jan 31 2010

%I A172108
%S A172108 0,0,0,0,0,0,1,3,3,1,1,8,18,16,5,1,18,78,136,105,30,1,38,288,856,1205,
%T A172108 810,210,1,78,978,4576,10305,12090,7140,1680,1,158,3168,22216,74405,
%U A172108 134370,134610,70560,15120,1,318,9978,101536,483105,1252650,1882860
%N A172108 The triangle T_4(n,m), where T_4(n,m) is the number of surjective multi-valued functions from {1,1,1,1,2,3,...,n-3} to {1,2,3,...,m} by rows (n>=1,1<=m<=n)
%C A172108 T_4(1,m)=T_4(2,m)=T_4(3,m)=0 by definition. T_4(n,m) also gives the number of ordered partitions of {1,1,1,1,2,3,...,n-3} into exactly m parts.
%F A172108 For n>=4, T_4(n,m)=Sum_{l=0..m}binomial(m,l)*binomial(l+3,4)*(-1)^(m-l)*l^(n-4)
%t A172108 f[r_, n_, m_] := Sum[Binomial[m, l] Binomial[l + r - 1, r] (-1)^(m - l) l^(n - r), {l, 1, m}]; For[n = 4, n <= 10, n++, Print[Table[f[4, n, m], {m, 1, n}]]]
%Y A172108 This is related to A019538, A172106 and A172107. The row sum is A172111.
%K A172108 nonn,new
%O A172108 1,8
%A A172108 Martin Griffiths (martin.griffiths(AT)manchester.ac.uk), Jan 25 2010

%I A172106
%S A172106 0,1,1,1,4,3,1,10,21,12,1,22,93,132,60,1,46,345,900,960,360,1,94,1173,
%T A172106 4980,9300,7920,2520,1,190,3801,24612,71400,103320,73080,20160,1,382,
%U A172106 11973,113652,480060,1048320,1234800,745920,181440,1,766,37065,502500
%N A172106 The triangle T_2(n,m), where T_2(n,m) is the number of surjective multi-valued functions from {1,1,2,3,...,n-1} to {1,2,3,...,m} by rows (n>=1,1<=m<=n)
%C A172106 T_2(1,m)=0 by definition. T_2(n,m) also gives the number of ordered partitions of {1,1,2,3,...,n-1} into exactly m parts.
%F A172106 For n>=2, T_2(n,m)=Sum_{l=0..m}binomial(m,l)*binomial(l+1,2)*(-1)^(m-l)*l^(n-2)
%e A172106 T_2(3,2)=4 since there are 4 ordered partitions of {1,1,2} into exactly 2 parts: (1) {{1},{1,2}} (2) {{1,2},{1}} (3) {{2},{1,1}} (4) {{1,1},{2}}
%t A172106 f[r_, n_, m_] := Sum[Binomial[m, l] Binomial[l + r - 1, r] (-1)^(m - l) l^(n - r), {l, 1, m}]; For[n = 2, n <= 10, n++, Print[Table[f[2, n, m], {m, 1, n}]]]
%Y A172106 This is related to A019538, A172107 and A172108. The row sum is A172109, and the row sum of is A000670 (the ordered Bell numbers).
%K A172106 nonn,new
%O A172106 1,5
%A A172106 Martin Griffiths (martin.griffiths(AT)manchester.ac.uk), Jan 25 2010

%I A172107
%S A172107 0,0,0,1,2,1,1,6,9,4,1,14,45,52,20,1,30,177,388,360,120,1,62,621,2260,
%T A172107 3740,2880,840,1,126,2049,11524,30000,39720,26040,6720,1,254,6525,54292,
%U A172107 207620,418320,460320,262080,60480,1,510,20337,243268,1309560,3755640
%N A172107 The triangle T_3(n,m), where T_3(n,m) is the number of surjective multi-valued functions from {1,1,1,2,3,...,n-2} to {1,2,3,...,m} by rows (n>=1,1<=m<=n)
%C A172107 T_3(1,m)=T_3(2,m)=0 by definition. T_3(n,m) also gives the number of ordered partitions of {1,1,1,2,3,...,n-2} into exactly m parts.
%F A172107 For n>=3, T_3(n,m)=Sum_{l=0..m}binomial(m,l)*binomial(l+2,3)*(-1)^(m-l)*l^(n-3)
%t A172107 f[r_, n_, m_] := Sum[Binomial[m, l] Binomial[l + r - 1, r] (-1)^(m - l) l^(n - r), {l, 1, m}]; For[n = 3, n <= 10, n++, Print[Table[f[3, n, m], {m, 1, n}]]]
%Y A172107 This is related to A019538, A172106 and A172108. The row sum is A172110.
%K A172107 nonn,new
%O A172107 1,5
%A A172107 Martin Griffiths (martin.griffiths(AT)manchester.ac.uk), Jan 25 2010

%I A172105
%S A172105 1,2,3,6,7,27,37,38,297,298,299,302,303,305,306,307,315,316
%N A172105 Numbers n such that nth partial sum of A167020-nth partial sum of A167021=0 (or A172103(n)-A172104(n)=0).
%C A172105 Where A167020 is binary sequence: a(n)=1 iff 6*n-1 is prime and A167021 is binary sequence: a(n)=1 iff 6*n+1 is prime.
%Y A172105 Cf. A167020, A167021, A172103, A172104.
%K A172105 nonn,new
%O A172105 1,2
%A A172105 Juri-Stepan Gerasimov (2stepan(AT)rambler.ru), Jan 25 2010

%I A172104
%S A172104 1,2,3,3,4,5,6,6,6,7,8,9,10,10,10,11,12,13,13,13,14,14,15,15,16,17,18,
%T A172104 18,18,19,19,20,21,21,22,22,23,24,24,25,25,25,25,25,26,27,28,28,28,28,
%U A172104 29,30,30,30,31,32,32,33,33,33,34,35,36,36,36,37,37,38,38,39,39,40,41
%N A172104 Partial sums of A167021.
%C A172104 Where A167021 is binary sequence: a(n)=1 iff 6*n+1 is prime.
%Y A172104 Cf. A002476, A167021.
%K A172104 nonn,new
%O A172104 1,2
%A A172104 Juri-Stepan Gerasimov (2stepan(AT)rambler.ru), Jan 25 2010

%I A172103
%S A172103 1,2,3,4,5,5,6,7,8,9,9,10,10,11,12,12,13,14,15,15,15,16,17,17,18,18,18,
%T A172103 19,20,21,21,22,23,23,23,23,23,24,25,26,26,27,28,29,30,30,31,31,32,32,
%U A172103 32,33,34,34,34,34,34,35,36,37,37,37,37,38,39,39,40,40,40,41,41,42,42
%N A172103 Partial sums of A167020.
%C A172103 Where A167020 is binary sequence: a(n)=1 iff 6*n-1 is prime.
%Y A172103 Cf. A007528, A167020.
%K A172103 nonn,new
%O A172103 1,2
%A A172103 Juri-Stepan Gerasimov (2stepan(AT)rambler.ru), Jan 25 2010

%I A172102
%S A172102 3,11,29,59,101,239,619,809,4253,5323,5923,6551,29131,37277,48341,54413,
%T A172102 58711,60937,70537,101063,110533,214993,224603,417203,445069,466537,
%U A172102 473867,511391,519089,534629,633449,686269,713771,741913,770767,1000537
%N A172102 Prime partial sums of Chen primes (starting with 1).
%C A172102 43 is the first prime which is not a Chen prime, hence this sequence begins the same as prime sums of the first n primes (see A013916). The subset consisting of Chen prime partial sums of Chen primes begins a(1) = 3 = A109611(2), a(2) = 11 = A109611(5), a(3) = 29 = A109611(10), a(4) = 59 = A109611(10), a(5) = 101 = A109611(21), a(6) = 239 = A109611(40), a(7) = 809 = A109611(95). Which are the next Chen prime partial sums of Chen primes?
%F A172102 {p: p prime and for some k, p = SUM [i=1..k] {q such that q + 2 is either a prime or a semiprime} = {p: p in A000040 and p in A118482}.
%e A172102 a(7) = 1+2+3+5+7+11+13+17+19+23+29+31+37+41+47+53+59+67+71+83 = 619 is prime, which is the sum of the first 19 Chen primes (starting with 1).
%p A172102 Contribution from R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Feb 07 2010: (Start)
%p A172102 isA001358 := proc(n) return ( numtheory[bigomega](n) = 2 ); end proc:
%p A172102 isA109611 := proc(n) isprime(n) and ( isprime(n+2) or isA001358(n+2) ); end proc:
%p A172102 A109611 := proc(n) option remember; local a; if n = 1 then 2; else a := nextprime( procname(n-1) ) ; while not isA109611(a) do a := nextprime(a) ; end do ; return a; end if; end proc:
%p A172102 A118482 := proc(n) option remember ; 1+add( A109611(j),j=1..n) ; end proc:
%p A172102 isA172102 := proc(n) if isprime(n) then for j from 1 do if A118482(j) > n then return false; elif A118482(j) = n then return true; end if; end do ; else false ; end if; end proc:
%p A172102 for n from 1 to 10000000 do if isA172102(n) then printf("%d,\n",n) ; end if; end do ; (End)
%Y A172102 Cf. A000040, A001358, A109611, A118482.
%K A172102 easy,nonn,new
%O A172102 1,1
%A A172102 Jonathan Vos Post (jvospost3(AT)gmail.com), Jan 25 2010
%E A172102 Extended by R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Feb 07 2010

%I A172101
%S A172101 1,0,1,0,1,1,0,1,1,1,0,1,2,2,1,0,1,2,4,2,1,0,1,3,6,6,3,1,0,1,3,9,9,9,3,
%T A172101 1,0,1,4,12,18,18,12,4,1,0,1,4,16,24,36,24,16,4,1,0,1,5,20,40,60,60,40,
%U A172101 20,5,1,0,1,5,25,50,100,100,100,50,25,5,1,0,1,6,30,75,150,200,200,150
%N A172101 Triangle, read by rows, given by [0,1,0,-1,0,1,0,-1,0,1,0,-1,0,...] DELTA [1,0,-1,0,1,0,-1,0,1,0,-1,0,1,...] where DELTA is the operator defined in A084938.
%C A172101 Number of symmetric Dyck paths of semilength n with k peaks. Diagonal sums : A088518, row sums : A001405.
%F A172101 Sum_{k, 0<=k<=n} T(n,k) = A001405(n).
%e A172101 Triangle begins : 1 ; 0,1 ; 0,1,1 ; 0,1,1,1 ; 0,1,2,2,1 ; 0,1,2,4,2,1 ; 0,1,3,6,6,3,1 ; ...
%Y A172101 Cf. A088855
%K A172101 nonn,tabl,new
%O A172101 0,13
%A A172101 Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Jan 25 2010

%I A172100
%S A172100 3,5,7,9,11,13,15,17,19,21,23,25,1,3,5,7,9,11,13,15,17,19,21,23,25,1
%N A172100 Diagonal of the 26x26 Caesar Shift table.
%H A172100 Anonymous, <a href="http://en.wikipedia.org/wiki/Caesar_cipher">Ceasar cipher</a>, Wikipedia.
%F A172100 a(n)=1+ (2*n mod 26).
%t A172100 Clear[a, n, m];
%t A172100 a = Table[1 + Mod[n + m, 26], {m, 1, 26}, {n, 1, 26}];
%t A172100 Table[a[[n, n]], {n, 1, Length[a]}]
%K A172100 nonn,less,fini,full,new
%O A172100 1,1
%A A172100 Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Jan 25 2010
%E A172100 Keywords added, spelling corrected by the Assoc. Edts. of the OEIS, Feb 02 2010

%I A172099
%S A172099 1,1,1,1,0,0,1,1,1,1,0,0,0,1,0,0,1,1,1,0,0,1,1,0,0,1,1,0,0,0,1,0,0,1,1,
%T A172099 1,1,0,0,0,1,0,0,1,2,0,0,1,1,0,0,1,1,0,0,0,1,0,0,1,1,1,0,0,1,1,0,0,1,1,
%U A172099 0,0,1,2,0,0,1,2,0,0,1,2,0,0,1,1,0,0,1,1,0,0,0,1,0,0,1,1,1,1,0,0,0,1,0
%N A172099 Coefficients of polynomial recursion with powers n^2-1: p(x, n) = x^(2*n - 1)*p(x, n - 1) + p(x, n - 2)
%C A172099 Row sums are:
%C A172099 {1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144,...}.
%F A172099 p(x, n) = x^(2*n - 1)*p(x, n - 1) + p(x, n - 2)
%e A172099 {1},
%e A172099 {1, 1},
%e A172099 {1, 0, 0, 1, 1},
%e A172099 {1, 1, 0, 0, 0, 1, 0, 0, 1, 1},
%e A172099 {1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 1, 1},
%e A172099 {1, 1, 0, 0, 0, 1, 0, 0, 1, 2, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 1, 1},
%e A172099 {1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 2, 0, 0, 1, 2, 0, 0, 1, 2, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 1, 1},
%e A172099 {1, 1, 0, 0, 0, 1, 0, 0, 1, 2, 0, 0, 1, 2, 0, 0, 2, 2, 0, 0, 1, 2, 0, 0, 2, 3, 0, 0, 1, 2, 0, 0, 1, 2, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 1, 1},
%e A172099 {1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 2, 0, 0, 2, 3, 0, 0, 1, 3, 0, 0, 2, 3, 0, 0, 2, 3, 0, 0, 2, 3, 0, 0, 2, 3, 0, 0, 2, 3, 0, 0, 1, 2, 0, 0, 1, 2, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 1, 1}, {1, 1, 0, 0, 0, 1, 0, 0, 1, 2, 0, 0, 1, 2, 0, 0, 2, 3, 0, 0, 2, 3, 0, 0, 3, 4, 0, 0, 2, 4, 0, 0, 3, 5, 0, 0, 2, 4, 0, 0, 3, 4, 0, 0, 2, 4, 0, 0, 3, 4, 0, 0, 2, 3, 0, 0, 2, 3, 0, 0, 1, 2, 0, 0, 1, 2, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 1, 1},
%e A172099 {1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 2, 0, 0, 2, 3, 0, 0, 2, 4, 0, 0, 2, 4, 0, 0, 3, 5, 0, 0, 3, 5, 0, 0, 4, 6, 0, 0, 4, 6, 0, 0, 4, 6, 0, 0, 3, 6, 0, 0, 4, 6, 0, 0, 3, 5, 0, 0, 3, 5, 0, 0, 3, 5, 0, 0, 3, 4, 0, 0, 2, 3, 0, 0, 2, 3, 0, 0, 1, 2, 0, 0, 1, 2, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 1, 1}
%t A172099 Clear[p, x, n, a];
%t A172099 p[x, 0] = 1; p[x, 1] = x + 1;
%t A172099 p[x_, n_] := p[x, n] = x^(2*n - 1)*p[x, n - 1] + p[x, n - 2];
%t A172099 a = Table[CoefficientList[p[x, n], x], {n, 0, 10}];
%t A172099 Flatten[a]
%K A172099 nonn,uned,new
%O A172099 0,45
%A A172099 Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Jan 25 2010

%I A172098
%S A172098 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,1,1,1,1,1,1,1,1,1,1,1,
%T A172098 1,2,2,2,1,2,1,1,1,1,1,1,1,1,1,1,1,2,1,2,2,2,2,2,3,2,2,1,2,1,1,1,1,1,1,
%U A172098 1,1,1,1,1,1,2,2,3,2,3,2,3,2,3,3,3,3,2,3,2,2,1,2,1,1,1,1,1,1,1,1,1,1,1
%N A172098 Coefficients of polynomial recursion with powers n*(n-1)/2: p(x, n) = x^(n - 1)*p(x, n - 1) + p(x, n - 2)
%C A172098 Row sums are:
%C A172098 {1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144,...}.
%F A172098 p(x, n) = x^(n - 1)*p(x, n - 1) + p(x, n - 2)
%e A172098 {1},
%e A172098 {1, 1},
%e A172098 {1, 1, 1},
%e A172098 {1, 1, 1, 1, 1},
%e A172098 {1, 1, 1, 1, 1, 1, 1, 1},
%e A172098 {1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1},
%e A172098 {1, 1, 1, 1, 1, 2, 2, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1},
%e A172098 {1, 1, 1, 1, 2, 1, 2, 2, 2, 2, 2, 3, 2, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1},
%e A172098 {1, 1, 1, 1, 1, 2, 2, 3, 2, 3, 2, 3, 2, 3, 3, 3, 3, 2, 3, 2, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1},
%e A172098 {1, 1, 1, 1, 2, 1, 2, 2, 3, 3, 3, 4, 3, 4, 3, 5, 3, 4, 3, 4, 3, 4, 4, 3, 3, 2, 3, 2, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1},
%e A172098 {1, 1, 1, 1, 1, 2, 2, 3, 2, 4, 3, 4, 3, 5, 4, 5, 5, 5, 6, 5, 6, 4, 6, 4, 6, 4, 5, 4, 5, 4, 4, 4, 3, 3, 2, 3, 2, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1}
%t A172098 Clear[p, x, n, a];
%t A172098 p[x, 0] = 1; p[x, 1] = x + 1;
%t A172098 p[x_, n_] := p[x, n] = x^(n - 1)*p[x, n - 1] + p[x, n - 2];
%t A172098 a = Table[CoefficientList[p[x, n], x], {n, 0, 10}];
%t A172098 Flatten[a]
%K A172098 nonn,uned,new
%O A172098 0,24
%A A172098 Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Jan 25 2010

%I A172095
%S A172095 7,11,13,19,27,37,53,107,163,243,2187,2917,4373,8747,1594323,86093443
%N A172095 Integers n such that n-1,n,n+1 have few distinct primes: n=p^r, p odd prime, and (n^2-1)/8 divisible by at most two distinct prime factors.
%C A172095 In the known values of this sequence, (n^2-1)/8 is odd unless n=7, and (n^2-1)/8 is the product of two distinct primes unless n=3^5, with 3^5-1= 2 11^2. Note the terms 3^3=27, 3^5=243, 3^7=2187, and 3^13=1594323. The other known entries are prime.
%Y A172095 Cf. A075081 A062547 A111974 A005105
%K A172095 hard,more,nonn,new
%O A172095 1,1
%A A172095 Dino Lorenzini (lorenzin(AT)uga.edu), Jan 25 2010

%I A172094
%S A172094 1,1,1,3,4,1,11,17,7,1,45,76,40,10,1,197,353,216,72,13,1,903,1688,1145,
%T A172094 458,113,16,1,4279,8257,6039,2745,829,163,19,1,20793,41128,31864,15932,
%U A172094 5558,1356,222,22,1,103049,207905,168584,90776,35318,10070,2066,290,25
%N A172094 Triangle , read by rows, given by [1,2,1,2,1,2,1,2,1,2,1,2,...] DELTA [1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938.
%C A172094 Riordan array (f(x), f(x)-1) where f(x) is the g.f. of A001003. Equals A122538*A007318.
%F A172094 T(0,0)=1, T(n,k)=0 if k>n, T(n,0)= T(n-1,0)+2*T(n-1,1), T(n,k)=T(n-1,k-1)+3*T(n-1,k)+2*T(n-1,k+1) for k>0. Sum_{k, 0<=k<=n} T(n,k) = A109980(n). Sum_{k, k>=0} T(m,k)*T(n,k)*2^k = T(m+n,0)= A001003(m+n).
%e A172094 Triangle begins : 1 ; 1,1 ; 3,4,1 ; 11,17,7,1 ; 45,76,40,10,1 ; ...
%K A172094 nonn,tabl,new
%O A172094 0,4
%A A172094 Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Jan 25 2010

%I A172093
%S A172093 1,1,1,1,3,1,1,99,99,1,1,8819,8915,8819,1,1,3034499,3043315,
%T A172093 3043315,3034499,1,1,4151231699,4154266195,4154274915,4154266195,
%U A172093 4151231699,1,1,22682342182499,22686493414195,22686496448595
%V A172093 1,1,1,1,-3,1,1,-99,-99,1,1,-8819,-8915,-8819,1,1,-3034499,-3043315,
%W A172093 -3043315,-3034499,1,1,-4151231699,-4154266195,-4154274915,-4154266195,
%X A172093 -4151231699,1,1,-22682342182499,-22686493414195,-22686496448595
%N A172093 Subtraction q form triangle:q=4;c(n)=Product[Sum[2^k, {k, 0, i}], {i, 1, n - 1}]; t(n,k) = -c(n) + c(n - k) + c(k, q)
%C A172093 Row sums are:
%C A172093 {1, 2, -2, -356, -86660, -90587564, -442533635468, -10372635857431772,
%C A172093 -1181791865462943686876, -659483322031096429636025180,
%C A172093 -1811325619257503185883288046965084,...}
%F A172093 q=4;c(n)=Product[Sum[2^k, {k, 0, i}], {i, 1, n - 1}];
%F A172093 t(n,k) = -c(n) + c(n - k) + c(k, q)
%e A172093 {1},
%e A172093 {1, 1},
%e A172093 {1, -3, 1},
%e A172093 {1, -99, -99, 1},
%e A172093 {1, -8819, -8915, -8819, 1},
%e A172093 {1, -3034499, -3043315, -3043315, -3034499, 1},
%e A172093 {1, -4151231699, -4154266195, -4154274915, -4154266195, -4151231699, 1},
%e A172093 {1, -22682342182499, -22686493414195, -22686496448595, -22686496448595, -22686493414195, -22682342182499, 1},
%e A172093 {1, -495563828620360499, -495586510962542995, -495586515113774595, -495586515116800275, -495586515113774595, -495586510962542995, -495563828620360499, 1},
%e A172093 {1, -43304349690907567762499, -43304845254736188122995, -43304845277418530305395, -43304845277422681528275, -43304845277422681528275, -43304845277418530305395, -43304845254736188122995, -43304349690907567762499, 1},
%e A172093 {1, -15136082740745886405358372499, -15136126045095577312926134995, -15136126045591141141546495395, -15136126045591163823888669075, -15136126045591163828036866275, -15136126045591163823888669075, -15136126045591141141546495395, -15136126045095577312926134995, -15136082740745886405358372499, 1}
%t A172093 Clear[c,t,n,k,t];
%t A172093 c[n_,q_]:=Product[Sum[q^k,{k,0,i}],{i,1,n-1}];
%t A172093 t[n_,k_,q_]=-c[n,q]+c[n-k,q]+c[k,q];
%t A172093 Table[Table[Table[t[n,k,q],{k,0,n}],{n,0,10}],{q,2,10}];
%t A172093 Table[Flatten[Table[Table[t[n,k,q],{k,0,n}],{n,0,10}]],{q,2,10}]
%K A172093 sign,tabl,uned,new
%O A172093 0,5
%A A172093 Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Jan 25 2010

%I A172092
%S A172092 1,1,1,1,2,1,1,47,47,1,1,2027,2072,2027,1,1,249599,251624,
%T A172092 251624,249599,1,1,91359839,91609436,91611416,91609436,91359839,
%U A172092 1,1,100039779839,100131139676,100131389228,100131389228
%V A172092 1,1,1,1,-2,1,1,-47,-47,1,1,-2027,-2072,-2027,1,1,-249599,-251624,
%W A172092 -251624,-249599,1,1,-91359839,-91609436,-91611416,-91609436,-91359839,
%X A172092 1,1,-100039779839,-100131139676,-100131389228,-100131389228
%N A172092 Subtraction q form triangle:q=3;c(n)=Product[Sum[2^k, {k, 0, i}], {i, 1, n - 1}]; t(n,k) = -c(n) + c(n - k) + c(k, q)
%C A172092 Row sums are:
%C A172092 {1, 2, 0, -92, -6124, -1002444, -457549964, -600604617484, -2298816299112204,
%C A172092 -25856055844713627404, -858811326017167374184204,...}
%F A172092 q=3;c(n)=Product[Sum[2^k, {k, 0, i}], {i, 1, n - 1}];
%F A172092 t(n,k) = -c(n) + c(n - k) + c(k, q)
%e A172092 {1},
%e A172092 {1, 1},
%e A172092 {1, -2, 1},
%e A172092 {1, -47, -47, 1},
%e A172092 {1, -2027, -2072, -2027, 1},
%e A172092 {1, -249599, -251624, -251624, -249599, 1},
%e A172092 {1, -91359839, -91609436, -91611416, -91609436, -91359839, 1},
%e A172092 {1, -100039779839, -100131139676, -100131389228, -100131389228, -100131139676, -100039779839, 1},
%e A172092 {1, -328330832269439, -328430872049276, -328430963409068, -328430963656640, -328430963409068, -328430872049276, -328330832269439, 1},
%e A172092 {1, -3231760682422271999, -3232089013254541436, -3232089113294321228, -3232089113385679040, -3232089113385679040, -3232089113294321228, -3232089013254541436, -3231760682422271999, 1},
%e A172092 {1, -95420966894492894054399, -95424198655175316326396, -95424198983506148595788, -95424198983606188373600, -95424198983606279483840, -95424198983606188373600, -95424198983506148595788, -95424198655175316326396, -95420966894492894054399, 1}
%t A172092 Clear[c,t,n,k,t];
%t A172092 c[n_,q_]:=Product[Sum[q^k,{k,0,i}],{i,1,n-1}];
%t A172092 t[n_,k_,q_]=-c[n,q]+c[n-k,q]+c[k,q];
%t A172092 Table[Table[Table[t[n,k,q],{k,0,n}],{n,0,10}],{q,2,10}];
%t A172092 Table[Flatten[Table[Table[t[n,k,q],{k,0,n}],{n,0,10}]],{q,2,10}]
%K A172092 sign,tabl,uned,new
%O A172092 0,5
%A A172092 Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Jan 25 2010

%I A172091
%S A172091 1,1,1,1,1,1,1,17,17,1,1,293,309,293,1,1,9449,9741,9741,9449,
%T A172091 1,1,605429,614877,615153,614877,605429,1,1,77514569,78119997,
%U A172091 78129429,78129429,78119997,77514569,1,1,19844960309,19922474877
%V A172091 1,1,1,1,-1,1,1,-17,-17,1,1,-293,-309,-293,1,1,-9449,-9741,-9741,-9449,
%W A172091 1,1,-605429,-614877,-615153,-614877,-605429,1,1,-77514569,-78119997,
%X A172091 -78129429,-78129429,-78119997,-77514569,1,1,-19844960309,-19922474877
%N A172091 Subtraction q form triangle:q=2;c(n)=Product[Sum[2^k, {k, 0, i}], {i, 1, n - 1}]; t(n,k) = -c(n) + c(n - k) + c(k, q)
%C A172091 Row sums are:
%C A172091 {1, 2, 1, -32, -893, -38378, -3055763, -467527988, -139304120393,
%C A172091 -81405588536318, -93713294552041343,...}
%F A172091 q=2;c(n)=Product[Sum[2^k, {k, 0, i}], {i, 1, n - 1}];
%F A172091 t(n,k) = -c(n) + c(n - k) + c(k, q)
%e A172091 {1},
%e A172091 {1, 1},
%e A172091 {1, -1, 1},
%e A172091 {1, -17, -17, 1},
%e A172091 {1, -293, -309, -293, 1},
%e A172091 {1, -9449, -9741, -9741, -9449, 1},
%e A172091 {1, -605429, -614877, -615153, -614877, -605429, 1},
%e A172091 {1, -77514569, -78119997, -78129429, -78129429, -78119997, -77514569, 1},
%e A172091 {1, -19844960309, -19922474877, -19923080289, -19923089445, -19923080289, -19922474877, -19844960309, 1},
%e A172091 {1, -10160775938249, -10180620898557, -10180698413109, -10180699018245, -10180699018245, -10180698413109, -10180620898557, -10160775938249, 1},
%e A172091 {1, -10404674406948149, -10414835182886397, -10414855027846689, -10414855105360965, -10414855105956945, -10414855105360965, -10414855027846689, -10414835182886397, -10404674406948149, 1}
%t A172091 Clear[c,t,n,k,t];
%t A172091 c[n_,q_]:=Product[Sum[q^k,{k,0,i}],{i,1,n-1}];
%t A172091 t[n_,k_,q_]=-c[n,q]+c[n-k,q]+c[k,q];
%t A172091 Table[Table[Table[t[n,k,q],{k,0,n}],{n,0,10}],{q,2,10}];
%t A172091 Table[Flatten[Table[Table[t[n,k,q],{k,0,n}],{n,0,10}]],{q,2,10}]
%K A172091 sign,tabl,uned,new
%O A172091 0,8
%A A172091 Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Jan 25 2010

%I A172090
%S A172090 1,1,1,1,2,1,1,1,1,1,1,1,0,1,1,1,1,0,0,1,1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,
%T A172090 1,1,1,0,0,0,0,0,1,1,1,1,0,0,0,0,0,0,1,1,1,1,0,0,0,0,0,0,0,1,1
%N A172090 Subtraction triangle:a(n)=If[PrimeQ[Abs[a(n - 1)]], a(n - 1)*n - 2, a(n - 1)*n/(n - 1)]; t(n,m)=-a(n)+a(n-k)+a(k)
%C A172090 Row sums are:
%C A172090 {1, 2, 4, 4, 4, 4, 4, 4, 4, 4, 4,...};
%C A172090 a(n) is:
%C A172090 {1, -2, -6, -9, -12, -15, -18, -21, -24, -27, -30, -33, -36, -39, -42, -45,
%C A172090 -48, -51, -54, -57, -60, -63, -66, -69, -72, -75, -78, -81, -84, -87, -90}
%F A172090 a(n)=If[PrimeQ[Abs[a(n - 1)]], a(n - 1)*n - 2, a(n - 1)*n/(n - 1)];
%F A172090 t(n,m)=-a(n)+a(n-k)+a(k)
%e A172090 {1},
%e A172090 {1, 1},
%e A172090 {1, 2, 1},
%e A172090 {1, 1, 1, 1},
%e A172090 {1, 1, 0, 1, 1},
%e A172090 {1, 1, 0, 0, 1, 1},
%e A172090 {1, 1, 0, 0, 0, 1, 1},
%e A172090 {1, 1, 0, 0, 0, 0, 1, 1},
%e A172090 {1, 1, 0, 0, 0, 0, 0, 1, 1},
%e A172090 {1, 1, 0, 0, 0, 0, 0, 0, 1, 1},
%e A172090 {1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1}
%t A172090 a[0]=1;a[1]=-2;
%t A172090 a[n_]:=a[n]=If[PrimeQ[Abs[a[n-1]]],a[n-1]*n-2,a[n-1]*n/(n-1)];
%t A172090 Table[a[n],{n,0,30}];
%t A172090 Table[Table[ -a[n]+a[n-k]+a[k],{k,0,n}],{n,0,10}];
%t A172090 Flatten[%]
%K A172090 nonn,tabl,uned,new
%O A172090 0,5
%A A172090 Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Jan 25 2010

%I A172089
%S A172089 1,1,1,1,2,1,2,3,3,2,3,8,6,8,3,8,15,20,20,15,8,15,48,45,80,45,48,15,48,
%T A172089 105,168,210,210,168,105,48,105,384,420,896,630,896,420,384,105,384,945,
%U A172089 1728,2520,3024,3024,2520,1728,945,384,945,3840,4725,11520,9450,16128
%N A172089 Factorial two triangle: t(n,m)=n!/(m!!*(n-m)!!)
%C A172089 Row sums are:
%C A172089 {1, 2, 4, 10, 28, 86, 296, 1062, 4240, 17202, 77088,...}
%F A172089 t(n,m)=n!/(m!!*(n-m)!!)
%e A172089 {1},
%e A172089 {1, 1},
%e A172089 {1, 2, 1},
%e A172089 {2, 3, 3, 2},
%e A172089 {3, 8, 6, 8, 3},
%e A172089 {8, 15, 20, 20, 15, 8},
%e A172089 {15, 48, 45, 80, 45, 48, 15},
%e A172089 {48, 105, 168, 210, 210, 168, 105, 48},
%e A172089 {105, 384, 420, 896, 630, 896, 420, 384, 105},
%e A172089 {384, 945, 1728, 2520, 3024, 3024, 2520, 1728, 945, 384},
%e A172089 {945, 3840, 4725, 11520, 9450, 16128, 9450, 11520, 4725, 3840, 945}
%t A172089 binomialn[n_, k_] = n!/(Factorial2[n - k]*Factorial2[k]);
%t A172089 Table[Table[binomialn[n, k], {k, 0, n}], {n, 0, 10}];
%t A172089 Flatten[%]
%K A172089 nonn,tabl,uned,new
%O A172089 0,5
%A A172089 Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Jan 25 2010

%I A172088
%S A172088 1,1,1,1,0,1,1,0,0,1,1,4,4,4,1,1,6,10,10,6,1,1,32,38,42,38,
%T A172088 32,1,1,56,88,94,94,88,56,1,1,278,334,366,368,366,334,278,1,1,560,
%U A172088 838,894,922,922,894,838,560,1,1,2894,3454,3732,3784,3810,3784,3732
%V A172088 -1,-1,-1,-1,0,-1,-1,0,0,-1,-1,4,4,4,-1,-1,6,10,10,6,-1,-1,32,38,42,38,
%W A172088 32,-1,-1,56,88,94,94,88,56,-1,-1,278,334,366,368,366,334,278,-1,-1,560,
%X A172088 838,894,922,922,894,838,560,-1,-1,2894,3454,3732,3784,3810,3784,3732
%N A172088 Factorial two (!!) subtraction triangle: t(n,m)=n!!-m!!-(n-m)!!
%C A172088 Row sums are:
%C A172088 {-1, -2, -2, -2, 10, 30, 180, 474, 2322, 6426, 31536,...}
%F A172088 t(n,m)=n!!-m!!-(n-m)!!
%e A172088 {-1},
%e A172088 {-1, -1},
%e A172088 {-1, 0, -1},
%e A172088 {-1, 0, 0, -1},
%e A172088 {-1, 4, 4, 4, -1},
%e A172088 {-1, 6, 10, 10, 6, -1},
%e A172088 {-1, 32, 38, 42, 38, 32, -1},
%e A172088 {-1, 56, 88, 94, 94, 88, 56, -1},
%e A172088 {-1, 278, 334, 366, 368, 366, 334, 278, -1},
%e A172088 {-1, 560, 838, 894, 922, 922, 894, 838, 560, -1},
%e A172088 {-1, 2894, 3454, 3732, 3784, 3810, 3784, 3732, 3454, 2894, -1}
%t A172088 t[n_, m_] = n!! - m!! - (n - m)!!;
%t A172088 Table[Table[t[n, m], {m, 0, n}], {n, 0, 10}];
%t A172088 Flatten[%]
%K A172088 sign,tabl,uned,new
%O A172088 0,12
%A A172088 Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Jan 25 2010

%I A172087
%S A172087 1,1,3,3,15,15,21,21,15,15,33,33,1365,1365,3,3,255,255
%N A172087 Main pure Bernoulli twin numbers (fourth part;see submitted A172079,A172083,A172086):denominators.Also double A001897.
%C A172087 Main pure Bernoulli twin numbers are 2,-2,1/3,-1/3,-1/15,1/15,1/21,-1/21,-1/15,1/15,5/33,-5/33,.Numerators are submitted A172086:2,-2,1,-1,-1,1,. Also reduced (2*A000367/A002445=2,-2,2/6,-2/6,-2/30,2/42,-2/42,-2/30,2/30,).
%K A172087 nonn,uned,new
%O A172087 0,3
%A A172087 Paul Curtz (bpcrtz(AT)free.fr), Jan 25 2010

%I A172085
%S A172085 0,1,41,212,660,1585,3241,5936,10032,15945,24145,35156,49556,67977,
%T A172085 91105,119680,154496,196401,246297,305140,373940,453761,545721,650992,
%U A172085 770800,906425,1059201,1230516,1421812,1634585,1870385,2130816,2417536
%N A172085 a(n)=(27*n^4+22*n^3-21*n^2-16*n)/12
%C A172085 Numbers:(0,1,2,3,4,5,6,7,8,9,10,..) A172082:(0,1,21,78,190,375,651, ..) 41 is in the sequence because 41=21*2-(1+0); 212=78*3-(21+1+0); 660=190*4-(78+21+1+0);
%F A172085 a(n)=n*(n+1)*(27*n^2-5*n-16)/12
%e A172085 For n=0, a(0)=0; n=1, a(1)=1; n=2, a(2)=41; n=3, a(3)=212
%Y A172085 Cf. A172082
%K A172085 nonn,new
%O A172085 0,3
%A A172085 Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Jan 25 2010

%I A172086
%S A172086 2,2,1,1,1,1,1,1,1,1,5,5,691,691,7,7,3617,3617
%V A172086 2,-2,1,-1,-1,1,1,-1,-1,1,5,-5,-691,691,7,-7,-3617,3617
%N A172086 Main pure Bernoulli twin numbers (third part) : (Bernoulli twin numbers (A051716/A051717) 1,-1/2,-1/3,-1/6,-1/30,1/30,) + companion(s) (submitted A172083/A051717) 1,-3/2,2/3,-1/6,-1/30,1/30. See A172079.
%C A172086 Denominators:1,1,3,3,15,15,21,21,15,15,33,33,1365,1365,=double A001897. See A000367/A002445.
%K A172086 nonn,uned,new
%O A172086 0,1
%A A172086 Paul Curtz (bpcrtz(AT)free.fr), Jan 25 2010

%I A172084
%S A172084 3,2,8,6,4,5,0,5,5,2,7,7,9,4,1,0,4,2,2,8,7,8,2,5,7,1,9,3,7,7,2,9,2,9,0,
%T A172084 6,5,3,1,4,7,4,4,5,2,1,4,0,2,6,7,4,2,2,4,4,0,3,0,5,5,1,8,7,7,4,4,6,8,3,
%U A172084 6,1,9,7,8,8,3,3,1,8,5,4,4,5,7,7,3,0,7,8,8,9,8,1,1,8,9,6,0,0,4,9,3,1,5
%N A172084 AGM(3,x)=Pi (AGM = arithmetic-geometric mean)
%C A172084 AGM(3,3.28645055277941042287...)=Pi
%H A172084 Gerd Lamprecht, <a href="http://freenet-homepage.de/gerdlamprecht/Roemisch_JAVA.htm##@N@C0]='50';@C1]=MitGenau('3.286450552779410422878257193772929',@C0]);@B0]='1.0';aD[0]='0.'+addstr('0',@U@C0])-2)+'1';IM=2;@N@Bi]=bigc(1,GetKoDezi(796,0,@U@C0])),bigc(19,'3.0',@C1]));@Bi]=bigc(2,@Bi],'2.045601998');@C1]=bigc(0,@C1],@Bi]);@Nbigc(5,bigc(6,@Bi],@C0]),aD[0])%3C0@N0@N1@Nif(i%3C2)i=2;">Iterationsrechner mit Algorithmus</a>
%H A172084 Gerd Lamprecht, <a href="http://freenet-homepage.de/gerdlamprecht/Zahlenfolgen.html">Zahlenfolgen (sequence)</a>
%F A172084 x=3.2864505; do{ Diff=AGM(3,x)-Pi; x=...; }loop until (abs(Diff)<1e-...
%e A172084 AGM(3,3.28645055277941042287...)=Pi
%o A172084 (Other) Gerd Lamprecht online Iterationsrechner: ##@N@C0]='50';@C1]=MitGenau('3.286450552779410422878257193772929',@C0]);@B0]='1.0';aD[0]='0.'+addstr('0',@U@C0])-2)+'1';IM=2;@N@Bi]=bigc(1,GetKoDezi(796,0,@U@C0])),bigc(19,'3.0',@C1]));@Bi]=bigc(2,@Bi],'2.045601998');@C1]=bigc(0,@C1],@Bi]);@Nbigc(5,bigc(6,@Bi],@C0]),aD[0])%3C0@N0@N1@Nif(i%3C2)i=2;
%Y A172084 Cf. A053004
%K A172084 cons,nonn,uned,new
%O A172084 1,1
%A A172084 Gerd Lamprecht (gerdlamprecht(AT)googlemail.com), Jan 25 2010

%I A172082
%S A172082 0,1,21,78,190,375,651,1036,1548,2205,3025,4026,5226,6643,8295,10200,
%T A172082 12376,14841,17613,20710,24150,27951,32131,36708,41700,47125,53001,
%U A172082 59346,66178,73515,81375,89776,98736,108273,118405,129150,140526
%N A172082 a(n)=(18*n^3+3*n^2-15*n)/6
%C A172082 Generated by formula: n*(n+1)*[2*d*n-(2*d-3)]/6, [with d=9]
%F A172082 a(n)=n*(n+1)*(18*n-15)/6
%e A172082 For n=0, a(0)=0; n=1, a(1)=1; n=2, a(2)=21; n=3, a(3)=78;
%K A172082 nonn,new
%O A172082 0,3
%A A172082 Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Jan 25 2010

%I A172083
%S A172083 1,3,2,1,1,1,1,1,1,1,5,5,691,691,7,7,3617,3617
%V A172083 1,-3,2,-1,-1,1,1,-1,-1,1,5,-5,-691,691,7,-7,-3617,3617
%N A172083 Numerators of differences of submitted A172079/(1,A027642). Also A051716 with a(1)=-3 and a(2)=2 instead of -1 and -1. Companion to A051716.Also with denominators.
%C A172083 Denominators:A051717. For main pure Bernoulli twin numbers (second part).
%K A172083 nonn,uned,new
%O A172083 0,2
%A A172083 Paul Curtz (bpcrtz(AT)free.fr), Jan 25 2010

%I A172081
%S A172081 8,9,6,9,4,6,3,8,7,4,2,4,6,0,6,1,7,2,9,1,2,6,0,0,3,7,1,0,6,8,7,6,5,4,
%T A172081 4,4,1,7,9,9,9,3,7,5,7,4,2,0,9,1,8,0,5,6,1,6,5,8,2,7,4,6,4,9,6,1,0,3,8,
%U A172081 1,4,1,5,4,0,6,2,4,2,0,8,2,2,4,1,3,4,6,3,5,6,7,1,9,7,5,3,1,4,4,4,7,4,0
%N A172081 Decimal expansion of the local minimum F(x) of the Fibonacci Function at x = A171909.
%C A172081 Define the Fibonacci Function F(x) and its derivative as in A171909.
%C A172081 The derivative is dF/dx = ( phi^x * log(phi) - cos(Pi*x) *log(phi)/ phi^x + Pi*sin(Pi*x)/ phi^x)/sqrt(5).
%C A172081 Set dF(x)/dx=0 to find the local minimum.
%H A172081 Gerd Lamprecht, <a href="http://freenet-homepage.de/gerdlamprecht/Roemisch_JAVA.htm##@Na=0;@N@B0]=GetKoDezi(-2,0,200);@Ni==0@N1@N0@N#">Iterationsrechner</a>
%H A172081 Gerd Lamprecht, <a href="http://freenet-homepage.de/gerdlamprecht/Zahlenfolgen.html">Zahlenfolgen (sequences)</a>
%H A172081 E. Weisstein, <a href="http://mathworld.wolfram.com/FibonacciNumber.html">Fibonacci Number</a>, Mathworld.
%e A172081 F(1.67668837258...)=0.896946387424606172912600371068765...
%p A172081 p := (1+sqrt(5))/2 ; F := (p^x - cos(Pi*x)/p^x )/sqrt(5);
%p A172081 Fpr := diff(F,x) ; Fpr2 := diff(Fpr,x) ;
%p A172081 Digits := 80 ; x0 := 1.67 ;
%p A172081 for n from 1 to 10 do
%p A172081 x0 := evalf(x0-subs(x=x0,Fpr)/subs(x=x0,Fpr2)) ;
%p A172081 print( evalf(subs(x=x0,F))) ;
%p A172081 end do : # R. J. Mathar, Feb 02 2010
%o A172081 (Other) Gerd Lamprecht online Iterationsrechner: #@P@Q5)*0.5+0.5,x)/@Q5)+@P@Q5)*0.5-0.5, x)*sin(PI*(x-0.5))/@Q5)@Na=0.19; b=1.6; @B2]=2; @N@B0]=Fx(b); @B1]=Fx(b-a); @B2]=Fx(b+a); if(@B0]%3C@B1]&&@B0]%3C@B2])a/=10; @Eif(@B1]%3C@B2])b-=a; @Eb+=a; @N@A@B1]-@B2])%3C1e-17@N1@N1@Nc=Fx(b);
%Y A172081 Cf. A171909, A001622, A030171, A172197
%K A172081 cons,nonn,new
%O A172081 0,2
%A A172081 Gerd Lamprecht (gerdlamprecht(AT)googlemail.com), Jan 25 2010
%E A172081 Edited, offset and leading zero normalized by R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Feb 02 2010

%I A172080
%S A172080 0,1,37,190,590,1415,2891,5292,8940,14205,21505,31306,44122,60515,81095,
%T A172080 106520,137496,174777,219165,271510,332710,403711,485507,579140,685700,
%U A172080 806325,942201,1094562,1264690,1453915,1663615,1895216,2150192,2430065
%N A172080 a(n)=(12*n^4+10*n^3-9*n^2-7*n)/6
%C A172080 Numbers: (0,1,2,3,4,5,6,7,8,9,10,..,) A172078:(0,1,19,70,170,335,581,...,) 37 is in the sequence because 37=19*2-(1+0); 190=70*3-(19+1+0); 590=170*4-(70+19+1+0);
%F A172080 a(n)=n*(n+1)*(12*n^2-2*n-7)/6
%e A172080 For n=0, a(0)=0, n=1, a(1)=1; n=2, a(2)=37; n=3, a(3)=190
%Y A172080 Cf. A172078
%K A172080 nonn,new
%O A172080 0,3
%A A172080 Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Jan 25 2010

%I A172079
%S A172079 0,1,1,1,0,1,0,1,0,1,0,5,0,691,0,7,0,3617,0,43867,0
%V A172079 0,1,-1,1,0,-1,0,1,0,-1,0,5,0,-691,0,7,0,-3617,0,43867,0
%N A172079 Companion of A164940=numerators of prolonged Bernoulli numbers (B0(n)=0,1,1/2,1/6,0,-1/30,0,1/42,) .Also A164940 with a(2)=-1 instead of 1.
%C A172079 Denominators are like for A164940: 1,A027642. For main pure Bernoulli twin numbers (first part).
%K A172079 nonn,uned,new
%O A172079 0,12
%A A172079 Paul Curtz (bpcrtz(AT)free.fr), Jan 25 2010

%I A172078
%S A172078 0,1,19,70,170,335,581,924,1380,1965,2695,3586,4654,5915,7385,9080,
%T A172078 11016,13209,15675,18430,21490,24871,28589,32660,37100,41925,47151,
%U A172078 52794,58870,65395,72385,79856,87824,96305,105315,114870,124986,135679
%N A172078 a(n)=(16*n^3+3*n^2-13*n)/6
%C A172078 Generated by formula: n*(n+1)*[2*d*n-(2*d-3)]/6, [with d=8]
%F A172078 a(n)=n*(n+1)*(16*n-13)/6
%e A172078 For n=0, a(0)=0; n=1, a(1)=1; n=2, a(2)=19; n=3, a(3)=70;
%K A172078 nonn,new
%O A172078 0,3
%A A172078 Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Jan 25 2010

%I A172077
%S A172077 0,1,33,168,520,1245,2541,4648,7848,12465,18865,27456,38688,53053,71085,
%T A172077 93360,120496,153153,192033,237880,291480,353661,425293,507288,600600,
%U A172077 706225,825201,958608,1107568,1273245,1456845,1659616,1882848,2127873
%N A172077 a(n)=(7*n^4+6*n^3-5*n^2-4*n)/4
%C A172077 Numbers: (0,1,2,3,4,5,6,7,8,9,10,..,) A172076:(0,1,17,62,150,295,511) 33 in this sequence because 33=17*2-(1+0); 168=62*3-(17+1+0); 520=150*4-(62+17+1+0);
%F A172077 a(n)=n*(n+1)*(21*n^2-3*n-12)/12
%e A172077 For n=0, a(0)=0; n=1, a(1)=1; n=2, a(2)=33; n=3, a(3)=168
%Y A172077 Cf. A172076
%K A172077 nonn,new
%O A172077 0,3
%A A172077 Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Jan 25 2010

%I A172076
%S A172076 0,1,17,62,150,295,511,812,1212,1725,2365,3146,4082,5187,6475,7960,9656,
%T A172076 11577,13737,16150,18830,21791,25047,28612,32500,36725,41301,46242,
%U A172076 51562,57275,63395,69936,76912,84337,92225,100590,109446,118807,128687
%N A172076 a(n)=(14*n^3+3*n^2-11*n)/6
%C A172076 Generated by formula: n*(n+1)*[2*d*n-(2*d-3)]/6, [with d=7]
%F A172076 a(n)=n*(n+1)*(14*n-11)/6
%e A172076 For n=0, a(0)=0; n=1, a(1)=1; n=2, a(2)=17; n=3, a(3)=62
%K A172076 nonn,new
%O A172076 0,3
%A A172076 Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Jan 25 2010

%I A172075
%S A172075 0,1,29,146,450,1075,2191,4004,6756,10725,16225,23606,33254,45591,61075,
%T A172075 80200,103496,131529,164901,204250,250250,303611,365079,435436,515500,
%U A172075 606125,708201,822654,950446,1092575,1250075,1424016,1615504,1825681
%N A172075 a(n)=(9*n^4+8*n^3-6*n^2-5*n)/6
%C A172075 Numbers: (0,1,2,3,4,5,6,7,8,9,10,..,); A172073:(0,1,15,54,130,255,441,70,..,) 29 in this sequence because 29=15*2-(1+0); 146=54*3-(15+1+0); 450=130*4-(54+15+1+0);
%F A172075 a(n)=n*(n+1)*(9*n^2-n-5)/6 = n*(n+1)*(18*n^2-2*n-10)/12
%e A172075 For n=0, a(0)=0; n=1, a(1)=1; n=2, a(2)=29; n=3, a(3)=146
%Y A172075 Cf. A172073
%K A172075 nonn,new
%O A172075 0,3
%A A172075 Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Jan 25 2010

%I A172073
%S A172073 0,1,15,54,130,255,441,700,1044,1485,2035,2706,3510,4459,5565,6840,8296,
%T A172073 9945,11799,13870,16170,18711,21505,24564,27900,31525,35451,39690,44254,
%U A172073 49155,54405,60016,66000,72369,79135,86310,93906,101935,110409,119340
%N A172073 a(n)=(4*n^3+n^2-3*n)/2
%C A172073 Pyramidal number tetradecagonals generated by formula: a(n)=n*(n+1)*[2*d*n-(2*d-3)]/6, [with d=6]
%F A172073 a(n)=n*(n+1)*(12*n-9)/6 = n*(n+1)*(4*n-3)/2
%e A172073 For n=0, a(0)=0; n=1, a(1)=1; n=2, a(2)=15
%K A172073 nonn,new
%O A172073 0,3
%A A172073 Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Jan 25 2010

%I A172072
%S A172072 1,5,6,7,8,9,10,12,14,15,16,17,19,21,22,24,27,28,29,30,33,37,38,40,41,
%T A172072 45,46,50,51,52,55,56,59,61,65,66,68,71,72,74,75,77,85,87,90,91,97,101,
%U A172072 103,104,106,108,109,111,112,114,118,119,120,124,130,131,134,144,145
%N A172072 Numbers n such that either prime(n)-5/2-+7/2 is prime.
%e A172072 a(1)=1 because prime(1)-5/2-7/2=2-5/2-7/2=-4(nonprime) and prime(1)-5/2+7/2=2-5/2+7/2=3(prime).
%Y A172072 Cf. A000040, A046117, A172071.
%K A172072 nonn,new
%O A172072 1,2
%A A172072 Juri-Stepan Gerasimov (2stepan(AT)rambler.ru), Jan 25 2010

%I A172074
%S A172074 161,1,4,11,1,1,3,6,1,13,8,1,6,1,4,1,1,2,1,1,1,1,13,2,1,3,8,1,2,19,1,54,
%T A172074 1,19,2,1,8,3,1,2,13,1,1,1,1,2,1,1,4,1,6,1,8,13,1,6,3,1,1,11,4,1,222
%N A172074 Continued fraction to calculate digits of the golden ratio.
%C A172074 The 62 trailing terms are repeated infinitely.
%C A172074 This is just one of an infinte set of cf's, related to the golden ratio,
%C A172074 and more specifically to the square root of 125,12500,1250000...
%C A172074 Mathematica:
%C A172074 ContinuedFraction[N[Sqrt[125],50000],6]
%C A172074 ContinuedFraction[N[Sqrt[12500],50000], 63]
%C A172074 ContinuedFraction[N[Sqrt[1250000],50000], 315]
%C A172074 ContinuedFraction[N[Sqrt[125000000],50000], 3041]
%C A172074 ContinuedFraction[N[Sqrt[12500000000],50000], 30217]
%C A172074 ContinuedFraction[N[Sqrt[1250000000000],313489], 304325]
%e A172074 223/222
%e A172074 1114/223
%e A172074 12477/1114
%e A172074 13591/12477
%t A172074 ContinuedFraction[N[Sqrt[12500],50000], 63]
%Y A172074 Cf. A010186
%K A172074 cofr,nonn,new
%O A172074 0,1
%A A172074 Shane Findley (divineprime(AT)yahoo.com), Jan 25 2010

%I A172071
%S A172071 2,11,13,17,19,23,29,37,43,47,53,59,67,73,79,89,103,107,109,113,137,157,
%T A172071 163,173,179,197,199,229,233,239,257,263,269,277,283,313,317,337,353,
%U A172071 359,373,379,389,439,449,463,467,509,547,563,569,577,593,599,607,613
%N A172071 Primes p such that either p-5/2-+7/2 is prime.
%C A172071 Two together with values of p+6 where (p,p+6) are both prime.
%e A172071 a(1)=2 because 2-5/2-7/2=-4(nonprime) and 2-5/2+7/2=3(prime).
%Y A172071 Cf. A000040, A046117.
%K A172071 nonn,new
%O A172071 1,1
%A A172071 Juri-Stepan Gerasimov (2stepan(AT)rambler.ru), Jan 25 2010

%I A172070
%S A172070 3,11,17,29,41,59,71,101,107,137,149,179,191,197,227,239,269,281,311,
%T A172070 347,419,431,461,521,569,599,617,641,659,809,821,827,857,881,1019,1031,
%U A172070 1049,1061,1091,1151,1229,1277,1289,1301,1319,1427,1451,1481,1487,1607
%N A172070 Primes p such that either p-1/2-+5/2 is prime.
%C A172070 Lesser of twin primes without 5.
%e A172070 a(1)=3 because 3-1/2-5/2=0(nonprime) and 3-1/2+5/2=5(prime).
%Y A172070 Cf. A001359.
%K A172070 nonn,new
%O A172070 1,1
%A A172070 Juri-Stepan Gerasimov (2stepan(AT)rambler.ru), Jan 25 2010

%I A172069
%S A172069 1,2,2,2,2,2,1,1,2,1,2,1,1,2,1,1,0,1,1,0,0,0,1,2,0,2
%N A172069 Each entry indicates the number of primes adjacent to A129912 entries (0,1 or 2)
%C A172069 A129912 by my definition starts at 2 (Editor added initial 1)
%e A172069 Both 415799 and 415801 are prime, so the entry at 415800 (n=26) is a 2 (twin primes). Neither 27719 nor 27721 are prime, so the entry at 27720 (n=17) is a 0.
%Y A172069 A129912
%K A172069 easy,nonn,new
%O A172069 1,2
%A A172069 Bill R McEachen (bmceache(AT)centralsan.org), Jan 24 2010

%I A172068
%S A172068 1,2,2,2,4,4,6,6,4,12,12,8,20,20,16,8,40,40,32,16,70,70,60,40,16,140,
%T A172068 140,120,80,32,252,252,224,168,96,32,504,504,448,336,192,64,924,924,840,
%U A172068 672,448,224,64
%N A172068 Triangular array T(n,k) = the number of n-step one dimensional walks that return to the origin exactly k times.
%C A172068 In a ballot count of n total votes cast for two candidates, T(n,k) is the number of counts in which exactly k ties occur during the counting process (disregarding the initial tie of 0 to 0) and considering every possible outcome of votes.
%D A172068 Feller,W An Introduction to Probability Theory and its Applications, Vol 1, 3rd ed. New York: Wiley, pp.67-97, 1968
%F A172068 T(n,0) = T(n,1) = 2 Binomial(n-1,Floor((n-1)/2)) T(2n,n) = 2^n
%e A172068 T(5,2) = 8 because there are eight possible vote count sequences in which five votes are cast and the count becomes tied two times during the counting process: {-1, 0, -1, 0, -1}, {-1, 0, -1, 0, 1}, {-1, 0, 1, 0, -1}, {-1, 0, 1, 0, 1}, {1, 0, -1, 0, -1}, {1, 0, -1, 0, 1}, {1, 0, 1, 0, -1}, {1, 0, 1, 0, 1} Triangle begins: 1; 2; 2,2; 4,4; 6,6,4; 12,12,8; 20,20,16,8; 40,40,32,16;
%t A172068 Table[Table[ Length[Select[Map[Accumulate, Strings[{-1, 1}, n]], Count[ #, 0] == k &]], {k, 0, Floor[n/2]}], {n, 0, 20}] // Grid
%Y A172068 The first two columns corresponding to k=0 and k=1 are A063886
%K A172068 nonn,new
%O A172068 0,2
%A A172068 Geoffrey Critzer (critzer.geoffrey(AT)usd443.org), Jan 24 2010

%I A172067
%S A172067 1,11,79,468,2486,12323,58277,266492,1188679,5202523,22436251,95630272,
%T A172067 403770544,1691678428,7042481236,29161852240,120212658034,493656394350,
%U A172067 2020590599710,8247228533780,33579755528278,136434358356201
%N A172067 Sequence whose G.f is given by: f(z)=(2/(3*sqrt(1-4*z)-1+4*z))*((1-sqrt(1-4*z))/(2*z))^k with k=10.
%C A172067 This sequence is the 10-th diagonal below the main diagonal (which itself is A026641) in the array which grows with "Pascal rule" given here by rows: 1,0,1,0,1,0,1,0,1,0,1,0,1,0, 1,1,1,1,1,1,1,1,1,1,1,1,1,1, 1,1,2,2,3,3,4,4,5,5,6,6,7,7, 1,2,4,6,9,12,16,20,25,30, 1,3,7,13,22,34,50,70,95. The MAPLE programs give the first diagonals of this array.
%F A172067 a(n)=sum('(-1)^(p)*binomial(2*n+k-p,n-p)',p=0..n) (with k=10).
%e A172067 a(4)=C(18,4)-C(17,3)+C(16,2)-C(15,1)+C(14,0)=60*51-680+120-15+1=2486.
%p A172067 for k from 0 to 20 do for n from 0 to 40 do a(n):=sum('(-1)^(p)*binomial(2*n-p+k, n-p)', p=0..n): od:seq(a(n), n=0..40):od; for k from 0 to 40 do taylor((2/(3*sqrt(1-4*z)-1+4*z))*((1-sqrt(1-4*z))/(2*z))^k, z=0, 40+k):od;
%Y A172067 Cf. A026641, A014300, A014301, A172025 and from A172061 to A172066.
%K A172067 easy,nonn,new
%O A172067 0,2
%A A172067 Richard Choulet (richardchoulet(AT)yahoo.fr), Jan 24 2010

%I A172066
%S A172066 1,10,67,376,1912,9142,41941,186880,815083,3498146,14827487,62236064,
%T A172066 259187048,1072567256,4415408372,18098359424,73915594466,300958990724,
%U A172066 1222228100590,4952609171080,20030298812596,80876902778482
%N A172066 Sequence whose G.f is given by: f(z)=(2/(3*sqrt(1-4*z)-1+4*z))*((1-sqrt(1-4*z))/(2*z))^k with k=9.
%C A172066 This sequence is the 9-th diagonal below the main diagonal (which itself is A026641) in the array which grows with "Pascal rule" given here by rows: 1,0,1,0,1,0,1,0,1,0,1,0,1,0, 1,1,1,1,1,1,1,1,1,1,1,1,1,1, 1,1,2,2,3,3,4,4,5,5,6,6,7,7, 1,2,4,6,9,12,16,20,25,30, 1,3,7,13,22,34,50,70,95. The MAPLE programs give the first diagonals of this array.
%F A172066 a(n)=sum('(-1)^(p)*binomial(2*n+k-p,n-p)',p=0..n) (with k=9).
%e A172066 a(4)=C(17,4)-C(16,3)+C(15,2)-C(14,1)+C(13,0)=17*4*5*7-16*5*7+105-14+1=5*7*(68-16)+92=1912.
%p A172066 for k from 0 to 20 do for n from 0 to 40 do a(n):=sum('(-1)^(p)*binomial(2*n-p+k, n-p)', p=0..n): od:seq(a(n), n=0..40):od; for k from 0 to 40 do taylor((2/(3*sqrt(1-4*z)-1+4*z))*((1-sqrt(1-4*z))/(2*z))^k, z=0, 40+k):od;
%Y A172066 Cf. A026641, A014300, A014301, A172025, A172061, A172062, A172063, A172064, A172065.
%K A172066 easy,nonn,new
%O A172066 0,2
%A A172066 Richard Choulet (richardchoulet(AT)yahoo.fr), Jan 24 2010

%I A172065
%S A172065 1,9,56,297,1444,6656,29618,128603,548591,2309467,9624964,39799813,
%T A172065 163556776,668796712,2723729944,11055878188,44753742226,180746332690,
%U A172065 728571706240,2932018571370,11783070278816,47297147250204
%N A172065 Sequence whose G.f is given by: f(z)=(2/(3*sqrt(1-4*z)-1+4*z))*((1-sqrt(1-4*z))/(2*z))^k with k=8.
%C A172065 This sequence is the 8-th diagonal below the main diagonal (which itself is A026641) in the array which grows with "Pascal rule" given here by rows: 1,0,1,0,1,0,1,0,1,0,1,0,1,0, 1,1,1,1,1,1,1,1,1,1,1,1,1,1, 1,1,2,2,3,3,4,4,5,5,6,6,7,7, 1,2,4,6,9,12,16,20,25,30, 1,3,7,13,22,34,50,70,95. The MAPLE programs give the first diagonals of this array.
%F A172065 a(n)=sum('(-1)^(p)*binomial(2*n+k-p,n-p)',p=0..n) (with k=8).
%e A172065 a(4)=C(16,4)-C(15,3)+C(14,2)-C(13,1)+C(12,0)=20*91-35*13+91-13+1=1820-455+79=1444.
%p A172065 for k from 0 to 20 do for n from 0 to 40 do a(n):=sum('(-1)^(p)*binomial(2*n-p+k, n-p)', p=0..n): od:seq(a(n), n=0..40):od; for k from 0 to 40 do taylor((2/(3*sqrt(1-4*z)-1+4*z))*((1-sqrt(1-4*z))/(2*z))^k, z=0, 40+k):od;
%Y A172065 Cf. A026641, A014300, A014301, A172025, A172061, A172062, A172063, A172064.
%K A172065 easy,nonn,new
%O A172065 0,2
%A A172065 Richard Choulet (richardchoulet(AT)yahoo.fr), Jan 24 2010

%I A172064
%S A172064 1,8,46,230,1068,4744,20476,86662,361711,1494384,6126818,24972326,
%T A172064 101320712,409609664,1651162688,6640469816,26655382802,106830738224,
%U A172064 427612715516,1709790470780,6830461107736,27266848437608
%N A172064 Sequence whose G.f is given by: f(z)=(2/(3*sqrt(1-4*z)-1+4*z))*((1-sqrt(1-4*z))/(2*z))^k with k=7.
%C A172064 This sequence is the 7-th diagonal below the main diagonal (which itself is A026641) in the array which grows with "Pascal rule" given here by rows: 1,0,1,0,1,0,1,0,1,0,1,0,1,0, 1,1,1,1,1,1,1,1,1,1,1,1,1,1, 1,1,2,2,3,3,4,4,5,5,6,6,7,7, 1,2,4,6,9,12,16,20,25,30, 1,3,7,13,22,34,50,70,95. The MAPLE programs give the first diagonals of this array.
%F A172064 a(n)=sum('(-1)^(p)*binomial(2*n+k-p,n-p)',p=0..n) (with k=7)
%e A172064 a(4)=C(15,4)-C(14,3)+C(13,2)-C(12,1)+C(11,0)=7*13*15-14*13*2+78-12+1=1068.
%p A172064 for k from 0 to 20 do for n from 0 to 40 do a(n):=sum('(-1)^(p)*binomial(2*n-p+k, n-p)', p=0..n): od:seq(a(n), n=0..40):od; for k from 0 to 40 do taylor((2/(3*sqrt(1-4*z)-1+4*z))*((1-sqrt(1-4*z))/(2*z))^k, z=0, 40+k):od;
%Y A172064 Cf. A026641, A014300, A014301, A172025, A172061, A172062, A172063.
%K A172064 easy,nonn,new
%O A172064 0,2
%A A172064 Richard Choulet (richardchoulet(AT)yahoo.fr), Jan 24 2010

%I A172063
%S A172063 1,7,37,174,771,3300,13820,57044,233108,945793,3817351,15347362,
%T A172063 61520899,246052888,982365976,3916739872,15599504614,62076995998,
%U A172063 246866382826,981218764540,3898442536366,15483778158792,61482966826992
%N A172063 Sequence whose G.f is given by: f(z)=(2/(3*sqrt(1-4*z)-1+4*z))*((1-sqrt(1-4*z))/(2*z))^k with k=6.
%C A172063 This sequence is the 6-th diagonal below the main diagonal (which itself is A026641) in the array which grows with "Pascal rule" given here by rows: 1,0,1,0,1,0,1,0,1,0,1,0,1,0, 1,1,1,1,1,1,1,1,1,1,1,1,1,1, 1,1,2,2,3,3,4,4,5,5,6,6,7,7, 1,2,4,6,9,12,16,20,25,30, 1,3,7,13,22,34,50,70,95. The MAPLE programs give the first diagonals of this array.
%F A172063 (n)=sum('(-1)^(p)*binomial(2*n+k-p,n-p)',p=0..n) (with k=6)
%e A172063 a(4)=C(14,4)-C(13,3)+C(12,2)-C(11,1)+C(10,0)=7*13*11-26*11+66-11+1=771.
%p A172063 for k from 0 to 20 do for n from 0 to 40 do a(n):=sum('(-1)^(p)*binomial(2*n-p+k, n-p)', p=0..n): od:seq(a(n), n=0..40):od; for k from 0 to 40 do taylor((2/(3*sqrt(1-4*z)-1+4*z))*((1-sqrt(1-4*z))/(2*z))^k, z=0, 40+k):od;
%Y A172063 Cf. A026641, A014300, A014301, A172025, A172061, A172062.
%K A172063 easy,nonn,new
%O A172063 0,2
%A A172063 Richard Choulet (richardchoulet(AT)yahoo.fr), Jan 24 2010

%I A172062
%S A172062 1,6,29,128,541,2232,9076,36568,146446,584082,2322967,9220544,36548573,
%T A172062 144732176,572756312,2265577184,8959034798,35421613196,140035644602,
%U A172062 553606049024,2188652065586,8653317051056,34216118389384
%N A172062 Sequence whose G.f is given by: f(z)=(2/(3*sqrt(1-4*z)-1+4*z))*((1-sqrt(1-4*z))/(2*z))^k with k=5.
%C A172062 This sequence is the 5-th diagonal below the main diagonal (which itself is A026641) in the array which grows with "Pascal rule" given here by rows: 1,0,1,0,1,0,1,0,1,0,1,0,1,0, 1,1,1,1,1,1,1,1,1,1,1,1,1,1, 1,1,2,2,3,3,4,4,5,5,6,6,7,7, 1,2,4,6,9,12,16,20,25,30, 1,3,7,13,22,34,50,70,95. The MAPLE programs give the first diagonals of this array.
%F A172062 a(n)=sum('(-1)^(p)*binomial(2*n+k-p,n-p)',p=0..n) (with k=5)
%e A172062 a(4)=C(13,4)-C(12,3)+C(11,2)-C(10,1)+C(9,0)=13*11*5-20*11+55-10+1=541
%p A172062 for k from 0 to 20 do for n from 0 to 40 do a(n):=sum('(-1)^(p)*binomial(2*n-p+k, n-p)', p=0..n): od:seq(a(n), n=0..40):od; for k from 0 to 40 do taylor((2/(3*sqrt(1-4*z)-1+4*z))*((1-sqrt(1-4*z))/(2*z))^k, z=0, 40+k):od;
%K A172062 easy,nonn,new
%O A172062 0,2
%A A172062 Richard Choulet (richardchoulet(AT)yahoo.fr), Jan 24 2010

%I A172061
%S A172061 1,5,22,91,367,1461,5776,22748,89402,350974,1377174,5403193,21201211,
%T A172061 83211277,326703424,1283211208,5042294926,19822108582,77958648604,
%U A172061 306739666198,1207433301046,4754874514690,18732340230592,73827134976216
%N A172061 Sequence whose G.f is given by: f(z)=(2/(3*sqrt(1-4*z)-1+4*z))*((1-sqrt(1-4*z))/(2*z))^k with k=4.
%C A172061 This sequence is the 4-th diagonal below the main diagonal (which itself is A026641) in the array which grows with "Pascal rule" given here by rows: 1,0,1,0,1,0,1,0,1,0,1,0,1,0, 1,1,1,1,1,1,1,1,1,1,1,1,1,1, 1,1,2,2,3,3,4,4,5,5,6,6,7,7, 1,2,4,6,9,12,16,20,25,30, 1,3,7,13,22,34,50,70,95. The MAPLE programs give the first diagonals of this array.
%F A172061 a(n)=sum('(-1)^(p)*binomial(2*n+k-p,n-p)',p=0..n) (with k=4)
%e A172061 a(4)=C(12,4)-C(11,3)+C(10,2)-C(9,1)+C(8,0)=55*9-55*3+45-9+1=367.
%p A172061 for k from 0 to 40 do for n from k to k+40 do a(n):=sum('(-1)^(p)*binomial(n-p-k+n,-k+n-p)',p=0..n-k): od:seq(a(n),n=k..40+k):od;
%Y A172061 Cf. A026641, A014300, A014301, A172025.
%K A172061 easy,nonn,new
%O A172061 0,2
%A A172061 Richard Choulet (richardchoulet(AT)yahoo.fr), Jan 24 2010

%I A172057
%S A172057 7,13,19,31,43,61,73,103,109,139,151,181,193,199,229,241,271,283,313,
%T A172057 349,421,433,463,523,571,601,619,643,661,811,823,883,1021,1033,1051,
%U A172057 1063,1093,1153,1231,1279,1291,1303,1321,1429,1453,1483,1489,1609,1821
%N A172057 Primes p such that either p-5/2-+1/2 is prime.
%C A172057 Greater of twin primes where twin primes of the form 6*k+1.
%F A172057 a(n)=A006512(n+1).
%Y A172057 Cf. A006512.
%K A172057 nonn,new
%O A172057 1,1
%A A172057 Juri-Stepan Gerasimov (2stepan(AT)rambler.ru), Jan 24 2010

%I A172060
%S A172060 0,2,14,76,374,1748,7916,35096,153254,661636,2831300,12030632,50826684,
%T A172060 213707336,894945944,3734901296,15540685574,64496348516,267060529364,
%U A172060 1103587381256,4552196053844,18747042089816,77092267322984
%N A172060 The number of returns to the origin in all possible one dimensional walks of length 2n.
%C A172060 a(n)/4^n is the expected number of times a gambler will return to his break even point while making 2n equal wagers on the outcome of a fair coin toss. Note the surpisingly low and slow growth of this expectation.
%D A172060 Feller,W "An Introduction to Probability Theory and its Applications", Vol 1, 3rd ed. New York: Wiley, pp. 67-97, 1968
%F A172060 a(n) = (2n+1)!/(n!)^2 -4^n a(n) = 4*a(n-1) + Binomial(2n,n) O.g.f. (1-(1-4x)^(1/2))/(1-4x)^(3/2)
%e A172060 a(2)= 14 because ther are 14 0's in the set of all possible walks of length 4: {{-1, -2, -3, -4}, {-1, -2, -3, -2}, {-1, -2, -1, -2}, {-1, -2, -1, 0}, {-1, 0, -1, -2}, {-1, 0, -1, 0}, {-1, 0, 1, 0}, {-1, 0, 1, 2}, {1, 0, -1, -2}, {1, 0, -1, 0}, {1, 0, 1, 0}, {1, 0, 1, 2}, {1, 2, 1, 0}, {1, 2, 1, 2}, {1, 2, 3, 2}, {1, 2, 3, 4}}
%t A172060 Table[Count[Flatten[Map[Accumulate, Strings[{-1, 1}, n]]], 0], {n, 0, 20, 2}]
%K A172060 nonn,new
%O A172060 0,2
%A A172060 Geoffrey Critzer (critzer.geoffrey(AT)usd443.org), Jan 24 2010

%I A172059
%S A172059 2,5,10,17,28,41,58,89,126,197,270,349,446,559,690,889,1200,1537,1910,
%T A172059 2643,3562,4553,1111111111111115664,11112222222222222226775
%N A172059 Partial sums of absolute primes A003459.
%C A172059 After a(1) = 2, a(2) = 5, a(4) = 17, a(6) = 41, a(8) = 89, a(10) = 197, and a(12) = 349, which are the next primes? This sequence, which alternates between even and odd, is arbitrary as it is in base 10; what are the analogues in other bases, and the diagonal of the array of such sequences?
%F A172059 a(n) = SUM[i=1..n] A003459(i) = SUM[i=1..n] (i-th p such that primes: every permutation of digits is a prime).
%e A172059 a(x) = 2 + 3 + 5 + 7 + 11 + 13 + 17 + 31 + 37 + 71 + 73 + 79 + 97 + 113 + 131 + 199 + 311 + 337 + 373 + 733 + 919 + 991 + 1111111111111111111 + 11111111111111111111111.
%K A172059 base,easy,more,nonn,new
%O A172059 1,1
%A A172059 Jonathan Vos Post (jvospost3(AT)gmail.com), Jan 24 2010

%I A172058
%S A172058 3,5,19,163,379,419,827,907,1427,1787,1979,1987,2083,2243,2339,2539,
%T A172058 2659,2699,3083,3643,3659,4723,5147,5443,5563,5779,6203,6299,6547,6619,
%U A172058 6803,6947,7043,7283,7499,7547,7883,7907,8219,8387,8539,8563,8627,8923
%N A172058 Prime numbers p such that every prime divisor of p-1 is a primitive root modulo p.
%C A172058 The sequence is probably infinite. If so, then there are infinitely many primes for which 2 is a primitive root (A001122).
%t A172058 m = 1; s = {}; While[Prime[m] < 10000, m = m + 1; p = Prime[m]; pf = FactorInteger[p - 1]; L = Length[pf]; j = 0; While[j < L, j = j + 1; q = First[pf[[j]]]; If[MultiplicativeOrder[q, p] == p - 1, , j = L + 1]; If[j == L, s = {s, p},] ] ]; s = Flatten[s]
%Y A172058 Cf. A001122
%K A172058 nonn,new
%O A172058 1,1
%A A172058 Vantieghem Emmanuel (manuvti(AT)hotmail.com), Jan 24 2010

%I A172056
%S A172056 59,61,103,109,149,151,163,257,313,389,401,449,479,541,569,571,673,677,
%T A172056 709,733,769,821,823,839,857,883,919,947,971,983,1061,1087,1093,1097,
%U A172056 1129,1151,1163,1181,1249,1283,1301,1319,1321,1381,1433,1489,1493,1549
%N A172056 Primes p such that 2*p+-1 and 2*p+-3 are all composites.
%e A172056 a(1)=59 because 2*59-1=117, 2*59+1=119, 2*59-3=115, 2*59+3=121 are all composites.
%t A172056 okQ[n_]:=Union[PrimeQ[{2n+1,2n-1,2n+3,2n-3}]]=={False}; Select[Prime[Range[250]],okQ] [From Harvey P. Dale (hpd1(AT)nyu.edu), Feb 07 2010]
%Y A172056 Cf. A000040, A002808, A068497.
%K A172056 nonn,new
%O A172056 1,1
%A A172056 Juri-Stepan Gerasimov (2stepan(AT)rambler.ru), Jan 24 2010
%E A172056 Corrected and extended by Harvey P. Dale (hpd1(AT)nyu.edu), Feb 07 2010

%I A172055
%S A172055 10,19,22,30,36,45,49,63,66,85,93,98,100,110,115,122,126,132,138,143,
%T A172055 155,158,168,171,178,185,187,198,206,213,217,229,231,236,239,243,248,
%U A172055 255,269,275,284,293,300,309,317,321,325,331,337,343,349,351,357,378
%N A172055 nth number k such that 6*k-1 is composite while 6*k+1 is prime plus nth number m such that 6*m-1 is prime while 6*m+1 is composite.
%F A172055 a(n)=A121765(n)+A121763(n).
%Y A172055 Cf. A121763, A171765.
%K A172055 nonn,new
%O A172055 1,1
%A A172055 Juri-Stepan Gerasimov (2stepan(AT)rambler.ru), Jan 24 2010

%I A172054
%S A172054 2,3,4,2,6,7,5,7,8,7,9,12,12,12,9,4,6,4,8,9,7,8,12,11,14,17,17,12,18,17,
%T A172054 19,13,13,10,11,9,8,7,15,17,18,13,12,13,13,11,11,15,19,19,23,23,19,12,
%U A172054 16,17,12,11,18,22,27,29,27,27,25,18,27,28,23,22,23,17,21,24,23,23,30
%N A172054 nth number k such that 6*k-1 is composite while 6*k+1 is prime minus nth number m such that 6*m-1 is prime while 6*m+1 is composite.
%C A172054 n(where A172054(n)<0)?
%F A172054 a(n)=A121765(n)-A121763(n).
%Y A172054 Cf. A121763, A171765.
%K A172054 nonn,new
%O A172054 1,1
%A A172054 Juri-Stepan Gerasimov (2stepan(AT)rambler.ru), Jan 24 2010

%I A172053
%S A172053 1,22,27,36,41,46,53,65,68,77,82,99,103,112,124,128,134,139,149,162,176,
%T A172053 183,193,206,225,232,237,243,249,276,282,287,293,301,330,339,346,351,
%U A172053 358,371,385,402,405,408,413,434,443,454,457,479,482,497,505,510,522
%N A172053 nth nonnegative number k such that neither 6*k+-1 is prime plus nth number m such that 6*m+-1 are both twin primes.
%F A172053 a(n)=A171696(n)+A002822(n).
%Y A172053 Cf. A002822, A171696.
%K A172053 nonn,new
%O A172053 1,2
%A A172053 Juri-Stepan Gerasimov (2stepan(AT)rambler.ru), Jan 24 2010

%I A172052
%S A172052 1,18,21,26,27,26,29,31,32,31,32,39,39,46,48,48,44,45,45,46,36,39,39,32,
%T A172052 35,32,31,29,29,6,8,11,7,7,10,5,4,3,6,13,25,24,25,26,27,42,41,40,39,57,
%U A172052 58,59,61,64,74,87,87,91,93,99,102,103,102,101,101,101,109,107,112,116
%N A172052 a(n)=abs(A171696(n)-A002822(n)).
%C A172052 Abs(nth nonnegative number such that neither 6*k+-1 is prime minus nth number such that 6*m+-1 are both twin primes).
%Y A172052 Cf. A002808, A171696.
%K A172052 nonn,new
%O A172052 1,2
%A A172052 Juri-Stepan Gerasimov (2stepan(AT)rambler.ru), Jan 24 2010

%I A172051
%S A172051 0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,
%T A172051 1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,
%U A172051 0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0
%N A172051 Decimal expansion of 1/999999.
%F A172051 a(n)=(1-((-1)^A172050(n+4)))/2. A similar formula is given by Hieronymus Fischer in A022003.
%F A172051 a(n)= if (n+1) mod 6 = 0 then 1 else 0.
%F A172051 a(n) = A079979(n+1). [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jan 28 2010]
%Y A172051 Cf. A022003, A172050.
%K A172051 nonn,new
%O A172051 0,1
%A A172051 Mats Granvik (mats.granvik(AT)abo.fi), Jan 24 2010

%I A172050
%S A172050 0,4,8,13,20,30,46,72,116,191,320,542,924,1580,2704,4625,7900,13470,
%T A172050 22922,38928,65980,111619,188488,317758,534840,898900,1508696,2528917,
%U A172050 4233956,7080606,11828710,19741272,32916164,54835655,91276304
%N A172050 A008585+A029907.
%Y A172050 Cf. A008585, A029907, (1-((-1)^a(n+4)))/2 = A172051.
%K A172050 nonn,new
%O A172050 0,2
%A A172050 Mats Granvik (mats.granvik(AT)abo.fi), Jan 24 2010

%I A172049
%S A172049 1,1,1,3,5,7,1,3,5,7,1,3,5,7,9,11,13,15,17,1,3,5,7,9,11,13,15,17,1,3,5,
%T A172049 7,9,11,13,15,17,19,21,23,25,27,29,31,1,3,5,7,9,11,13,15,17,19,21,23,25,
%U A172049 27,29,31,1,3,5,7,9,11,13,15,17,19,21,23,25,27,29,31,33,35,37,39,41,43
%N A172049 Second bisection of (A172002=1,2,3,4,8,9,7,10)=2,4,9,10,11, - first bisection =1,3,8,7,6,: A172002(2n)-A172002(2n-1). Doubled triangle: 1; 1; 1,3,5,7; 1,3,5,7,; .Number of terms of every subsequence:1,1,4,4,9,9,16,16,=A008794(n+2).See A168342.
%C A172049 Fractal sequence (odds) from compact symmetric Janet table in A172002.Greatest number of every subsequence (twice):1,7,17,31,49,71,=A056220(n+1). See A158405.
%K A172049 nonn,uned,new
%O A172049 1,4
%A A172049 Paul Curtz (bpcrtz(AT)free.fr), Jan 24 2010

%I A172048
%S A172048 2,14,23,32,38,41,50,53,59,68,74,77,83,86,95,98,104,113,116,122,128,131,
%T A172048 137,140,143,149,158,167,173,176,179,182,185,188,194,200,203,212,215,
%U A172048 218,221,230,233,239,242,248,254,257,263,266,275,278,281,284,293
%N A172048 A104275(n) + A014076(n).
%C A172048 Alternatively: the sequence of the numbers 3*k-1 for all nonprime 2*k-1, k>=1.
%t A172048 Flatten[Table[If[PrimeQ[2*n - 1], {}, 3*n - 1], {n, 1, 100}]]
%Y A172048 Cf. A104275 , A014076
%K A172048 nonn,easy,new
%O A172048 1,1
%A A172048 Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Jan 24 2010
%E A172048 The two equivalent defns. separated by the Assoc. Edts. of the OEIS - Feb 02 2010

%I A172047
%S A172047 0,1,25,124,380,905,1841,3360,5664,8985,13585,19756,27820,38129,51065,
%T A172047 67040,86496,109905,137769,170620,209020,253561,304865,363584,430400,
%U A172047 506025,591201,686700,793324,911905,1043305,1188416,1348160,1523489
%N A172047 a(n)=(15*n^4+14*n^3-9*n^2-8*n)/12
%C A172047 Numbers:(0,1,2,3,4,5,6,7,8,9,10,..,) A007587 (0,1,13,46,110,215,371,588,...,) 25 in this sequence because: 25=13*2-(1+0); 124=46*3-(13+1+0); 380=110*4-(46+13+1+0);
%F A172047 a(n)=n*(n+1)*(15*n^2-n-8)/12
%K A172047 nonn,new
%O A172047 0,3
%A A172047 Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Jan 24 2010

%I A172045
%S A172045 0,1,17,80,240,565,1141,2072,3480,5505,8305,12056,16952,23205,31045,
%T A172045 40720,52496,66657,83505,103360,126560,153461,184437,219880,260200,
%U A172045 305825,357201,414792,479080,550565,629765,717216,813472,919105,1034705
%N A172045 a(n)=(9*n^4+10*n^3-3*n^2-4*n)/12
%C A172045 Numbers=(0,1,2,3,4,5,6,7,8,9,10 ... ) A002414=(0,1,9,30.70,135,231,364, 540 ..) sequence: 0=0*0-0; 1=1*1-0=1; 17=9*2-(1+0); 80=30*3-(9+1+0); 240=70*4-(30+9+1+0);
%K A172045 nonn,new
%O A172045 0,3
%A A172045 Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Jan 24 2010

%I A175087
%S A175087 1,1,1,0,1,1,1,1,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,1,1,1,1,1,1,0,1,1,1,
%T A175087 0,1,1,1,1,1,1,1,1,1,1,1,1,0,1
%N A175087 Number of numbers whose product of perfect divisors to n.
%C A175087 Perfect divisor of n is divisor d such that d^k = n for some k >= 1. See A175068 (product of perfect divisors of n), A175084 (possible values for product of perfect divisors of n) and A171085 (numbers m such that product of perfect divisors of x = m has no solution). a(n) = 0 or 1 for all n.
%K A175087 nonn,new
%O A175087 1,1
%A A175087 Jaroslav Krizek (jaroslav.krizek(AT)atlas.cz), Jan 24 2010

%I A175086
%S A175086 1,8,16,27,64,81,125,128,216,256,343,625,729
%N A175086 Perfect powers m such that product of perfect divisors of x = m has solution.
%C A175086 Perfect divisor of n is divisor d such that d^k = n for some k >= 1. Subsequence of A001597 (perfect powers). See A175068 (product of perfect divisors of n) and A175086 (numbers m such that product of perfect divisors of x = m has no solution).
%K A175086 nonn,new
%O A175086 1,2
%A A175086 Jaroslav Krizek (jaroslav.krizek(AT)atlas.cz), Jan 24 2010

%I A172041
%S A172041 5,29,536870909,
%T A172041 13803492693581127574869511724554050904902217944340773110325048447598589
%N A172041 Primes of the form 2^p-3 with p also prime.
%C A172041 Subsequence of A050415. p = 3,5,29,233,.. are the primes in A050414.
%K A172041 nonn,new
%O A172041 1,1
%A A172041 Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Jan 24 2010
%E A172041 Definition clarified by R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jan 28 2010

%I A175085
%S A175085 4,9,25,32,36,49,100,121,144,169,196,225,243,289,324,361,400,441,484,
%T A175085 512,529,576,676,784,841,900,961
%N A175085 Numbers m such that product of perfect divisors of x = m has no solution.
%C A175085 Perfect divisor of n is divisor d such that d^k = n for some k >= 1. Subsequence of A001597 (perfect powers). Complement of A157084.
%K A175085 nonn,new
%O A175085 1,1
%A A175085 Jaroslav Krizek (jaroslav.krizek(AT)atlas.cz), Jan 24 2010

%I A175084
%S A175084 1,2,3,5,6,7,8,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,26,27,28,29,
%T A175084 30,31,33,34,35,37,38,39,40,41,42,43,44,45,46,47,48,50,51,52,53,54,55,
%U A175084 56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78
%N A175084 Possible values for product of perfect divisors of n.
%C A175084 Perfect divisor of n is divisor d such that d^k = n for some k >= 1. All terms of this sequence occur only once. See A089723 (number of perfect divisors of n) and A175068 (product of perfect divisors of n).
%C A175084 Complement of A175085. [From Jaroslav Krizek (jaroslav.krizek(AT)atlas.cz), Jan 30 2010]
%K A175084 nonn,new
%O A175084 1,2
%A A175084 Jaroslav Krizek (jaroslav.krizek(AT)atlas.cz), Jan 24 2010

%I A172042
%S A172042 1,4,8,16,32,64,96,120,240,192,288,384,384,528,1056,1344,896,960,960,
%T A172042 1152,1728,1920,3200,2560,2560,2560,3328,3744,3456,4032,3456,6144,5632,
%U A172042 6336,5760,5760,7776,8856,13776,14784,8448,8640,9216,10752,10080,8640
%N A172042 A000010(A083553(n)).
%C A172042 Except for first term, a(n) is divisible by 4.
%p A172042 A083553 := proc(n) (ithprime(n+1)-1)*(ithprime(n)-1) ; end proc: A172042 := proc(n) numtheory[phi](A083553(n)) ; end proc: seq(A172042(n),n=1..80) ; [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jan 30 2010]
%Y A172042 Cf. A000010, A083553.
%K A172042 easy,nonn,new
%O A172042 1,2
%A A172042 Giovanni Teofilatto (g.teofilatto(AT)tiscalinet.it), Jan 24 2010
%E A172042 a(2) inserted and terms beyond a(10) added by R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jan 30 2010

%I A160748
%S A160748 2,3,5,7,37,73,337,353,373,727,733,757,3257,3373,3527,3733,7253,7523,
%T A160748 7577,7757,32233,32257,32323,32353,32377,32537,33223,33533,35227,35257,
%U A160748 35323,35327,35353,35537,35753,37273,37573,72227,72253,72337,72353
%N A160748 Primes whose digits are primes and reverse is prime.
%K A160748 nonn,base,new
%O A160748 1,1
%A A160748 Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Jan 24 2010

%I A175083
%S A175083 1,1,1,0,1,2,1,0,0,2,1,2,1,1,1,0,1,1,1,1,1,2,1,1,0,1,0,1,1,3,1,0,1,2,1,
%T A175083 0,1,1,1,1,1,2,1,1,1,1,1,1,0,1
%N A175083 Number of numbers whose sum of perfect divisors to n.
%C A175083 Perfect divisor of m is divisor d such that d^k = m for some k >= 1. See A175067 (sum of perfect divisors of n) and A175081 (values taken by the sum of perfect divisors of n (A175067) sorted into ascending order).
%K A175083 nonn,new
%O A175083 1,6
%A A175083 Jaroslav Krizek (jaroslav.krizek(AT)atlas.cz), Jan 24 2010

%I A175082
%S A175082 1,2,3,5,6,7,10,11,12,13,14,15,17,18,19,20,21,22,23,24,26,28,29,30,31,
%T A175082 33,34,35,37,38,39,40,41,42,43,44,45,46,47,48,50,51,52,53,54,55,56,57,
%U A175082 58,59,60,61,62,63,65,66,67,68,69,70,71,72,73,74,75
%N A175082 Possible values for sum of perfect divisors of n.
%C A175082 Perfect divisor of n is divisor d such that d^k = n for some k >= 1. See A175067 (sum of perfect divisors of n) and A175081 (values taken by the sum of perfect divisors of n (A175067) sorted into ascending order).
%C A175082 Complement of A001597(n+1) for n >= 1 (perfect powers >= 4). a(n) = A007916(n-1) for n >= 2. [From Jaroslav Krizek (jaroslav.krizek(AT)atlas.cz), Jan 30 2010]
%K A175082 nonn,new
%O A175082 1,2
%A A175082 Jaroslav Krizek (jaroslav.krizek(AT)atlas.cz), Jan 24 2010

%I A175081
%S A175081 1,2,3,5,6,6,7,10,10,11,12,12,13,14,15,17,18,19,20,21,22,22,23,24,26,28,
%T A175081 29,30,30,30,31,33,34,34,35,37,38,39,40,41,42,42,43,44,45,46,47,48,50,
%U A175081 51,52,53,54,55,56,56,57,58,59,60,61,62,63,65,66,67,68,69,70,71,72,73
%N A175081 Values taken by the sum of perfect divisors of n (A175067) sorted into ascending order.
%C A175081 Perfect divisor of n is divisor d such that d^k = n for some k >= 1. See A089723 (number of perfect divisors of n) and A175067 (sum of perfect divisors of n).
%K A175081 nonn,new
%O A175081 1,2
%A A175081 Jaroslav Krizek (jaroslav.krizek(AT)atlas.cz), Jan 24 2010

%I A171791
%S A171791 1,1,4,25,194,1603,15264,122316,1897710,8845133,1169435932,
%T A171791 52853978047,3193246498792,205347570309000,14534295599537024,
%U A171791 1115833257773950536,92445637289048967654,8219735646409095418617
%V A171791 1,1,-4,25,-194,1603,-15264,122316,-1897710,-8845133,-1169435932,
%W A171791 -52853978047,-3193246498792,-205347570309000,-14534295599537024,
%X A171791 -1115833257773950536,-92445637289048967654,-8219735646409095418617
%N A171791 G.f. A(x) satisfies: [x^n] A(x)^((n+1)^2) = 0 for n>1 with a(0)=a(1)=1.
%e A171791 G.f.: A(x) = 1 + x - 4*x^2 + 25*x^3 - 194*x^4 + 1603*x^5 +...
%e A171791 The coefficients in the square powers of g.f. A(x) begin:
%e A171791 A^1: [1, 1, -4, 25, -194, 1603, -15264, 122316, -1897710, ...];
%e A171791 A^4: [1, 4, -10, 56, -427, 3360, -33546, 218880, -5179834, ...];
%e A171791 A^9: [1, 9, 0, 21, -252, 1701, -25992, -2970, -7903413, ...];
%e A171791 A^16: [1, 16, 56, 0, -84, -784, -18656, -384896, -13426530, ...];
%e A171791 A^25: [1, 25, 200, 525, 0, -2695, -38600, -878150, -26292375, ...];
%e A171791 A^36: [1, 36, 486, 3000, 7821, 0, -101322, -1916352, -52357590, ...];
%e A171791 A^49: [1, 49, 980, 10241, 58898, 170079, 0, -4515000, -108626140, ...];
%e A171791 A^64: [1, 64, 1760, 27136, 256048, 1500352, 4979712, 0, -234893352,...];
%e A171791 A^81: [1, 81, 2916, 61425, 838026, 7720839, 48097152, 184870512, 0,...]; ...
%e A171791 Note how the coefficient of x^n in A(x)^((n+1)^2) = 0 for n>1.
%o A171791 (PARI) {a(n)=local(A=[1,1]);for(m=3,n+1,A=concat(A,0);A[ #A]=-Vec(Ser(A)^(m^2))[m]/m^2);A[n+1]}
%K A171791 sign,new
%O A171791 0,3
%A A171791 Paul D. Hanna (pauldhanna(AT)juno.com), Jan 24 2010

%I A171780
%S A171780 1,1,3,12,55,273,1431,7837,44726,266381,1658300,10768609,72407500,
%T A171780 498510748,3477131466,24461950817,174793982029,1294469021982,
%U A171780 10177610535232,85391010070808,741460526149745,6291905077685633
%V A171780 1,-1,3,-12,55,-273,1431,-7837,44726,-266381,1658300,-10768609,72407500,
%W A171780 -498510748,3477131466,-24461950817,174793982029,-1294469021982,
%X A171780 10177610535232,-85391010070808,741460526149745,-6291905077685633
%N A171780 G.f. F_1(x) satisfies: x = Sum_{n>=1} F_{n}(x)^n, where the n-th iteration of the g.f. is defined by: F_{n}(x) = F_{n-1}( F_1(x) ) with F_0(x) = x.
%e A171780 Let F_{n}(x) denote the n-th iteration of g.f. F_1(x), then:
%e A171780 (1) x = F_1(x)^1 + F_2(x)^2 + F_3(x)^3 + F_4(x)^4 + F_5(x)^5 +...
%e A171780 (2) F_n(x) = F_{n+1}(x) + F_{n+2}(x)^2 + F_{n+3}(x)^3 + F_{n+4}(x)^4 +...
%e A171780 where initial terms of F_n(x) for n=1..8 begin:
%e A171780 F_1(x)^1 = x - x^2 + 3*x^3 - 12*x^4 + 55*x^5 - 273*x^6 + 1431*x^7 +...
%e A171780 F_2(x)^2 = x^2 - 4*x^3 + 20*x^4 - 112*x^5 + 672*x^6 - 4226*x^7 +...
%e A171780 F_3(x)^3 = x^3 - 9*x^4 + 72*x^5 - 567*x^6 + 4485*x^7 - 35817*x^8 +...
%e A171780 F_4(x)^4 = x^4 - 16*x^5 + 192*x^6 - 2080*x^7 + 21504*x^8 +...
%e A171780 F_5(x)^5 = x^5 - 25*x^6 + 425*x^7 - 6150*x^8 + 81700*x^9 +...
%e A171780 F_6(x)^6 = x^6 - 36*x^7 + 828*x^8 - 15552*x^9 + 260400*x^10 +...
%e A171780 F_7(x)^7 = x^7 - 49*x^8 + 1470*x^9 - 34937*x^10 + 723632*x^11 +...
%e A171780 F_8(x)^8 = x^8 - 64*x^9 + 2432*x^10 - 71552*x^11 + 1802240*x^12 +...
%e A171780 The initial iterations of the g.f. begin:
%e A171780 F_2(x) = x - 2*x^2 + 8*x^3 - 40*x^4 + 224*x^5 - 1345*x^6 +...
%e A171780 F_3(x) = x - 3*x^2 + 15*x^3 - 90*x^4 + 595*x^5 - 4184*x^6 +...
%e A171780 F_4(x) = x - 4*x^2 + 24*x^3 - 168*x^4 + 1280*x^5 - 10306*x^6 +...
%e A171780 F_5(x) = x - 5*x^2 + 35*x^3 - 280*x^4 + 2415*x^5 - 21895*x^6 +...
%e A171780 F_6(x) = x - 6*x^2 + 48*x^3 - 432*x^4 + 4160*x^5 - 41923*x^6 +...
%e A171780 F_7(x) = x - 7*x^2 + 63*x^3 - 630*x^4 + 6699*x^5 - 74270*x^6 +...
%e A171780 F_8(x) = x - 8*x^2 + 80*x^3 - 880*x^4 + 10240*x^5 - 123844*x^6 +...
%o A171780 (PARI) {a(n)=local(a_n=0,G=x,F=x-x^2+sum(k=3,n-1,a(k)*x^k));if(n<1,0,if(n==1,1, for(k=2,n,G=x;for(i=1,k,G=subst(F,x,G+x*O(x^n)));a_n=a_n-polcoeff(G^k,n));a_n))}
%K A171780 sign,new
%O A171780 1,3
%A A171780 Paul D. Hanna (pauldhanna(AT)juno.com), Jan 24 2010

%I A172040
%S A172040 1,0,2,0,2,4,0,6,8,8,0,22,28,24,16,0,90,112,96,64,32,0,394,484,416,288,
%T A172040 160,64,0,1806,2200,1896,1344,800,384,128,0,8558,10364,8952,6448,4000,
%U A172040 2112,896,256,0,41586,50144,43392,31616,20160,11264,5376,2048,512,0
%N A172040 Triangle T(n,k), read by rows, given by [0,1,2,1,2,1,2,1,2,1,2,...] DELTA [1,0,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938.
%C A172040 Riordan array (1, 2x*f(x)) where f(x) is the g.f. of A001003 . Riordan production matrix is : (0, (2-x)/(1-x)).
%F A172040 Sum_[k, 0<=k<=n} T(n,k)= A006318(n).
%e A172040 Triangle begins : 1 ; 0,2 ; 0,2,4 ; 0,6,8,8 ; 0,22,28,24,16 ; ...
%Y A172040 Cf. A104219
%K A172040 nonn,tabl,new
%O A172040 0,3
%A A172040 Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Jan 23 2010

%I A171790
%S A171790 1,1,3,15,91,612,4389,32890,254475,2017356,16301164,133767543,
%T A171790 1111731933,9338434700,79155435870,676196049060,5815796869995,
%U A171790 50318860986108,437662920058980,3824609516638444,33563127932394060
%V A171790 1,1,-3,15,-91,612,-4389,32890,-254475,2017356,-16301164,133767543,
%W A171790 -1111731933,9338434700,-79155435870,676196049060,-5815796869995,
%X A171790 50318860986108,-437662920058980,3824609516638444,-33563127932394060
%N A171790 G.f. A(x) satisfies: A(x*(1+x)^3) = 1 + x.
%F A171790 G.f. A(x) satisfies: [x^n] A(x)^(3*n+k) = 0 for k=1..n-1, n>1.
%F A171790 G.f. A(x) satisfies: [x^n] A(x)^(3*n) = 3*(-1)^(n-1) for n>0.
%F A171790 G.f. A(x) satisfies: [x^n] A(x)^(4*n) = 4 for n>0.
%F A171790 G.f. A(x) satisfies: [x^n] A(x)^(5*n) = 5*C(2n-1,n) for n>0.
%F A171790 G.f. A(x) = 1 + Series_Reversion(x*(1+x)^3).
%F A171790 G.f. A(x) = [x/Series_Reversion(x*(1+x)^3)]^(1/3).
%F A171790 a(n) = 3*(-1)^(n-1)*C(4*n-1,n-1)/(4*n-1) = 3*(-1)^(n-1)*A006632(n) for n>0.
%e A171790 G.f.: A(x) = 1 + x - 3*x^2 + 15*x^3 - 91*x^4 + 612*x^5 +...
%e A171790 The coefficients in the successive powers of g.f. A(x) begin:
%e A171790 A^1: [1, 1, -3, 15, -91, 612, -4389, 32890, -254475, ...];
%e A171790 A^2: [1, 2, -5, 24, -143, 952, -6783, 50600, -390195, ...];
%e A171790 A^3: [1, 3, -6, 28, -165, 1092, -7752, 57684, -444015, ...];
%e A171790 A^4: [1, 4, -6, 28, -165, 1092, -7752, 57684, -444015, ...];
%e A171790 A^5: [1, 5, -5, 25, -150, 1001, -7140, 53295, -411125, ...];
%e A171790 A^6: [1, 6, -3, 20, -126, 858, -6188, 46512, -360525, ...];
%e A171790 A^7: [1, 7, 0, 14, -98, 693, -5096, 38760, -302841, ...];
%e A171790 A^8: [1, 8, 4, 8, -70, 528, -4004, 31008, -245157, ...];
%e A171790 A^9: [1, 9, 9, 3, -45, 378, -3003, 23868, -191862, ...];
%e A171790 A^10: [1, 10, 15, 0, -25, 252, -2145, 17680, -145350, ...];
%e A171790 A^11: [1, 11, 22, 0, -11, 154, -1452, 12584, -106590, ...];
%e A171790 A^12: [1, 12, 30, 4, -3, 84, -924, 8580, -75582, ...];
%e A171790 A^13: [1, 13, 39, 13, 0, 39, -546, 5577, -51714, ...];
%e A171790 A^14: [1, 14, 49, 28, 0, 14, -294, 3432, -34034, ...];
%e A171790 A^15: [1, 15, 60, 50, 0, 3, -140, 1980, -21450, ...];
%e A171790 A^16: [1, 16, 72, 80, 4, 0, -56, 1056, -12870, ...];
%e A171790 A^17: [1, 17, 85, 119, 17, 0, -17, 510, -7293, ...];
%e A171790 A^18: [1, 18, 99, 168, 45, 0, -3, 216, -3861, ...];
%e A171790 A^19: [1, 19, 114, 228, 95, 0, 0, 76, -1881, ...];
%e A171790 A^20: [1, 20, 130, 300, 175, 4, 0, 20, -825, ...];
%e A171790 A^21: [1, 21, 147, 385, 294, 21, 0, 3, -315, ...];
%e A171790 A^22: [1, 22, 165, 484, 462, 66, 0, 0, -99, ...];
%e A171790 A^23: [1, 23, 184, 598, 690, 161, 0, 0, -23, ...];
%e A171790 A^24: [1, 24, 204, 728, 990, 336, 4, 0, -3, ...];
%e A171790 ...
%o A171790 (PARI) {a(n)=polcoeff(1+serreverse(x*(1+x+O(x^(n+1)))^3),n)}
%o A171790 (PARI) {a(n)=if(n==0,1,3*(-1)^(n-1)*binomial(4*n-1,n-1)/(4*n-1))}
%Y A171790 Cf. A006632, A001764.
%K A171790 sign,new
%O A171790 0,3
%A A171790 Paul D. Hanna (pauldhanna(AT)juno.com), Jan 23 2010

%I A172039
%S A172039 5,7,11,13,17,19,23,29,31,37,43,47,53,59,61,67,71,73,83,89,97,101,103,
%T A172039 107,109,113,127,131,137,139,149,151,157,163,167,173,179,181,191,193,
%U A172039 197,199,211,223,227,229,233,251,257,163,271,277,181,283,293,307,311
%N A172039 Petoukhov primes, generated by M*H*M; M = 2^n circulant matrices generated from A164281, H = all inequivalent Hadamard matrices of order 2^n
%C A172039 The basic idea for the sequence was conceived by Sergey Petoukhov; coupled with
%C A172039 the strategy of using circulant matrices in M.
%D A172039 NJA Sloane, Tables of Hadamard Matrices.
%F A172039 Let M = 2^n x 2^n circulant matrices generated from rows of A164281: (1; 1,2; 1,2,4,2; 1,2,4,2,4,8,4,2;...) and H = inequivalent Hadamard matrices of order 2^n. A172039 consists of the primes extracted from the products M*H*M using all of the Hadamard matrices in orders 2^n. Last, change and (-) signs to (+).
%e A172039 The 4x4 circulant matrix using A164281 = [1,2,4,2; 2,1,2,4; 4,2,1,2; 2,4,2,1]
%e A172039 = M. The 4x4 inequivalent Hadamard matrix = [ ++++; +-+-; ++--; +--+ ] = H.
%e A172039 The product M*H*M =
%e A172039 ...
%e A172039 -7,. 1, 29, 13;
%e A172039 .1, 23, 37, 11;
%e A172039 29, 37, 47, 31;
%e A172039 13, 11, 13, 17;
%e A172039 ... Then extract all terms that are primes, becoming the ordered set, A172039.
%e A172039 Similarly, with order 16 we create a 16x16 circulant matrix M using the
%e A172039 terms (1, 2, 4, 2, 4, 8, 4, 2, 4, 8, 16, 8, 4, 8, 4, 2), (Cf. A164281).
%e A172039 Using (16.4 Hadamard matrix = H; Cf. Tables of Hadamard Matrices); we take
%e A172039 the product M*H*M, extracting the primes and putting them into the ordered
%e A172039 set. The top row of that product = (487, 95, 197, 637, 31, 241, 1085, 109,
%e A172039 355, 227, 55, 313, 31, 97, 341, 443), with the the primes: 487, 197, 31, 241,
%e A172039 109, 227, 313, 31, 97, and 443.
%Y A172039 Cf. A000040, A164281
%K A172039 nonn,new
%O A172039 1,1
%A A172039 Gary W. Adamson (qntmpkt(AT)yahoo.com), Jan 23 2010

%I A172038
%S A172038 0,1,1,0,2,1,3,1,0,1,5,2,6,1,1,0,8,1,9,4,2,1,11,1,0,1,3,6,14,1,
%T A172038 15,2,4,1,1,0,18,1,5,3,20,1,21,10,2,1,23,1,0,1,7,12,26,1,3,5,8,1,
%U A172038 29,2,30,1,1,0,4,1,33,16,10,1,35,3,36,1,5,18,2,1,39,1,0,1,41,4,6
%V A172038 0,-1,1,0,2,-1,3,1,0,-1,5,2,6,-1,1,0,8,-1,9,4,2,-1,11,1,0,-1,3,6,14,-1,
%W A172038 15,2,4,-1,1,0,18,-1,5,3,20,-1,21,10,2,-1,23,1,0,-1,7,12,26,-1,3,5,8,-1,
%X A172038 29,2,30,-1,1,0,4,-1,33,16,10,-1,35,3,36,-1,5,18,2,-1,39,1,0,-1,41,4,6
%N A172038 Smallest non-negative integer such that n+(a(n))^2 is a perfect square, or -1 if impossible
%C A172038 a(n) is -1 for all n = 4m+2
%e A172038 a(7) is 3 because 7+1=8 and 7+4=11 are not perfect squares, but 7+9=16 is.
%p A172038 A172038 := proc(n) local r,kpa,kma,a,k ; r := {} ; for kpa in numtheory[divisors](n) do kma := n/kpa ; if type(kpa-kma,'even') then a := (kpa-kma)/2 ; k := kpa- a; if a >= 0 and k >= 0 and kpa+kma = 2*k then r := r union {a}; end if; end if; end do ; if r <> {} then return min(op(r)) ; else return -1 ; end if; end proc: for n from 1 to 100 do printf("%d,",A172038(n)) ; end do : [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Feb 07 2010]
%K A172038 nonn,new
%O A172038 1,5
%A A172038 J. Lowell (jhbubby(AT)mindspring.com), Jan 23 2010
%E A172038 More terms from R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Feb 07 2010

%I A172037
%S A172037 2,5,73,167,2423,7621
%N A172037 Prime partial sums of Sophie Germain primes A005384.
%C A172037 a(1) and a(2) are themselves Sophie Germain primes. What are the next Sophie Germain prime partial sums of Sophie Germain primes? There are no more through A066819(44).
%F A172037 A000040 INTERSECTION A066819 = {p such that p is prime and SUM[i=1..k]A005384(k) is prime} = {p such that p is prime and SUM[i=1..k]{p is prime and 2p+1 is prime}.}.
%e A172037 a(1) = 2 = first Sophie Germain prime A005384(1). a(2) = 5 = sum of first two Sophie Germain primes = 2+3. a(3) = 73 = sum of first six Sophie Germain primes = 2+3+5+11+23+29.
%Y A172037 Cf. A000040, A005384, A005385, A066819, A172036.
%K A172037 easy,more,nonn,new
%O A172037 1,1
%A A172037 Jonathan Vos Post (jvospost3(AT)gmail.com), Jan 23 2010

%I A172036
%S A172036 5,23,2267,9157,26437
%N A172036 Prime partial sums of safe primes A005385.
%C A172036 a(2) = 23 is not only the 3rd partial sum of safe primes, but is itself the 4th safe prime. What are the next safe prime partial sums of safe primes [no more through 66869(41)]?
%F A172036 A000040 INTERSECTION A066869 = A000040 INTERSECTION = SUM[i=1..n]A005385 (i) = {p such that p prime and (p-1)/2 prime and p is an element of SUM[i=1..n]{p prime and (p-1)/2 prime}.}.
%e A172036 a(1) = 5 = A066869(1) is prime. a(2) = 23 = A066869(3) is prime. a(3) = 2267 = A066869(15) is prime.
%Y A172036 Cf. A005384, A005385, A066869.
%K A172036 easy,more,nonn,new
%O A172036 1,1
%A A172036 Jonathan Vos Post (jvospost3(AT)gmail.com), Jan 23 2010

%I A175080
%S A175080 2,2999,3299,5147,5981,8999,9587,10037,10427,10559,10937,11579,12889,
%T A175080 13367,14143,14591,14621,15859,16301,16871,18041,18839,18947,19661,
%U A175080 21059,21557,22229,22343,22853,23399,23957,24317,24659,25523,27179
%N A175080 Primes q (except greater of twin primes) with result 2 under iterations of {r mod (max prime p < r)} starting at r = q.
%C A175080 Subsequence of A175075. Union of a(n) and sequence A006512 (greater of twin primes) is A175075.
%e A175080 Iteration procedure for a(2) = 2999: 2999 mod 2971 = 28, 28 mod 23 = 5, 5 mod 3 = 2.
%K A175080 nonn,new
%O A175080 1,1
%A A175080 Jaroslav Krizek (jaroslav.krizek(AT)atlas.cz), Jan 23 2010

%I A175079
%S A175079 1,3,10,123,1357324
%N A175079 The smallest natural numbers m with first occurence 0, 1, 2, 3, ... for number of steps of iterations of {r mod (max prime p < r)} needed to reach 1 or 2 starting at r = m.
%C A175079 I offer a prize of 100 liters of Pilsner Urquell to the discoverer of a(5). Conjecture: a(n) is not equal A135543(n) + 1 for all n >= 1. See A175071 (natural numbers m with result 1) and A175072 (natural numbers m with result 2). See A175077 (results 1 or 2 under iterations) and A175078 (number of steps of iterations).
%e A175079 Iteration for a(4) = 1357324 has 4 steps: 1357324 mod 1357201 = 123, 123 mod 113 = 10, 10 mod 7 = 3, 3 mod 2 = 1.
%K A175079 nonn,new
%O A175079 0,2
%A A175079 Jaroslav Krizek (jaroslav.krizek(AT)atlas.cz), Jan 23 2010
%E A175079 Jaroslav Krizek (jaroslav.krizek(AT)atlas.cz), Jan 30 2010

%I A175078
%S A175078 0,0,1,1,1,1,1,1,1,2,2,1,1,1,1,2,2,1,1,1,1,2,2,1,1,2,2,2,2,1,1,1,1,2,2,
%T A175078 2,2,1,1,2,2,1,1,1,1,2,2,1,1,2,2,2,2,1,1,2,2,2,2,1,1,1,1,2,2,2,2,1,1,2,
%U A175078 2,1,1,1,1,2,2,2,2,1,1,2,2,1,1,2,2,2,2,1,1,2,2,2,2,2,2,1,1,2
%N A175078 Number of steps of iterations of {r mod (max prime p < r)} needed to reach 1 or 2 starting at r = n.
%C A175078 a(123) = 3 (first occurence of value 3), a(1357324) = 4 (first occurence of value 4). I offer a prize of 100 liters of Pilsner Urquell to the discoverer of value of first occurence of value 5. See A175071 (natural numbers m with result 1) and A175072 (natural numbers m with result 2). See A175077 = results 1 or 2 under iterations of {r mod (max prime p < r)} starting at r = n.
%C A175078 Essentially the same as A121561. [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jan 28 2010]
%e A175078 a(123) = 3; iteration procedure for n = 123: 123 mod 113 = 10, 10 mod 7 = 3, 3 mod 2 = 1.
%K A175078 nonn,new
%O A175078 1,10
%A A175078 Jaroslav Krizek (jaroslav.krizek(AT)atlas.cz), Jan 23 2010

%I A175077
%S A175077 1,2,1,1,2,1,2,1,2,1,1,1,2,1,2,1,1,1,2,1,2,1,1,1,2,1,1,2,1,1,2,1,2,1,1,
%T A175077 2,1,1,2,1,1,1,2,1,2,1,1,1,2,1
%N A175077 Results 1 or 2 under iterations of {r mod (max prime p < r)} starting at r = n.
%C A175077 See A175071 (natural numbers m with result 1) and A175072 (natural numbers m with result 2).
%e A175077 Iteration procedure for n = 6: 6 mod 5 = 1. Iteration procedure for n = 7: 7 mod 5 = 2.
%K A175077 nonn,uned,new
%O A175077 1,2
%A A175077 Jaroslav Krizek (jaroslav.krizek(AT)atlas.cz), Jan 23 2010

%I A172035
%S A172035 5,0,2,2,9,3,2,5,3,2,7,2,4,5,2,2,5,2,3,6,2,2,2,2,4,8,4,2,2,4,2,8,2,3,2,
%T A172035 2,4,4,6,2,4,2,10,3,4,2,3,2,4,3,5,6,3,4,4,2,2,2,2,2,3,4,3,3,3,5,3,3,8,2,
%U A172035 3,12,2,3,2,5,2,3,8,16,8,3,4,2,3,2,4,2,2,5,7,4,3,8,3,2,6,2,3,6,2,2,10
%N A172035 Smallest exponent k > 1 that sum of digits of k-th power of the n-th prime is a prime (n=1,2,...) or 0 if no such k exists
%C A172035 k = 1 is the "trivial" case: sod(prime(n)) = prime(m)
%C A172035 n = 2, prime(2) = 3: 3^k is for k > 1 a multiple of 3^2
%D A172035 M. Fujiwara, Y. Ogawa: Introduction to truly beautiful Mathematics, Chikuma Shobo, Tokyo 2005
%D A172035 Theo Kempermann, Zahlentheoretische Kostproben, Harri Deutsch, 2. aktualisierte Auflage 2005
%D A172035 Hans Schubart: Einfuehrung in die klassische und moderne Zahlentheorie, Vieweg, Braunschweig 1974
%e A172035 sod(2^5)=5, sod(5^2)=7, sod(7^2)=13, sod(11^9)=53, sod(13^3)=19, sod(17^2)=19
%e A172035 sod(19^5)=37, sod(23^3)=17, sod(29^2)=13, sod(31^7)=31, sod(37^2)=19
%e A172035 sod(41^4)=31, sod(43^5)=31, sod(47^2)=13, sod(53^2)=19, sod(59^5)=47
%e A172035 sod(61^2)=13, sod(67^3)=19, sod(71^6)=37, sod(73^2)=19, sod(79^2)=13
%e A172035 sod(83^2)=31, sod(89^2)=19, sod(97^4)=43, sod(101^8)=67, sod(103^4)=31
%e A172035 sod(107^2)=19, sod(109^2)=19, sod(113^4)=31, sod(127^2)=19, sod(131^8)=61
%e A172035 sod(137^2)=31, sod(139^3)=37, sod(149^2)=7, sod(151^2)=13, sod(157^4)=31
%e A172035 sod(163^4)=37, sod(167^6)=73, sod(173^2)=31, sod(179^4)=37, sod(181^2)=19
%e A172035 sod(191^10)=97, sod(193^3)=37, sod(197^4)=37, sod(199^2)=19, sod(211^3)=37
%e A172035 sod(223^2)=31, sod(227^4)=43, sod(229^3)=37
%o A172035 (MAGMA) S:=[ 5, 0 ]; for n in [3..103] do j:=2; while not IsPrime(&+Intseq(NthPrime(n)^j)) do j+:=1; end while; Append(~S, j); end for; S; [From Klaus Brockhaus (klaus-brockhaus(AT)t-online.de), Jan 29 2010]
%Y A172035 Cf. A046704, A007605
%Y A172035 Cf. A172216. [From Klaus Brockhaus (klaus-brockhaus(AT)t-online.de), Jan 29 2010]
%K A172035 easy,nonn,new
%O A172035 1,1
%A A172035 Ulrich Krug (leuchtfeuer37(AT)gmx.de), Jan 23 2010
%E A172035 More terms from Klaus Brockhaus (klaus-brockhaus(AT)t-online.de), Jan 29 2010

%I A175076
%S A175076 9,15,21,25,28,33,36,39,45,49,52,55,58,63,66,69,75,78,81,85,88,91,94,96,
%T A175076 99,105,111,115,118,120,122,126,129,133,136,141,144,146,148
%N A175076 Composites c which end at 2 under iterations of {r mod (max prime p < r)} starting at r = c.
%C A175076 Subsequence of A175072. Union of a(n) and A175075 is A175072.
%e A175076 Iteration procedure for a(2) = 15: 15 mod 13 = 2. Iteration procedure for a(5) = 28: 28 mod 23 = 5, 5 mod 3 = 2.
%K A175076 nonn,new
%O A175076 1,1
%A A175076 Jaroslav Krizek (jaroslav.krizek(AT)atlas.cz), Jan 23 2010

%I A133736
%S A133736 1,1,2,3,7,15,52,236,2018,33044,1181670,87720798,12886156666,
%T A133736 3633055848955,1944000061673516,1967881435350411681,
%U A133736 3768516013573481061951,13670271805989797561408684
%N A133736 Number of graphs on n unlabeled nodes that have an Eulerian cycle, i.e. a cycle that goes through every edge in the graph exactly once.
%C A133736 Any such graph consists of a single connected Euler graph (see A003049) plus a number of isolated vertices.
%F A133736 a(n) = Sum_{k=1..n} A003049(k).
%Y A133736 A variant of A002854. See also A003049.
%K A133736 nonn,new
%O A133736 1,3
%A A133736 N. J. A. Sloane (njas(AT)research.att.com), based on email from Max Alekseyev, Jan 28 2010
%E A133736 Edited and extended by Max Alekseyev (maxale(AT)gmail.com), Jan 28 2010

%I A170907
%S A170907 1,4,8,13,20,28,37,47,59,73,87,101,118,137,156,176,198,223,248,271,299,
%T A170907 328,357,386,418,454,489,522,558,598,638,678,720,766,812,858,907,956,
%U A170907 1004,1048,1104,1161,1217,1268,1325,1386,1446,1505,1567,1635,1703,1765
%N A170907 Rows sums in triangle A170906.
%H A170907 David Applegate, Omar E. Pol and N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/tooth.pdf">The Toothpick Sequence and Other Sequences from Cellular Automata</a>
%H A170907 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/toothlist.html">Catalog of Toothpick and Cellular Automata Sequences in the OEIS</a>
%K A170907 nonn,new
%O A170907 1,2
%A A170907 N. J. A. Sloane (njas(AT)research.att.com), Jan 24 2010
%E A170907 More terms from R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jan 28 2010

%I A170906
%S A170906 1,1,2,1,1,2,2,2,1,1,2,2,4,1,2,1,1,2,2,4,2,2,3,3,1,1,2,2,4,2,4,5,4,
%T A170906 1,2,1,1,2,2,4,2,4,6,6,1,2,3,3,1,1,2,2,4,2,4,6,8,1,2,3,5,3,3,1,1,2,
%U A170906 2,4,2,4,6,8,2,2,3,5,5,3,5,4,1,1,2,2,4,2,4,6,8,2,4,5,6,7,6,6,4,1,2,1
%N A170906 Triangle read by rows: T(n,k) = number of cells that at turned from OFF to ON at stage k of the cellular automaton in the 30-60-90 triangle of hexagons defined in Comments.
%C A170906 Consider the tiling of the plane by hexagons, where each cell has 6 neighbors, as in the A151723, A151724, A170905.
%C A170906 Assume the hexagons are oriented so that each one has a pair of vertical edges.
%C A170906 Consider the (30 deg., 60 deg., 90 deg.) triangle of hexagons with n hexagons along the short side, along the X-axis, 2n-1 hexagons along the hypothenuse and n hexagons separated by single edges along the middle side, along the Y-axis.
%C A170906 Initially all cells are OFF. At stage 1, the cell in the 60 degree corner is turned ON; thereafter, a cell is turned ON if it has exactly one ON neighbour in the triangle. Once a cell is ON it stays ON.
%C A170906 T(n,k) is the number of cells that are turned from OFF to ON at stage k (1 <= k <= 2n-1).
%C A170906 The rows converge to A170905. The rows sums give A170907.
%H A170906 David Applegate, Omar E. Pol and N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/tooth.pdf">The Toothpick Sequence and Other Sequences from Cellular Automata</a>
%H A170906 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/toothlist.html">Catalog of Toothpick and Cellular Automata Sequences in the OEIS</a>
%e A170906 Triangle begins:
%e A170906 1
%e A170906 1 2 1
%e A170906 1 2 2 2 1
%e A170906 1 2 2 4 1 2 1
%e A170906 1 2 2 4 2 2 3 3 1
%e A170906 1 2 2 4 2 4 5 4 1 2 1
%e A170906 1 2 2 4 2 4 6 6 1 2 3 3 1
%e A170906 1 2 2 4 2 4 6 8 1 2 3 5 3 3 1
%e A170906 1 2 2 4 2 4 6 8 2 2 3 5 5 3 5 4 1
%e A170906 1 2 2 4 2 4 6 8 2 4 5 6 7 6 6 4 1 2 1
%e A170906 ...
%e A170906 Row n = 4, [1 2 2 4 1 2 1], corresponds to the sequence of cells being turned ON shown in the following triangle (X denotes a cell that stays OFF). The hexagons have to be imagined.
%e A170906 7
%e A170906 .6
%e A170906 6.5
%e A170906 .X.4
%e A170906 X.4.3
%e A170906 .4.X.2
%e A170906 4.3.2.1
%Y A170906 Cf. A151723, A151724, A170905, A170907.
%K A170906 nonn,tabf,new
%O A170906 1,3
%A A170906 N. J. A. Sloane (njas(AT)research.att.com), Jan 24 2010

%I A168252
%S A168252 1,3,5,7,10,11,13,14,15,17,19,21,22,26,29,31,33,34,35,38,39,41,43,46,
%T A168252 51,55,57,58,59,61,62,65,66,69,70,71,73,74,77,78,82,85,86,87,91,93,94,
%U A168252 95,101,103,105,106,107,109,110,111,115,118,119,122,123,129,130,133
%N A168252 Numbers n such that n is square-free and n is non-single or nonisolated.
%Y A168252 Cf. A000469 (nonprime square-free numbers), A005117 (square-free numbers), A120944 (composite square-free numbers), A167707 (non-single or nonisolated numbers).
%K A168252 nonn,new
%O A168252 1,2
%A A168252 Juri-Stepan Gerasimov (2stepan(AT)rambler.ru), Nov 21 2009
%E A168252 Edited by N. J. A. Sloane, Jan 23 2010

%I A175075
%S A175075 2,5,7,13,19,31,43,61,73,103,109,139,151,181,193,199,229,241,271,283,
%T A175075 313,349,421,433,463,523,571,601,619,643,661,811,823,829,859,883,1021
%N A175075 Primes q with result 2 under iterations of {r mod (max prime p < r)} starting at r = q.
%C A175075 a(1) = 2, a(n) = A006512(n-1) for 2 <= n <= 82, a(83) = 2999. Sequence is union of sequence A006512 (greater of twin primes) and sequence {2, 2999, 3299, 5147, 5981, 8999, 9587, 10037, 10427, ... }. Subsequence of A175072. Primes q with some results of {2, 28, 36, 52, 58, 66, ... } under first step of iteration of {r mod (max prime p < r)} starting at r = q, i.e. number 2 and primes q such that diference q and previous prime is equal to some of the values 2, 28, 36, 52, 58, 66, ...
%C A175075 Union a(n) and A175076 is A175172. [From Jaroslav Krizek (jaroslav.krizek(AT)atlas.cz), Jan 30 2010]
%e A175075 Iteration procedure for a(5) = 19: 19 mod 17 = 2. Iteration procedure for a(83) = 2999: 2999 mod 2971 = 28, 28 mod 23 = 5, 5 mod 3 = 2.
%K A175075 nonn,new
%O A175075 1,1
%A A175075 Jaroslav Krizek (jaroslav.krizek(AT)atlas.cz), Jan 23 2010

%I A175074
%S A175074 1,4,6,8,10,12,14,16,18,20,22,24,26,27,30,32,34,35,38,40,42,44,46,48,50,
%T A175074 51,54,56,57,60,62,64,65,68,70,72,74,76,77,80,82,84,86,87,90,92,93,95,
%U A175074 98,100,102
%N A175074 Nonprimes b with result 1 under iterations of {r mod (max prime p < r)} starting at r = b.
%C A175074 Subsequence of A175071. Union of a(n) and A175073 is A175071.
%C A175074 Union of a(n) and A175073 is A175071. [From Jaroslav Krizek (jaroslav.krizek(AT)atlas.cz), Jan 30 2010]
%e A175074 Iteration procedure for a(5) = 10: 10 mod 7 = 3, 3 mod 2 = 1.
%K A175074 nonn,new
%O A175074 1,2
%A A175074 Jaroslav Krizek (jaroslav.krizek(AT)atlas.cz), Jan 23 2010

%I A175073
%S A175073 3,11,17,23,29,37,41,47,53,59,67,71,79,83,89,97,101,107,113,127,131,137,
%T A175073 149,157,163,167,173,179,191,197,211,223,227,233,239,251,257,263,269,
%U A175073 277,281,293,307
%N A175073 Primes q with result 1 under iterations of {r mod (max prime p < r)} starting at r = q.
%C A175073 Subsequence of A175071.
%C A175073 Union of a(n) and A175074 is A175071. [From Jaroslav Krizek (jaroslav.krizek(AT)atlas.cz), Jan 30 2010]
%C A175073 The terms in A025584 but not in here are 2, 2999, 3299, 5147, 5981, 8999, 9587... , apparently those listed in A175080. [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Feb 01 2010]
%e A175073 Iteration procedure for a(2) = 11: 11 mod 7 = 4, 4 mod 3 = 1.
%K A175073 nonn,new
%O A175073 1,1
%A A175073 Jaroslav Krizek (jaroslav.krizek(AT)atlas.cz), Jan 23 2010

%I A175072
%S A175072 2,5,7,9,13,15,19,21,25,28,31,33,36,39,43,45,49,52,55,58,61,63,66,69,73,
%T A175072 75,78,81,85,88,91,94,96,99,103,105,109,111,115,118,120,122,126,129,133,
%U A175072 136,139,141,144,146,148,151
%N A175072 Natural numbers m with result 2 under iterations of {r mod (max prime p < r)} starting at r = m.
%C A175072 Complement of A175071.
%C A175072 Union of A175072 and A175176. [From Jaroslav Krizek (jaroslav.krizek(AT)atlas.cz), Jan 30 2010]
%e A175072 Iteration procedure for a(6) = 15: 15 mod 13 = 2. Iteration procedure for a(10) = 28: 28 mod 23 = 5, 5 mod 3 = 2.
%K A175072 nonn,new
%O A175072 1,1
%A A175072 Jaroslav Krizek (jaroslav.krizek(AT)atlas.cz), Jan 23 2010

%I A175071
%S A175071 1,3,4,6,8,10,11,12,14,16,17,18,20,22,23,24,26,27,29,30,32,34,35,37,38,
%T A175071 40,41,42,44,46,47,48,50,51,53,54,56,57,59,60,62,64,65,67,68,70,71,72,
%U A175071 74,76,77,79,80,82,83,84,86,87,89,90,92,93,95,97,98,100,101,102
%N A175071 Natural numbers m with result 1 under iterations of {r mod (max prime p < r)} starting at r = m.
%C A175071 Complement of A175072. Union of A175073 and A175074. [From Jaroslav Krizek (jaroslav.krizek(AT)atlas.cz), Jan 30 2010]
%e A175071 Iteration procedure for a(6) = 10: 10 mod 7 = 3, 3 mod 2 = 1. Iteration procedure for a(7) = 11: 11 mod 7 = 4, 4 mod 3 = 1.
%K A175071 nonn,new
%O A175071 1,2
%A A175071 Jaroslav Krizek (jaroslav.krizek(AT)atlas.cz), Jan 23 2010

%I A175070
%S A175070 0,0,0,2,0,0,0,2,3,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,5,0,3,0,0,0,0,2,0,0,0,
%T A175070 6,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,14,0,0,0,0,0,0,
%U A175070 0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,10
%N A175070 a(n) = sum of perfect divisors of n - n. Perfect divisor of n is divisor d such that d^k = n for some k >= 1.
%C A175070 a(n) = A175067(n) - n. a(1) = 0, for n >=2: a(n) = sum of perfect divisors of n less than n. a(n) > 0 for perfect powers n = A001597(m) for m > 2.
%K A175070 nonn,new
%O A175070 1,4
%A A175070 Jaroslav Krizek (jaroslav.krizek(AT)atlas.cz), Jan 23 2010

%I A175069
%S A175069 1,1,1,2,1,1,1,2,3,1,1,1,1,1,1,8,1,1,1,1,1,1,1,1,5,1,3,1,1,1,1,2,1,1,1,
%T A175069 6,1,1,1,1,1,1,1,1,1,1,1,1,7,1,1,1,1,1,1,1,1,1,1,1,1,1,1,64,1,1,1,1,1,1,
%U A175069 1,1,1,1,1,1,1,1,1,1,27,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,10
%N A175069 a(n) = product of perfect divisors of n / n. Perfect divisor of n is divisor d such that d^k = n for some k >= 1.
%C A175069 a(n) = A175068(n) / n. a(n) > 1 for perfect powers n = A001597(m) for m > 2.
%K A175069 nonn,new
%O A175069 1,4
%A A175069 Jaroslav Krizek (jaroslav.krizek(AT)atlas.cz), Jan 23 2010

%I A175068
%S A175068 1,2,3,8,5,6,7,16,27,10,11,12,13,14,15,128,17,18,19,20,21,22,23,24,125,
%T A175068 26,81,28,29,30,31,64,33,34,35,216,37,38,39,40,41,42,43,44,45,46,47,48,
%U A175068 343,50,51,52,53,54,55,56,57,58,59,60,61,62,63,4096,65,66,67,68,69,70
%N A175068 a(n) = product of perfect divisors of n. Perfect divisor of n is divisor d such that d^k = n for some k >= 1.
%C A175068 a(n) > n for perfect powers n = A001597(m) for m > 2.
%F A175068 a(n) = A175069(n) * n. [From Jaroslav Krizek (jaroslav.krizek(AT)atlas.cz), Jan 24 2010]
%e A175068 For n = 8: a(8) = 10; there are two perfect divisors of 8: 2 and 8; their product is 16.
%K A175068 nonn,new
%O A175068 1,2
%A A175068 Jaroslav Krizek (jaroslav.krizek(AT)atlas.cz), Jan 23 2010

%I A175067
%S A175067 1,2,3,6,5,6,7,10,12,10,11,12,13,14,15,22,17,18,19,20,21,22,23,24,30,26,
%T A175067 30,28,29,30,31,34,33,34,35,42,37,38,39,40,41,42,43,44,45,46,47,48,56,
%U A175067 50,51,52,53,54,55,56,57,58,59,60,61,62,63,78,65,66,67,68,69,70,71,72
%N A175067 a(n) = sum of perfect divisors of n. Perfect divisor of n is divisor d such that d^k = n for some k >= 1.
%C A175067 a(n) > n for perfect powers n = A001597(m) for m > 2.
%F A175067 a(n) = A175070(n) + n. [From Jaroslav Krizek (jaroslav.krizek(AT)atlas.cz), Jan 24 2010]
%e A175067 For n = 8: a(8) = 10; there are two perfect divisors of 8: 2 and 8; their sum is 10.
%K A175067 nonn,new
%O A175067 1,2
%A A175067 Jaroslav Krizek (jaroslav.krizek(AT)atlas.cz), Jan 23 2010

%I A175066
%S A175066 1,2,3,2,3,2,2,3,3,2,2,5,3,2,2,3,3,2,2,2,3,2,3,2,3,4,2,2,3,2,2,2,2,5,2,
%T A175066 2,3,2,5,2,2,2,2,3,5,2,2,2,3,3,2,2,2,3,2,2,2,3,2,2,2,2,3,2,3,3,2,2,3,2,
%U A175066 2,2,3,2,2,2,3,2,2,2,2,3,2,2,2,3,2,2,7,2,2,2,2,2,2,3,2,2,2,2,2,2,2,3,2
%N A175066 a(1) = 1, for n >= 2: a(n) = number of of ways h to write perfect powers A117453 (n) as m^k (m >= 2, k >= 2).
%C A175066 Perfect powers with first occurrence of h >= 2: 16, 64, 65536, 4096, ...
%e A175066 For n = 12; A117453 (12) = 5; there are 5 ways to write 4096 as m^k: 64^2 = 16^3 = 8^4 = 4^6 = 2^12.
%e A175066 729=27^2=9^3=3^6 and 1024=32^2=4^5=2^10 yield a(8)=a(9)=3. [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jan 24 2010]
%K A175066 nonn,new
%O A175066 1,2
%A A175066 Jaroslav Krizek (jaroslav.krizek(AT)atlas.cz), Jan 23 2010
%E A175066 Corrected and extended by R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jan 24 2010

%I A175065
%S A175065 4,16,64,65536,4096
%N A175065 Perfect powers m with first occurrence of number of ways n to write perfect powers as m^k (m >= 2, k >= 1) for n >= 2.
%C A175065 a(7) = ?, a(8) = 16777216. See A175064, A001597.
%e A175065 For n = 6; a(6) = 4096; there are 6 ways to write 4096 as m^k: 4096^1 = 64^2 = 16^3 = 8^4 = 4^6 = 2^12.
%K A175065 nonn,new
%O A175065 2,1
%A A175065 Jaroslav Krizek (jaroslav.krizek(AT)atlas.cz), Jan 23 2010

%I A175064
%S A175064 1,2,2,2,3,2,2,2,2,2,4,3,2,2,2,2,2,2,2,2,2,2,4,2,2,2,2,2,2,2,3,2,2,3,2,
%T A175064 4,2,2,2,2,2
%N A175064 a(1) = 1 for n >= 2: a(n) = number of ways h to write perfect powers A001597(n) as m^k (m >= 2, k >= 1).
%C A175064 Perfect powers with first occurrence of h >= 2: 4, 16, 64, 65536, 4096, ...
%e A175064 For n = 11: A001597(11) = 64; there are 4 ways to write 64 as m^k: 64^1 = 8^2 = 4^3 = 2^6.
%K A175064 nonn,new
%O A175064 1,2
%A A175064 Jaroslav Krizek (jaroslav.krizek(AT)atlas.cz), Jan 23 2010

%I A172034
%S A172034 23,52,111,172,239,310,389,472,581,718,857,1006,1199,1426,1659,1898,
%T A172034 2149,2406,2675,2946,3223,3516,3823,4134,4451,4810,5189,5572,5961,6358,
%U A172034 6759,7178,7609,8058,8519,8982,9449,9928,10427,10930,11451,12008,12571
%N A172034 Partial sums of Pillai primes (A063980).
%C A172034 The values alternate between odd and even. The first prime partial sum of Pillai primes is a(5) = 23 + 29 + 59 + 61 + 67 = 239. The second prime partial sum is a(7) = 389. The next such primes are a(11) = 857 (= the 72nd Pillai prime), a(23) = 3823, a(25) = 4451, a(27) = 5189. The coincidence which prompted this sequence is that the 266th Pillai prime is a(23), the sum of the first 23 Pillai primes. Curiously, 23 is the smallest Pillai prime. What are the next such Pillai primes in the partial sum?
%F A172034 a(n) = SUM[i=i..n]A063980(i) = SUM[i=i..n] {p: p prime and there exists an integer m such that m!+1 is 0 mod p and p is not 1 mod m}.
%e A172034 a(1) = 23 because 23 is the first Pillai prime A063980(1). a(2) = 52 because 23+29 = 52 is the sum of the first two Pillai primes A063980(1)+A063980(2).
%K A172034 nonn,new
%O A172034 1,1
%A A172034 Jonathan Vos Post (jvospost3(AT)gmail.com), Jan 23 2010
%E A172034 More terms from R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jan 24 2010

%I A172033
%S A172033 1,1,1,1,1,1,2,2,2,2,3,3,4,4,5,5,6,6,8,8,10,10,12,12,15,15,18,18,22,22,
%T A172033 27,27,32,32,38,38,46,46,54,54,64,64,76,76,89,89,104,104,122,122,142,
%U A172033 142,165,165,192,192,222,222,256,256,296,296,340,340,390,390,448,448
%N A172033 Number of partitions of n into distinct parts that are 1 or even, i.e. into distinct terms of A004277.
%C A172033 A000009 repeated: a(n) = A000009(floor(n/2)).
%e A172033 a(12) = #{12, 10+2, 8+4, 6+4+2} = 4;
%e A172033 a(13) = #{12+1, 10+2+1, 8+4+1, 6+4+2+1} = 4;
%e A172033 a(14) = #{14, 12+2, 10+4, 8+6, 8+4+2} = 5.
%Y A172033 Cf. A025065.
%K A172033 nonn,new
%O A172033 0,7
%A A172033 Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Jan 23 2010

%I A172032
%S A172032 0,1,3,19,19,379,379,3539,3539,42461,42461
%N A172032 a(n) is numerator of c(n+1)-2c(n)=(A027641/A0267642):0,1,3/2,19/6,19/3,379/30,379/15,3539/70,3539/35,42461/210,42461/105, .
%C A172032 Denominators of submitted A172030/A172031 are the same. Definition in A172031 must be corrected (A164555 instead of A027641),but values are good. (With or without denominators A171031) A172030 and a(n) are companions. From Bernoulli numbers to A166687=0,A131577=0,0,A000079 (third part: ( (A172030/A172031)=0,1,5/2,31/6,31/3, - (a(n)/A172031)=0,1,3/2,19/6,19/3, )=A166687 ).
%K A172032 nonn,uned,frac,new
%O A172032 0,3
%A A172032 Paul Curtz (bpcrtz(AT)free.fr), Jan 23 2010

%I A172031
%S A172031 1,1,2,6,3,30,15,70,35,210
%N A172031 a(n) is denominator of c(n+1)-2c(n)=(A027641/A027642 companion of A164555/A27642):0,1,5/2,31/6,31/3,619/30,619/15, .Numerators:submitted A172030.
%C A172031 From Bernoulli numbers to A166687=0,A131577=0,0,A000079 (second part).
%K A172031 nonn,uned,frac,new
%O A172031 0,3
%A A172031 Paul Curtz (bpcrtz(AT)free.fr), Jan 23 2010

%I A172029
%S A172029 3,37,4219,53412500,8558685628987501
%N A172029 Numbers n such that [a(n-1)]^3+a(n) is a cube
%e A172029 3^3+37=4^3; 37^3+4219=38^3
%K A172029 nonn,new
%O A172029 1,1
%A A172029 Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Jan 23 2010

%I A172030
%S A172030 0,1,5,31,31,619,619,5779,5779,69341,69341
%N A172030 a(n) is numerator of c(n)-2c(n+1)=(A164555/A027642):0,1,5/2,31/6,31/3,619/30,619/15,5779/70,5779/35, .
%C A172030 Denominators:1,1,2,6,3,30,15,70,35,210,105, . From Bernoulli numbers to A166687=0,A131577=0,0,A000079 (first part).
%K A172030 nonn,uned,frac,new
%O A172030 0,3
%A A172030 Paul Curtz (bpcrtz(AT)free.fr), Jan 23 2010

%I A172028
%S A172028 2,19,1141,3909070,45842496521911,6304603462084403923798841497
%N A172028 Numbers n such that [a(n-1)]^3+a(n) is a cube
%F A172028 a(n) = 1+3*a(n-1)*(a(n-1)+1). [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jan 25 2010]
%e A172028 2^3+19=3^3; 19^3+1141=20^3
%p A172028 A172028 := proc(n) option remember; if n <=2 then op(n,[2,19]) : else 1+3*procname(n-1)*(procname(n-1)+1); end if; end: seq(A172028(n),n=1..8) ; [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jan 25 2010]
%K A172028 nonn,new
%O A172028 1,1
%A A172028 Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Jan 23 2010

%I A172027
%S A172027 0,1,7,169,86191,22286924017,1490120946485455020919
%N A172027 Numbers n such that [a(n-1)]^3+a(n) is a cube
%K A172027 nonn,new
%O A172027 0,3
%A A172027 Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Jan 23 2010

%I A172026
%S A172026 1,0,1,1,0,1,0,2,0,1,3,0,3,0,1,0,7,0,4,0,1,12,0,12,0,5,0,1,0,30,0,18,0,
%T A172026 6,0,1,55,0,55,0,25,0,7,0,1,0,143,0,88,0,33,0,8,0,1,273,0,273,0,130,0,
%U A172026 42,0,9,0,1,0,728,0,455,0,182,0,52,0,10,0,1,1428,0,1428,0,700,0,245,0
%N A172026 Riordan array (f(x^2), x*f(x^2)) where f(x) is the g.f. of A001764.
%C A172026 Another version of A110616. Riordan production matrix is: (x/(1-x^2), 1/(1-x^2)).
%F A172026 Sum_{k, 0<=k<=n} T(n,k)= A047749(n+1).
%e A172026 Triangle begins : 1 ; 0,1 ; 1,0,1 ; 0,2,0,1 ; 3,0,3,0,1 ; 0,7,0,4,0,1 ; 12,0,12,0,5,0,1 ; ...
%Y A172026 Cf. A047749, A092276, A143603
%K A172026 nonn,tabl,new
%O A172026 0,8
%A A172026 Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Jan 23 2010

%I A172025
%S A172025 1,4,16,62,239,920,3544,13672,52834,204528,793092,3080226,11980667,
%T A172025 46662704,181971248,710454896,2776717742,10863073784,42537035408,
%U A172025 166704021596,653827252022,2566222449104,10079023179536,39611016586832
%N A172025 Sequence whose G.f is given by: f(z)=(2/(3*sqrt(1-4*z)-1+4*z))*((1-sqrt(1-4*z))/(2*z))^k with k=3.
%C A172025 This sequence is the third diagonal below the main diagonal (which itself is A026641) in the array which grows with "Pascal rule" given here by rows:
%C A172025 1,0,1,0,1,0,1,0,1,0,1,0,1,0,
%C A172025 1,1,1,1,1,1,1,1,1,1,1,1,1,1,
%C A172025 1,1,2,2,3,3,4,4,5,5,6,6,7,7,
%C A172025 1,2,4,6,9,12,16,20,25,30,
%C A172025 1,3,7,13,22,34,50,70,95.
%C A172025 The MAPLE programs give the first diagonals of this array.
%F A172025 a(n)=sum('(-1)^(p)*binomial(2*n+k-p,n-p)',p=0..n) (with k=3)
%e A172025 a(4)=C(11,4)-C(10,3)+C(9,2)-C(8,1)+C(7,0)=330-120+36-8+1=239
%p A172025 for k from 0 to 20 do for n from 0 to 40 do a(n):=sum('(-1)^(p)*binomial(2*n-p+k,n-p)',p=0..n): od:seq(a(n),n=0..40):od; for k from 0 to 40 do taylor((2/(3*sqrt(1-4*z)-1+4*z))*((1-sqrt(1-4*z))/(2*z))^k,z=0,40+k):od;
%Y A172025 Cf. A026641, A014300, A014301
%K A172025 easy,nonn,new
%O A172025 0,2
%A A172025 Richard Choulet (richardchoulet(AT)yahoo.fr), Jan 23 2010

%I A171777
%S A171777 1,1,3,23,473,27057,4102027,1539365191,1365364095921,2783117747148641,
%T A171777 12795599930746180499,130882205973999096722679,
%U A171777 2946911413331842739385098377,144807670567304192694224250060817
%N A171777 E.g.f.: A(x) = exp( Sum_{n>=1} 2^(n(n-1)/2) * x^n/n ).
%e A171777 E.g.f.: A(x) = 1 + x + 3*x^2/2! + 23*x^3/3! + 473*x^4/4! +...
%e A171777 log(A(x)) = x + 2*x^2/2 + 2^3*x^3/3 + 2^6*x^4/4 + 2^10*x^5/5 +...
%o A171777 (PARI) {a(n)=n!*polcoeff(exp(sum(m=1, n+1, 2^(m*(m-1)/2)*x^m/m)+x*O(x^n)), n)}
%Y A171777 Cf. A171776, A155200.
%K A171777 nonn,new
%O A171777 0,3
%A A171777 Paul D. Hanna (pauldhanna(AT)juno.com), Jan 23 2010

%I A172024
%S A172024 1,2,3,4,7,9,10,11,12,13,14,15,16,20,21,23,26,27,30,35,36,37,39,40,43,
%T A172024 53,55,67,72,85,97,98,123,130,131,132,138,141,146,180,182,185,188,192,
%U A172024 201,225,231,236,240,248,252,254,276,300,322,326,346,372,401,413,424
%N A172024 a(n) = smallest number > a(n-1) such that a(1)*a(2)*...*a(n) + 1 or a(1)*a(2)*...*a(n) - 1 is prime.
%H A172024 <a href="http://www.primecurios.com">Prime Curios!</a>
%t A172024 For[n = 2; res = {1}; z = 1, n <= 1000, n++, If[PrimeQ[n z + 1] || PrimeQ[n z - 1], AppendTo[res, n]; z *= n]] res [From J. Mulder (jasper.mulder(AT)planet.nl), Jan 28 2010]
%K A172024 nonn,new
%O A172024 1,2
%A A172024 G. L. Honaker, Jr. (honak3r(AT)gmail.com), Jan 22 2010
%E A172024 Terms beyond a(15) from R. J. Mathar (mathar(AT)strw.leidenuniv.nl) and J. Mulder (jasper.mulder(AT)planet.nl), Jan 25 2010

%I A172023
%S A172023 1,1,1,3,5,13,38,133,534
%N A172023 Number of Sona graphs.
%C A172023 See Table 1, column 2. See the paper by E.D.Demaine et al., for details.
%D A172023 E.D.Demaine, M.L. Demaine, P. Taslakian and G.T. Toussaint, "Sand drawings and Gaussian graphs", Journal of Mathematics and the Arts, 1 (2007), 125-133.
%H A172023 Author unknown, <a href="http://www.informaworld.com/smpp/title~content=t755420531~db=all">Link to all free volumes and issues</a>
%e A172023 The number of Sona graphs with nine faces is 534.
%K A172023 nonn,new
%O A172023 1,4
%A A172023 Parthasarathy Nambi (PachaNambi(AT)yahoo.com), Jan 22 2010

%I A172022
%S A172022 1,4,4,8,10,14,16,20,22,30,30,38,40,44,46,54,58,62,66,72,72,80,82,90,96,
%T A172022 102,102,108,108,114,126,132,136,140,148,152,156,164,166,174,178,182,
%U A172022 190,194,196,200,210,224,226,230,232,240,240,252,256,264,268,272,276
%N A172022 a(n)=prime(n)+(-1)^n.
%C A172022 1 together with A014687.
%F A172022 a(n+1)=A014687(n).
%e A172022 a(1)=prime(1)+(-1)^1=2-1=1.
%Y A172022 Cf. A000027, A000040, A014687.
%K A172022 nonn,new
%O A172022 1,2
%A A172022 Juri-Stepan Gerasimov (2stepan(AT)rambler.ru), Jan 22 2010

%I A172017
%S A172017 119,121,143,145,185,187,203,205,215,217,287,289,299,301,413,415,515,
%T A172017 517,527,529,533,535,551,553,581,583,623,625,695,697,779,781,791,793,
%U A172017 815,817,869,871,893,895,899,901,959,961,1055,1057,1079,1081,1133,1135
%N A172017 Twin semiprimes (or twin biprimes): nonprimes of the form 6*k-+1 where 6*k-+1 are both semiprime (or both biprime).
%Y A172017 Cf. A171697 (twin natural nonprimes).
%K A172017 nonn,new
%O A172017 1,1
%A A172017 Juri-Stepan Gerasimov (2stepan(AT)rambler.ru), Jan 22 2010

%I A172012
%S A172012 2,3,15,54,207,783,2970,11259,42687,161838,613575,2326239,8819442,33437043,
%T A172012 126769455,480619494,1822166847,6908359023,26191577610,99299809899,376474162527,
%U A172012 1427321917278,5411388239415,20516130470079,77782556128482,294896059795683
%N A172012 Expansion of (2-3*x)/(1-3*x-3*x^2) .
%C A172012 The case k=3 in a family of sequences a(n) = L(k,n), L(k,n)=k*(L(k,n-1)+L(k,n-2)), L(k,0)=2 and L(k,1)=k.
%C A172012 The case k=1 is A000032 (classic Lucas sequence), k=2 is A080040, this here is essentially A085480.
%H A172012 <a href="Sindx_Rea.html#recLCC">Index entries for sequences related to linear recurrences with constant coefficients</a>, signature (3,3).
%F A172012 a(n) = 3*( a(n-1)+a(n-2) ) = 2*A030195(n+1)-3*A030195(n).
%F A172012 L(k,n) = c^n+b^n where c=(k+d)/2 ; b=(k-d)/2; d=sqrt(k*(k+4)) (Binet formula).
%K A172012 nonn,easy,new
%O A172012 0,1
%A A172012 Claudio Peruzzi (claudio.peruzzi(AT)gmail.com), Jan 22 2010
%E A172012 Edited and extended by R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jan 23 2010

%I A172011
%S A172011 0,12,24,72,192,528,1440,3936,10752,29376,80256,219264,599040,1636608,
%T A172011 4471296,12215808,33374208,91180032,249108480,680577024,1859371008,5079896064,
%U A172011 13878534144,37916860416,103590789120,283015299072,773212176384,2112454950912
%N A172011 12*A002605(n).
%C A172011 The case k=2 in a family of sequences a(n)=G(k,n), G(k,0)=0, G(k,1)=k*(k+4), G(k,n)=k*G(k,n-1)+k*G(k,n-2).
%C A172011 The Binet formula is G(k,n) = (c^n-b^n)*d where d=sqrt(k*(k+4)); c=(k+d)/2; b=(k-d)/2.
%C A172011 The generating functions are k*(k+4)*x/(1-k*x-k*x^2).
%C A172011 The case k=1 is A022088.
%H A172011 <a href="Sindx_Rea.html#recLCC">Index entries for sequences related to linear recurrences with constant coefficients</a>, signature (2,2).
%F A172011 Binet formula: a(n) = 2*2^n*((-1+3^(1/2))^(-n)-(-1)^n*(1+3^(1/2))^(-n))*3^(1/2) .
%F A172011 G.f.: 12*x/(1-2*x-2*x^2). a(n) = 2*a(n-1)+2*a(n-2).
%K A172011 nonn,easy,new
%O A172011 0,2
%A A172011 Claudio Peruzzi (claudio.peruzzi(AT)gmail.com), Jan 22 2010
%E A172011 Edited and extended by R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jan 23 2010

%I A170905
%S A170905 0,1,2,2,4,2,4,6,8,2,4,6,10,10,8,14,16,2,4,6,10,10,10,18,26,18,8,14,24,28,20,
%T A170905 32,32,2,4,6,10,10,10,18,26,18,10,18,30,38,34,42,58,34,8,14,24,28,28,44,
%U A170905 68,60,28,32,56,70,50,70,64,2,4,6,10,10,10,18,26,18,10,18,30,38,34,42
%N A170905 Consider the hexagonal cellular automaton defined in A151723, A151724; a(n) = number of cells that go from OFF to ON at stage n, if we only look at a 60 degree wedge (including the two bounding edges).
%H A170905 David Applegate, Omar E. Pol and N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/tooth.pdf">The Toothpick Sequence and Other Sequences from Cellular Automata</a>
%H A170905 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/toothlist.html">Catalog of Toothpick and Cellular Automata Sequences in the OEIS</a>
%F A170905 a(n) = A170898(n-2)+1 for n >= 2.
%K A170905 nonn,new
%O A170905 0,3
%A A170905 N. J. A. Sloane (njas(AT)research.att.com), Jan 22 2010

%I A172021
%S A172021 1,1,2,2,1,2,4,6,6,1,2,4,8,14,20,20,1,2,4,8,16,30,50,70,70,1,2,4,8,16,
%T A172021 32,62,112,182,252,252,1,2,4,8,16,32,64,126,238,420,672,924,924,1,2,4,8,
%U A172021 16,32,64,128,254,492,912,1584,2508,3432,3432
%N A172021 Start with the triangle A171661, reverse its rows, add missing powers of 2 at beginning of each row.
%C A172021 Rows sum up to A030662
%C A172021 Triangle is a (mirrored) interspaced binomial transform of 1^n (see example). [From M. Dols (markdols99(AT)yahoo.com), Jan 24 2010]
%e A172021 Triangle begins:
%e A172021 ......1
%e A172021 ....1,2,2
%e A172021 ..1,2,4,6,6
%e A172021 1,2,4,8,14,20,20
%e A172021 Contribution from M. Dols (markdols99(AT)yahoo.com), Jan 24 2010: (Start)
%e A172021 Interspaced binomial transform of 1^n:
%e A172021 1...1...1...1...1...1...
%e A172021 ..2...2...2...2...2...2.
%e A172021 2...4...4...4...4...4...
%e A172021 ..6...8...8...8...8...8.
%e A172021 6.. 14..16..16..16..16..
%e A172021 ..20..30..32..32..32..32
%e A172021 20..50..62..64..64..64..
%e A172021 (End)
%Y A172021 Cf. A030662, A171661, A171698
%Y A172021 Cf. A004070 [From M. Dols (markdols99(AT)yahoo.com), Jan 24 2010]
%K A172021 nonn,tabl,new,more
%O A172021 1,3
%A A172021 M. Dols (markdols99(AT)yahoo.com), Jan 22 2010
%E A172021 Definition rewritten by N. J. A. Sloane, Jan 23 2010
%E A172021 More terms from M. Dols (markdols99(AT)yahoo.com), Jan 24 2010

%I A172020
%S A172020 1,1,2,4,8,16,28,49,84,144,252,441,777,1369,2405,4225,7410,12996,22800,
%T A172020 40000,70200,123201,216216,379456,665896,1168561,2050657,3598609,
%U A172020 6315113,11082241,19448018,34128964
%N A172020 Number of subsets S of {1,2,3,...,n} with the property that if x is a member of S then at least one of x-2 and x+2 is also a member of S.
%C A172020 It is interesting that, for k>0, it appears that a(2k) is the square of A005251(k+2). (This has since been proved by Andrew Weimholt; see link).
%C A172020 If we denote by d2 the second difference of {a(n)}, it appears that d2(2k) is the square of A005314(k).
%H A172020 <a href="http://list.seqfan.eu/pipermail/seqfan/2010-January/003502.html">Proof by Andrew Weimholt</a>
%F A172020 Andrew Weimholt has shown that a(2*n) = A005251(n+2) ^ 2, and a(2*n+1) = A005251(n+2) * A005251(n+3). (See the link.)
%Y A172020 Cf. A005251, A005314.
%K A172020 nonn,new
%O A172020 1,3
%A A172020 John W. Layman (layman(AT)math.vt.edu), Jan 22 2010

%I A172019
%S A172019 5,8,10,12,13,15,16,17,20,21,24,25,26,28,29,30,32,33,34,35,36,37,39,40,
%T A172019 41,42,44,45,48,50,51,52,53,55,56,57,58,60,61,63,64,65,66,68,69,70,72,
%U A172019 73,74,75,76,77,78,80,82,84,85,87,88,89,90,91,92,93,95,96,97,99,100,101
%N A172019 Numbers n such that A000010(n) = 4k
%C A172019 Complement A097987
%Y A172019 Cf. A000010, A097987, A066499, A066498, A066500, A066502.
%K A172019 easy,nonn,new
%O A172019 1,1
%A A172019 Giovanni Teofilatto (g.teofilatto(AT)tiscalinet.it), Jan 22 2010

%I A172016
%S A172016 35,37,65,67,77,79,95,97,125,127,155,157,161,163,209,211,221,223,275,
%T A172016 277,305,307,329,331,335,337,365,367,371,373,377,379,395,397,407,409,
%U A172016 437,439,455,457,485,487,497,499,539,541,545,547,575,577,605,607,611
%N A172016 The twin nonprime-prime numbers: numbers of the form 6*k-+1 such that 6*k-1=nonprime and 6*k+1=prime.
%C A172016 A171688 U A171697 U A172015 U A172016 = A007310.
%Y A172016 Cf. A007310(the twin numbers), A171688(the twin nontrivial primes), A171697(the twin natural nonprimes), A172015(the prime-nonprime numbers).
%K A172016 nonn,new
%O A172016 1,1
%A A172016 Juri-Stepan Gerasimov (2stepan(AT)rambler.ru), Jan 22 2010

%I A172015
%S A172015 23,25,47,49,53,55,83,85,89,91,113,115,131,133,167,169,173,175,233,235,
%T A172015 251,253,257,259,263,265,293,295,317,319,353,355,359,361,383,385,389,
%U A172015 391,401,403,443,445,449,451,467,469,479,481,491,493,503,505,509,511
%N A172015 The twin prime-nonprime numbers: numbers of the form 6*k-+1 such that 6*k-1=prime and 6*k+1=nonprime.
%C A172015 A171688 U A171697 U A172015 U A172016 = A007310.
%Y A172015 Cf. A007310(the twin numbers), A171688(the twin nontrivial primes), A171697(the twin natural nonprimes), A172016(the nonprime-prime numbers).
%K A172015 nonn,new
%O A172015 1,1
%A A172015 Juri-Stepan Gerasimov (2stepan(AT)rambler.ru), Jan 22 2010

%I A172004
%S A172004 1,1,3,4,3,9,15,15,9,24,47,59,47,24,61,136,195,195,136,61,145,360,580,
%T A172004 663,580,360,145,333,904,1586,2032,2032,1586,904,333,732,2152,4077,5684,
%U A172004 6350,5684,4077,2152,732,1565,4927,9948,14938,18123,18123,14938,9948
%N A172004 Let y = y(u,v) be implicitly defined by g(u,v,y(u,v)) = 0. Read as a triangle by rows k = 1,2,..., the sequence represents the number of terms a(i,k-i) in the expansion of the partial derivatives d^k y/du^i dv^{k-i} in terms of partial derivatives of g.
%C A172004 The sequence starts with a(1,0),a(0,1),a(2,0),a(1,1),a(0,2),a(3,0),...
%C A172004 The subsequences a(1,0),a(2,0),a(3,0),... and a(0,1),a(0,2),a(0,3),... coincide with the sequence A003262, which is the corresponding sequence for univariate implicit functions.
%D A172004 Wilde, T., Implicit higher derivatives and a formula of Comtet and Fiolet, preprint, 2008. Available at http://arxiv.org/abs/0805.2674
%F A172004 Let E = N^3 \ {(0,0,0), (0,0,1)} be a set of triples of natural numbers. The number of terms a(m,n) is the coefficient of u^m * v^n * y^{m+n-1} in
%F A172004 prod_{(r,s,t) in E} (1 - u^r * v^s * y^{r+s+t-1})^{-1}
%e A172004 The formulas dy/du = -g_u/g_y,
%e A172004 d^2y/du^2 = -g_yy g_u^2/g_y^3 + 2g_uy g_u/g_y^2 - g_uu/g_y,
%e A172004 d^2y/dudv = -2g_yy g_u g_v / g_y^3 + g_uy g_v/g_y^2 + g_vy g_u/g_y^3 - g_uv/g_y
%e A172004 imply that a(1,0) = 1, a(2,0) = 3, and a(1,1) = 4.
%o A172004 (Other) # Upon executing the following code in Sage 4.2 (using Singular as a backend), it
%o A172004 # computes the number of terms a(n1,n2) and stores it in the entry A[n1][n2] of the
%o A172004 # double list A.
%o A172004 N = 9
%o A172004 E1 = N
%o A172004 E2 = N
%o A172004 p = [[[0 for i1 in range(E1+1)] for i2 in range(E2+1)] for j in range(E1 + E2)]
%o A172004 q = [[[0 for i1 in range(E1+1)] for i2 in range(E2+1)] for j in range(E1 + E2)]
%o A172004 .
%o A172004 for m in range(1, E1 + E2):
%o A172004 ....for d in range(1, m+1):
%o A172004 ........quotient, remainder = divmod(m, d)
%o A172004 ........if remainder == 0:
%o A172004 ............for i1 in range(quotient + 1 + 1):
%o A172004 ................for i2 in range(quotient + 1 - i1 + 1):
%o A172004 ....................if d*i1 <= E1 and d*i2 <= E2:
%o A172004 ........................q[m][i1*d][i2*d] += 1/d
%o A172004 .
%o A172004 for i1 in range(E1 + 1):
%o A172004 ....for i2 in range(E2 + 1):
%o A172004 ........p[0][i1][i2] = 1
%o A172004 .
%o A172004 for n in range(1, E1 + E2):
%o A172004 ....for s in range(n+1):
%o A172004 ........for k1 in range(E1+1):
%o A172004 ............for k2 in range(E2+1):
%o A172004 ................for i1 in range(k1 + 1):
%o A172004 ....................for i2 in range(k2 + 1):
%o A172004 ........................p[n][k1][k2] += 1/n * s * q[s][k1-i1][k2-i2] * p[n-s][i1][i2]
%o A172004 .
%o A172004 A = [[ p[n1+n2-1][n1][n2] for n1 in range(0,E1+1)] for n2 in range(0,E2+1)]
%Y A172004 Cf. A003262, which is the univariate variant of this sequence.
%Y A172004 Cf. A172003, which is the analogous sequence for implicit divided differences, and A162326 for its univariate variant.
%K A172004 nonn,tabl,new
%O A172004 1,3
%A A172004 Georg Muntingh (georg.muntingh(AT)gmail.com), Jan 22 2010

%I A172010
%S A172010 0,1,4,20,96,464,2240,10816,52224,252160,1217536
%N A172010 Extended Fibonacci number F(n,i) with index (dimension) n F(n,i)=n*(F(n,i-1)+F(n,i-2))=(a^i-b^i)/d where d=sqrt(n(n+4)); a=(n+d)/2; b=(n-d)/2
%F A172010 a(n) = A057087(n-1), n>0. [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jan 28 2010]
%e A172010 n=1 (A000045)(classical Fibonacci sequence) n=2 (A002605) n=3 (A030195) n=4 above n=5+ new
%e A172010 F(n,0)=0 F(n,1)=1
%K A172010 nonn,uned,new
%O A172010 0,3
%A A172010 Claudio Peruzzi (claudio.peruzzi(AT)gmail.com), Jan 22 2010

%I A172003
%S A172003 1,1,3,5,3,13,33,33,13,71,245,351,245,71,441,1921,3597,3597,1921,441,
%T A172003 2955,15525,35931,46709,35931,15525,2955,20805,127905,352665,563821,
%U A172003 563821,352665,127905,20805,151695,1067925,3417975,6483285,7963151
%N A172003 Let y = y(u,v) be implicitly defined by g(u,v,y(u,v)) = 0. Read as a triangle by rows, the sequence represents the number of terms a(i,k-i) in the expansion of the bivariate divided difference [u_0,...,u_i; v_0,...,v_{k-i}]y in terms of trivariate divided differences of g.
%C A172003 The sequence starts with a(1,0),a(0,1),a(2,0),a(1,1),a(0,2),a(3,0),...
%F A172003 Let E = N^3 \ {(0,0,0), (0,0,1)} be a set of triples of natural numbers. The number of terms a(m,n) is the coefficient of u^m * v^n * y^{m+n-1} of the generating function
%F A172003 - log(1 - \sum_{(r,s,t) in E} u^r * v^s * y^{r+s+t-1})
%F A172003 = \sum_{q >= 1} (\sum_{(r,s,t) in E} u^r * v^s * y^{r+s+t-1})^q / q
%e A172003 The subsequences a(1,0),a(2,0),a(3,0),... and a(0,1),a(0,2),a(0,3),... coincide with the sequence A162326.
%e A172003 For (m,n) = (1,1), one expresses [u_0,u_1;v_0,v_1]y as a sum of 5 terms,
%e A172003 [01;01]y =
%e A172003 - [0;0;(0,0),(1,0),(1,1)]g * [01;0;(1,0)]g * [1;01;(1,1)]g /
%e A172003 ( [0;0;(0,0),(1,1)]g * [0;0;(0,0),(1,0)]g * [1;0;(1,0),(1,1)]g )
%e A172003 + [01;0;(1,0),(1,1)]g * [1;01;(1,1)]g /
%e A172003 ( [0;0;(0,0),(1,1)]g * [1;0;(1,0),(1,1)]g )
%e A172003 - [01;01;(1,1)]g / [0;0;(0,0),(1,1)]g
%e A172003 - [0;0;(0,0),(0,1),(1,1)]g * [0;01;(0,1)]g * [01;1;(1,1)]g /
%e A172003 ( [0;0;(0,0),(1,1)]g * [0;0;(0,0),(0,1)]g * [0;1;(0,1),(1,1)]g )
%e A172003 + [0;01;(0,1),(1,1)]g * [01;1;(1,1)]g /
%e A172003 ( [0;0;(0,0),(1,1)]g * [0;1;(0,1),(1,1)]g ),
%e A172003 where the numbers refer to the indices of the corresponding variable, e.g.
%e A172003 [1;01;(1,1)]g = [u_1;v_0,v_1;y(u_1,v_1)]g.
%o A172003 (Other) # To be executed in Sage 4.2 with Singular as a backend.
%o A172003 def P(n1,n2,q):
%o A172003 ....E = CartesianProduct(range(n1+1), range(n2+1), range(n1+n2+1))
%o A172003 ....E = [(i1,i2,j) for (i1,i2,j) in E if (i1,i2,j) != (0,0,0) and \
%o A172003 ..............(i1,i2,j) != (0,0,1) and i1 + i2 + j <= n1 + n2 and \
%o A172003 ..............2*(i1 + i2) + j - 1 <= 2*(n1+n2) - q]
%o A172003 ....return sum([X1^s1 * X2^s2 * Y^(s1+s2+t-1) for (s1,s2,t) in E])
%o A172003 .
%o A172003 R.<X1,X2,Y> = PolynomialRing(ZZ,3)
%o A172003 .
%o A172003 n1, n2 = 3, 3
%o A172003 L = [[0 for i1 in range(n1 + 1)] for i2 in range(n2 + 1)]
%o A172003 .
%o A172003 h = expand(1 + sum([((P(n1,n2,q))^q)/q for q in range(1,2*(n1+n2))]))
%o A172003 for k1 in range(0, n1+1):
%o A172003 ....for k2 in range(0, k1+1):
%o A172003 ........if (k1, k2) != (0, 0):
%o A172003 ............print k1, k2, h.coefficient({X1:k1, X2:k2, Y:k1+k2-1})
%Y A172003 Cf. A162326, which is the univariate variant of this sequence.
%Y A172003 Cf. A172004, which is the analogous sequence for implicit derivatives, and A003262 for its univariate variant.
%K A172003 nonn,tabl,new
%O A172003 1,3
%A A172003 Georg Muntingh (georg.muntingh(AT)gmail.com), Jan 22 2010

%I A172009
%S A172009 1,1,2,2,2,2,2,2,6,12,4,8,2,4,4,10,4,2,12,4,2,6,8,10,2,2,2,2,2,2,10,2,8,
%T A172009 2,14,22,8,2,18,8,4,14,4,12,2,4,2,8,6,2,6,6,10,2,2,4,6,4,4,2,4,2,22,8,2,
%U A172009 4,2,2,4,6,24,6,2,2,12,2,12,4,2,2,6,6,12,18,6,4,6,6,2,2,2,2,8,12,2,2,2
%N A172009 Count of minimal SNUSP programs using +, -, @ and # to yield N
%C A172009 SNUSP is a programming language where each command is an individual letter. The four of concern here are +, -, @ and #. + increments the current data value, - decrements it, @ is a "subroutine call" and # is a "return". When an @ is encountered, a record of the location is put on a stack and execution continues. When a # is encountered, if there is a return point on the stack, the execution continues at that a single character beyond that return point. If there is no return point on the stack, execution terminates.
%C A172009 Thus "@@++#" would put the first two "@" return points on the stack, increment data twice, return from the second "@" to the last "+", increment the data once more, return from the first "@" to the first "+", increment the data two more times and finally terminate when it hits the "#" with no return points on the stack. The data is always initialized to zero so this effectively puts 5 into the data. In order to place a particular value into the data, there one or more minimal strings of these characters. The i'th element of the sequence gives the number of minimal SNUSP programs using only these characters. After 2, all sequences end in either +++ or @++ both of which are equivalent so that all values above a(2) are even.
%H A172009 <a href="http://www.esolangs.org/wiki/SNUSP">SNUSP - Esolang</a>
%H A172009 <a href="http://c2.com/cgi/wiki?SnuspLanguage">SNUSP Language</a>
%e A172009 There are 12 minimal programs which yield 10: +@+++++# @@-++++# -@@++++# -@+@+++# +@-@+++# ++@@+++# +@++@++# @@-+@++# -@@+@++# -@+@@++# +@-@@++# ++@@@++# Thus a(10) = 12.
%o A172009 (Other) See A172005.
%Y A172009 Cf. A172005, A172006, A172007, A172008
%K A172009 nonn,new
%O A172009 1,3
%A A172009 Darrell Plank (jar_czar(AT)msn.com), Jan 22 2010

%I A172008
%S A172008 1,1,2,2,2,2,2,2,4,4,4,4,2,4,2,2,4,2,6,2,2,2,8,8,4,2,2,2,2,2,2,6,2,2,10,
%T A172008 10,4,2,8,8,2,2,4,8,2,2,2,6,14,2,4,4,8,8,2,4,2,4,2,2,2,4,6,6,2,2,2,2,4,
%U A172008 4,18,18,2,2,4,2,8,2,10,2,2,2,4,4,6,4,4,4,2,2,2,4,6,6,2,2,2,8,2,2,6,6,2
%N A172008 Count of minimal SNUSP programs using +, @ and # to yield N
%C A172008 Shortest SNUSP representation of a number using only + and @ %C A172005 SNUSP is a programming language where each command is an individual letter. The three of concern here are + and @ and #. + increments the current data value, @ is a "subroutine call" and # is a "return". When an @ is encountered, a record of the location is put on a stack and execution continues. When a # is encountered, if there is a return point on the stack, the execution continues at that a single character beyond that return point. If there is no return point on the stack, execution terminates.
%C A172008 Thus "@@++#" would put the first two "@" return points on the stack, increment data twice, return from the second "@" to the last "+", increment the data once more, return from the first "@" to the first "+", increment the data two more times and finally terminate when it hits the "#" with no return points on the stack. The data is always initialized to zero so this effectively puts 5 into the data. In order to place a particular value into the data, there one or more minimal strings of these characters. The i'th element of the sequence gives the number of minimal SNUSP programs using only these characters. After 2, all sequences end in either +++ or @++ both of which are equivalent so that all values above a(2) are even.
%H A172008 <a href="http://www.esolangs.org/wiki/SNUSP">SNUSP - Esolang</a>
%H A172008 <a href="http://c2.com/cgi/wiki?SnuspLanguage">SNUSP Language</a>
%e A172008 19 can be represented minimally in 6 ways using @, + and #: @+@+++++# +@@@++++# @++@@+++# @+@++@++# +@@@+@++# @++@@@++# Thus a(19) = 6.
%p A172008 See A172005
%Y A172008 A172005, A172006, A172007
%K A172008 nonn,new
%O A172008 1,3
%A A172008 Darrell Plank (jar_czar(AT)msn.com), Jan 22 2010

%I A172007
%S A172007 25,32,40,49,50,51,52,54,62,64,67,72,79,81,82,85,92,96,100,102,122,127,
%T A172007 128,129
%N A172007 Numbers which require a - in their minimal SNUSP representation
%C A172007 SNUSP is a programming language where each command is an individual letter. The four of concern here are +, -, @ and #. + increments the current data value, - decrements it, @ is a "subroutine call" and # is a "return". When an @ is encountered, a record of the location is put on a stack and execution continues. When a # is encountered, if there is a return point on the stack, the execution continues at that a single character beyond that return point. If there is no return point on the stack, execution terminates.
%C A172007 Thus "@@++#" would put the first two "@" return points on the stack, increment data twice, return from the second "@" to the last "+", increment the data once more, return from the first "@" to the first "+", increment the data two more times and finally terminate when it hits the "#" with no return points on the stack. The data is always initialized to zero so this effectively puts 5 into the data. In order to place a particular value into the data, there is a minimal string of these characters. In some cases, allowing the '-' command can shorten this minimal string. This sequence is a list of numbers which require a - in their minimal sequence. All the numbers represented in the above sequence save at most 2 characters by allowing the -. Whether this is a maximum savings and whether the savings can be arbitrarily large isn't known (at least not to me).
%H A172007 <a href="http://www.esolangs.org/wiki/SNUSP">SNUSP - Esolang</a>
%H A172007 <a href="http://c2.com/cgi/wiki?SnuspLanguage">SNUSP Language</a>
%e A172007 Using both + and -, 25 can be represented as @-@@@+++# but if we only allow +, the minimal program is @++@@++++# so we only need 8 characters if we allow both + and - but 9 if we allow only + so that 25 requires a - in its minimal representation. It is the first value with this property and so is the first value in our sequence.
%p A172007 See A172005.
%Y A172007 A172005, A172006
%K A172007 nonn,new
%O A172007 1,1
%A A172007 Darrell Plank (jar_czar(AT)msn.com), Jan 22 2010

%I A172006
%S A172006 1,2,3,4,4,5,5,5,6,7,6,7,6,7,7,8,7,7,8,8,7,8,8,9,8,8,8,8,8,9,9,9,9,8,9,
%T A172006 10,9,9,10,10,9,10,9,10,9,10,9,10,10,10,10,10,10,10,9,10,10,10,10,10,10,
%U A172006 10,11,11,10,11,10,10,10,11,11,11,10,11,11,10,11,11,11,11,11,11,11,12
%N A172006 Shortest SNUSP representation of a number using only +, - and @
%C A172006 SNUSP is a programming language where each command is an individual letter. The four of concern here are +, -, @ and #. + increments the current data value, - decrements it, @ is a "subroutine call" and # is a "return". When an @ is encountered, a record of the location is put on a stack and execution continues. When a # is encountered, if there is a return point on the stack, the execution continues at that a single character beyond that return point. If there is no return point on the stack, execution terminates.
%C A172006 Thus "@@++#" would put the first two "@" return points on the stack, increment data twice, return from the second "@" to the last "+", increment the data once more, return from the first "@" to the first "+", increment the data two more times and finally terminate when it hits the "#" with no return points on the stack. The data is always initialized to zero so this effectively puts 5 into the data. In order to place a particular value into the data, there is a minimal st ring of these characters. The i'th element of the sequence gives the minimal number of characters (excluding the "#" which is always the last character) to produce an SNUSP program which sets the data to i. The string above is a minimal string to produce 5 and has four characters before the # so the 5th item in the sequence is 4.
%C A172006 Sequence A172005 is the same as this one but disallows the '-' command. Many values have smaller sequences by allowing the -. There are some sequences that can cut up to two characters off by using the -. I don't know if larger savings are possible or if the savings can become arbitrarily large.
%H A172006 <a href="http://www.esolangs.org/wiki/SNUSP">SNUSP - Esolang</a>
%H A172006 <a href="http://c2.com/cgi/wiki?SnuspLanguage">SNUSP Language</a>
%e A172006 To produce 10, there are 4 minimal sequences, each of length 7 (as always, excluding the #): +@+++++# ++@@+++# +@++@++# ++@@@++# Thus a(10)=7. The first value that requires a - in its minimal representation is 25 which requires 8 characters. If we disallow the '-' command (as in sequence A172005), it requires 9 characters.
%p A172006 See A172005.
%Y A172006 A172005
%K A172006 nonn,new
%O A172006 1,2
%A A172006 Darrell Plank (jar_czar(AT)msn.com), Jan 22 2010

%I A172005
%S A172005 1,2,3,4,4,5,5,5,6,7,6,7,6,7,7,8,7,7,8,8,7,8,8,9,9,8,8,8,8,9,9,10,9,8,9,
%T A172005 10,9,9,10,11,9,10,9,10,9,10,9,10,11,11,11,12,10,11,9,10,10,10,10,10,10,
%U A172005 11,11,12,10,11,11,10,10,11,11,12,10,11,11,10,11,11,12,11,12,13,11,12
%N A172005 Shortest SNUSP representation of a number using only + and @
%C A172005 SNUSP is a programming language where each command is an individual letter. The two of concern here are + and @ and #. + increments the current data value, @ is a "subroutine call" and # is a "return". When an @ is encountered, a record of the location is put on a stack and execution continues. When a # is encountered, if there is a return point on the stack, the execution continues at that a single character beyond that return point. If there is no return point on the stack, execution terminates.
%C A172005 Thus "@@++#" would put the first two "@" return points on the stack, increment data twice, return from the second "@" to the last "+", increment the data once more, return from the first "@" to the first "+", increment the data two more times and finally terminate when it hits the "#" with no return points on the stack. The data is always initialized to zero so this effectively puts 5 into the data. In order to place a particular value into the data, there is a minimal string of these characters. The i'th element of the sequence gives the minimal number of characters (excluding the "#" which is always the last character) to produce an SNUSP program which sets the data to i. The string above is a minimal string to produce 5 and have four characters before the # so the 5th item in the sequence is 4.
%C A172005 All sequences for values >= 3 end in @++ or +++, both of which are equivalent so there are an even number of sequences for every value, half of which end in +++ and half of which end in @++. There are several variations to this sequence which I'll also enter in. They include allowing "-" to decrement the data, the number of minimal sequences and numbers which require a "-" in their minimal expression.
%H A172005 <a href="http://www.esolangs.org/wiki/SNUSP">SNUSP - Esolang</a>
%H A172005 <a href="http://c2.com/cgi/wiki?SnuspLanguage">Snusp Language</a>
%e A172005 To produce 10, there are 4 minimal sequences, each of length 7 (as always, excluding the #): +@+++++# ++@@+++# +@++@++# ++@@@++# Thus a(10)=7.
%K A172005 nonn,new
%O A172005 1,2
%A A172005 Darrell Plank (jar_czar(AT)msn.com), Jan 22 2010

%I A172002
%S A172002 1,2,3,4,8,9,7,10,6,11,5,12,16,17,15,18,14,19,13,20,29,30,28,31,27,32,
%T A172002 26,33,25,34,24,35,23,36,22,37,21,38,47,48,46,49,45,50,44,51,43,52,42,
%U A172002 53,41,54,40,55,39,56,72,73,71,74,70,75,69,76,68,77,67,78,66,79,65,80
%N A172002 A permutation of natural numbers A000027 via Janet distribution A137583=2,2,8,8,18,18,32,32,.
%C A172002 From a periodic elements table. 120 terms Janet table is mathematically the fundamental periodic table of the elements (see A138509 and A171710) . 118 terms tables come after. Here this 120 (or extended) table is compact,without spaces between terms, and symmetric.An axis shares the table in two parts (60+60 terms).The reading starts from (main) first (see A131941(n+1)) and second columns.See A167384. 120 terms table first 6 rows : Row 1: (15 spaces),1,2,(15 spaces); row 2: (15 spaces),3,4,(15 spaces); row 3: (12 spaces),5,6,7,8,9,10,11,12,(12 spaces); row 4: (12 spaces),13,14,15,16,17,18,19,20,(12 spaces); row 5:(7 spaces),21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,(7 spaces); row 6:(7 spaces),39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,(7 spaces);
%K A172002 nonn,uned,new
%O A172002 1,2
%A A172002 Paul Curtz (bpcrtz(AT)free.fr), Jan 22 2010

%I A172001
%S A172001 34,136,146,178,194,205,221,305,306,377,386,410,466,482,505,514,544,545,
%T A172001 562,584,674,689,706,712,745,776,793,802,820,850,866,884,890,898,905,
%U A172001 1154,1186,1202,1205,1220,1224,1234,1282,1314,1345,1346,1394,1405,1469
%N A172001 Non-square positive integers n representable as the sum of two square, while "negative" Pellian equation x^2 - n*y^2 = -1 has no integer solutions.
%C A172001 The set difference of A000415 and A172000 (note that A172000 is a subsequence of A000415).
%C A172001 Also, the set difference of A087643 and A022544 (note that A022544 is a subsequence of A087643).
%K A172001 nonn,new
%O A172001 1,1
%A A172001 Max Alekseyev (maxale(AT)gmail.com), Jan 21 2010

%I A172000
%S A172000 2,5,8,10,13,17,18,20,26,29,32,37,40,41,45,50,52,53,58,61,65,68,72,73,
%T A172000 74,80,82,85,89,90,97,98,101,104,106,109,113,116,117,122,125,128,130,
%U A172000 137,145,148,149,153,157,160,162,164,170,173,180,181,185,193,197,200
%N A172000 Non-square positive integers n for which "negative" Pellian equation x^2 - n*y^2 = -1 has integer solutions.
%C A172000 Non-square positive integers n for which the period length of continued fraction of sqrt(n) is odd.
%C A172000 Complement of A087643 in the non-square integers A000037.
%C A172000 Subsequence of A000415, their set difference is given by A172001.
%o A172000 (PARI) { for(n=1,1000, if(issquare(n),next); if( norm(bnfinit(x^2-n).fu[1])==-1, print1(n,", ")) ) }
%K A172000 nonn,new
%O A172000 1,1
%A A172000 Max Alekseyev (maxale(AT)gmail.com), Jan 21 2010

%I A171999
%S A171999 1,1,2,1,3,6,1,4,6,12,1,5,10,20,30,1,6,15,20,30,60,90,1,7,21,35,42,105,
%T A171999 140,210,1,8,28,56,70,168,280,420,560,1,9,36,72,84,126,252,504,630,756,
%U A171999 1260,1680,1,10,45,90,120,210,252,360,840,1260,2520,3150,4200,1,11,55
%N A171999 Triangle of trinomial coefficients.
%C A171999 Number of numbers in row n is given by A086753.
%C A171999 Row sums: A092255.
%C A171999 See A046816, Pascal's pyramid of trinomial coefficients.
%e A171999 First six rows:
%e A171999 1
%e A171999 1...2
%e A171999 1...3...6
%e A171999 1...4...6....12
%e A171999 1...5...10...20...30
%e A171999 1...6...15...20...30...60...90
%Y A171999 Cf. A036038, A046816, A092255.
%K A171999 nonn,tabl,new
%O A171999 1,3
%A A171999 Clark Kimberling (ck6(AT)evansville.edu), Jan 21 2010

%I A171998
%S A171998 1,1,1,1,3,1,5,7,6,1,65,15,25,10,1,455,455,0,65,15,1,1295,4725,
%T A171998 1715,140,140,21,1
%V A171998 1,1,1,1,3,1,-5,7,6,1,-65,-15,25,10,1,-455,-455,0,65,15,1,-1295,-4725,
%W A171998 -1715,140,140,21,1
%N A171998 A(n,k,m) is the (n,k)-th entry of the matrix inverting the matrix consisting of (-1)^(n-k) times the number of permutations of an n-set with k disjoint cycles of length less than or equal to m as the (n,k)-th entry, called the m-restrained Stirling numbers of the second kind. The example above shows the case m=3.
%C A171998 A(n,k,m) also can be expanded for nonpositive integers n and k using the multi-restrained Stirling numbers of the first kind.
%D A171998 Multi-restrained Stirling numbers (just submitted)
%F A171998 Explicit Formula A(n,k,m)= A(n-1,k-1,m)- Sum_{i=1}^{m-1} (-1)^{i}(k)...(k+i-1) A(n, k+i,m) A(n,k,m) = A(n-1,k-1,m) + k A(n-1,k,m) + (-1)^m k(k+1)...(k+m-1)A(n,k+m,m)
%e A171998 A(1,1,3)=1,A(1,2,3)=0,A(1,3,3)=0,A(1,4,3)=0,... A(2,1,3)=1,A(2,2,3)=1,A(2,3,3)=0,A(2,4,3)=0,... A(3,1,3)=1,A(3,2,3)=3,A(3,3,3)=1,A(3,4,3)=0,... A(4,1,3)=-5,A(4,2,3)=7,A(4,3,3)=6,A(4,4,3)=1,...
%Y A171998 Cf. A111246, 144633
%K A171998 nonn,new
%O A171998 1,5
%A A171998 Ji Young Choi (jychoi(AT)ship.edu), Jan 21 2010

%I A171882
%S A171882 1,1,0,1,1,1,1,2,1,0,1,3,4,1,1,1,4,27,16,1,0,1,5,256,7625597484987,
%T A171882 65536,1,1,1,6,3125,
%U A171882 13407807929942597099574024998205846127479365820592393377723561443721764030073546976801874298166903427690031858186486050853753882811946569946433649006084096
%N A171882 Square array, read by antidiagonals, where T(n,k)=n^^k for n>=0, k>=0.
%C A171882 n^^k defined the right-associative way: n^^2=n^n, n^^3=n^(n^n), n^^4=n^(n^(n^n)), etc.
%C A171882 n^^0=1 by convention, so that n^^(k+1) = n^(n^^k) for all k>=0.
%C A171882 More terms on Munafo website.
%C A171882 Array begins:
%C A171882 1,0,1,0,1,0,1,...
%C A171882 1,1,1,1,1,1,1,...
%C A171882 1,2,4,16,65536,...
%C A171882 1,3,27,7625597484987,...
%C A171882 1,4,256,...
%C A171882 1,5,3125,...
%C A171882 1,6,46656,...
%H A171882 R. Munafo, <a href="http://mrob.com/pub/math/hyper4.html">Hyper4 Iterated Exponential Function</a>
%Y A171882 Cf. A171881.
%K A171882 nonn,tabl,new
%O A171882 0,8
%A A171882 Robert Munafo (mrob27(AT)gmail.com), Jan 21 2010

%I A171881
%S A171881 0,1,1,2,1,1,3,4,1,1,4,27,16,1,1,5,256,19683,256,1,1,6,3125,4294967296,
%T A171881 7625597484987,65536,1,1,7,46656,298023223876953125,
%U A171881 340282366920938463463374607431768211456
%N A171881 Square array, read by antidiagonals, where T(n,k)=n^^k for n>=0, k>=1.
%C A171881 n^^k is defined the left-associative way: n^^2=n^n, n^^3=(n^n)^n=n^(n^2), n^^4=((n^n)^n)^n=n^(n^3), and in general n^^k=n^(n^(k-1)).
%C A171881 More terms on Munafo website.
%C A171881 Array begins:
%C A171881 0,1,1,1,1,1,...
%C A171881 1,1,1,1,1,1,...
%C A171881 2,4,16,256,65536,...
%C A171881 3,27,19683,...
%C A171881 4,256,4294967296,...
%C A171881 5,3125,...
%C A171881 6,46656,...
%H A171881 R. Munafo, <a href="http://mrob.com/pub/math/hyper4.html">Hyper4 Iterated Exponential Function</a>
%Y A171881 Cf. A171882.
%K A171881 nonn,tabl,new
%O A171881 0,4
%A A171881 Robert Munafo (mrob27(AT)gmail.com), Jan 21 2010

%I A171880
%S A171880 0,0,0,1,1,1,2,4,7,16,46,166,1014,47066,12348246366,
%T A171880 66716521529543607970475115226
%N A171880 a(n) = a(n-1) + a(n-2)*a(n-3) + a(n-4)*a(n-5)^a(n-6)
%C A171880 First 6 terms are {0,0,0,1,1,1}; thereafter apply the recurrence. Note that 0^0=1.
%H A171880 R. Munafo, <a href="http://mrob.com/pub/math/seq-accelerate.html">Accelerating Sequences</a>
%K A171880 nonn,new
%O A171880 0,7
%A A171880 Robert Munafo (mrob27(AT)gmail.com), Jan 21 2010

%I A171879
%S A171879 0,0,1,1,1,1,3,5,9,25,73,313,3263,1502337,278472902914281,
%T A171879 11984387434132924341157279996736444304839056033321
%N A171879 a(n) = a(n-1) + a(n-2)*a(n-3) + a(n-4)*a(n-5)^a(n-6)
%C A171879 First 6 terms are {0,0,1,1,1,1}; thereafter apply the recurrence. Note that 0^0=1.
%H A171879 R. Munafo, <a href="http://mrob.com/pub/math/seq-accelerate.html">Accelerating Sequences</a>
%K A171879 nonn,new
%O A171879 0,7
%A A171879 Robert Munafo (mrob27(AT)gmail.com), Jan 21 2010

%I A171878
%S A171878 0,0,0,0,1,2,3,6,13,33,120,765,4831534,55040353993453427047,
%T A171878 410186270246002225336426103593500672000000000000055040353997149550557
%N A171878 a(n) = a(n-1) + a(n-2)*a(n-3) + a(n-4)^a(n-5)
%C A171878 First 5 terms are {0,0,0,0,1}; thereafter apply the recurrence. Note that 0^0=1.
%H A171878 R. Munafo, <a href="http://mrob.com/pub/math/seq-accelerate.html">Accelerating Sequences</a>
%K A171878 nonn,new
%O A171878 0,6
%A A171878 Robert Munafo (mrob27(AT)gmail.com), Jan 21 2010

%I A171877
%S A171877 0,0,1,1,1,3,5,9,25,73,423,61297,3814697357801,
%T A171877 38288777744833624093154249190851262684887027625
%N A171877 a(n) = a(n-1) + a(n-2)*a(n-3) + a(n-4)^a(n-5)
%C A171877 First 5 terms are {0,0,1,1,1}; thereafter apply the recurrence. Note that 0^0=1.
%H A171877 R. Munafo, <a href="http://mrob.com/pub/math/seq-accelerate.html">Accelerating Sequences</a>
%K A171877 nonn,new
%O A171877 0,6
%A A171877 Robert Munafo (mrob27(AT)gmail.com), Jan 21 2010

%I A171876
%S A171876 1,1,1,1,1,3,3,1,1,4,6,19,27,50,56,1,1,5,10,47,131,472,1326,3779,9013,
%T A171876 19963,38073,65664,98804,133576,158658,1,1,6,16,103,497,3253,19735,
%U A171876 120843,681474,3561696
%N A171876 Mutual solutions to two classification counting problems: binary block codes of wordlength J with N used words; and classifications of N elements by J partitions.
%C A171876 This connection was conjectured by Robert Munafo, then proved by Andrew Weimholt.
%C A171876 A(n) counts 2-colorings of a J-dimensional hypercube with N red vertices and 2^J-N black, each edge has at most one red vertex. (Andrew Weimholt, Dec 30 2009)
%C A171876 This sequence contains terms of A039754 that are found in A171871/A171872. They occur in blocks of length 2^(J-1) as shown here:
%C A171876 1
%C A171876 1,1
%C A171876 1,1,3,3
%C A171876 1,1,4,6,19,27,50,56
%C A171876 1,1,5,10,47,131,472,1326,3779,9013,19963,38073,65664,98804,133576,158658
%H A171876 Harald Fripertinger, <a href="http://www.mathe2.uni-bayreuth.de/frib/html2/construction/blockcodes_2.html">Enumeration of block codes</a>
%H A171876 R. Munafo, <a href="http://mrob.com/pub/math/seq-a005646.html">Classifications of N Elements</a>
%Y A171876 Cf. A039754, A171872, A171871, A005646
%K A171876 nonn,new
%O A171876 0,6
%A A171876 Robert Munafo (mrob27(AT)gmail.com), Jan 21 2010

%I A171875
%S A171875 0,0,1,3,17,74,358,1631,7563,34751,160807
%N A171875 Subdiagonal of triangle A171871: Classifications of N elements containing exactly N-2 binary partitions.
%H A171875 R. Munafo, <a href="http://mrob.com/pub/math/seq-a005646.html">Classifications of N Elements</a>
%K A171875 nonn,new
%O A171875 2,4
%A A171875 Robert Munafo (mrob27(AT)gmail.com), Jan 21 2010

%I A171996
%S A171996 1,1,1,2,3,1,0,11,6,1,0,20,35,10,1,0,40,135,85,15,1,0,0,490,525,
%T A171996 175,21,1
%V A171996 1,-1,1,2,-3,1,0,11,-6,1,0,-20,35,-10,1,0,40,-135,85,-15,1,0,0,490,-525,
%W A171996 175,-21,1
%N A171996 A(n,k,m) is the number of permutations of an n-set with k disjoint cycles of length less than or equal to m, called the (n,k)-th m-restrained Stirling numbers of the first kind, and denoted by mS_1(n,k). The example above shows the case of m=3.
%C A171996 A(n,k,m) is also the (n,k)-th entry in the matrix inverting the matrix consisting of the m-restrained Stirling numbers of the second kind.
%D A171996 Multi-restrained Stirling numbers (Just submitted to a journal)
%F A171996 Explicit formula A(n,k,m) = Sum {(-1)^{n-k}n!} /{1^{k_1}2^{k_2}...m^{k_m}(k_1!)(k_2!)...(k_m!)}, where k_1+2k_2+\cdots+mk_m=n and k_1+k_2+\cdots+k_m=k Recurrence relations A(n,k,m)= Sum_{i=1}^{m} (-1)^{i-1}[n-1]_{i-1} A(n-i,k-1,m), A(n,k,m)= A(n-1,k-1,m)-(n-1)A(n-1,k,m) - (-1)^m(n-1)(n-2)...(n-m) A(n-m-1,k-1,m) Generating function f(t) = (1+t-{t^2}/2+ {t^3}/3+...+(-1)^{m-1} {t^m}/m )^x, for an indeterminate x ===> the n-th derivative of f(t) at t=0, f^(n)(0)= Sum_{k=1}^{n} A(n,k,m)[x]_k, where [x]_k is the k-th falling factorial
%e A171996 A(1,1,3)=1,A(1,2,3)=0,A(1,3,3)=0,A(1,4,3)=0,... A(2,1,3)=-1,A(2,2,3)=1,A(2,3,3)=0,A(2,4,3)=0,... A(3,1,3)=2,A(3,2,3)=-3,A(3,3,3)=1,A(3,4,3)=0,... A(4,1,3)=0,A(4,2,3)=11,A(4,3,3)=-6,A(4,4,3)=1,...
%Y A171996 Cf. A111246, A144633
%K A171996 nonn,new
%O A171996 1,4
%A A171996 Ji Young Choi (jychoi(AT)ship.edu), Jan 21 2010

%I A171910
%S A171910 1,6,92,406,1549,5361,12546,41908,141121
%N A171910 Roots of the Mertens function M(x) for 0<x<10^k where M(x) is the matching summatory function for the Moebius function
%C A171910 m(k) is the number of x in interval[1,10^k] defined as M(x) = 0, for k >=1, and is equal to {1,6,92,...}. It's well known that the function M oscillate infinitely around 0 when x tend towards infinity. M(x) = Sum_{n < = x} moebius(n)
%D A171910 E. Grosswald, Oscillation theorems of arithmetical functions, Trans. AMS 126 (1967), 1-28
%H A171910 <a href="http://www.research.att.com/~njas/sequences/Sindx_Mo.html"></a> (moebius transform)
%H A171910 <a href="http://mathworld.wolfram.com/MoebiusFunction.html"></a>
%e A171910 For k = [1,..,10], m(1) = 1 For x = [1,..,100], m(2) =6 For x = [1, ..., 1000], m(3) = 92
%K A171910 nonn
%O A171910 1,2
%A A171910 Michel LAGNEAU (mn.lagneau2(AT)orange.fr), Dec 31 2009

%I A171909
%S A171909 1,6,7,6,6,8,8,3,7,2,5,8,1,5,8,4,1,9,2,6,2,3,3,8,4,7,4,4,6,1,6,0,2,6,0,
%T A171909 7,7,8,5,9,0,8,9,3,4,0,6,1,1,7,5,2,0,3,4,7,5,1,6,5,6,5,0,6,5,2,5,0,3,2,
%U A171909 1,0,4,8,9,6,8,1,5,8,2,1,5,7,8,9,7,9,2,4,9,6,6,9,8,0,7,5,9,5,0,1,5,7,4
%N A171909 Decimal expansion of the abscissa x of a local minimum of the Fibonacci Function F(x).
%C A171909 Define the Fibonacci Function F(x) = ( phi^x - cos(Pi*x) / phi^x )/sqrt(5) as an interpolation
%C A171909 of the Fibonacci numbers, with phi = A001622, Pi = A000796.
%C A171909 The derivative is dF/dx = ( phi^x * log(phi) - cos(Pi*x) *log(phi)/ phi^x + Pi*sin(Pi*x)/ phi^x)/sqrt(5).
%C A171909 Set dF(x)/dx=0 to find local minima and maxima.
%H A171909 Gerd Lamprecht, <a href="http://freenet-homepage.de/gerdlamprecht/Roemisch_JAVA.htm">Iterationsrechner mit Algorithmus</a>
%H A171909 Gerd Lamprecht, <a href="http://freenet-homepage.de/gerdlamprecht/Zahlenfolgen.html">Zahlenfolgen (sequence)</a>
%H A171909 E. Weisstein, <a href="http://mathworld.wolfram.com/FibonacciNumber.html">Fibonacci Number</a>, Mathworld.
%H A171909 A. Stakhov and Boris Rozin, <a href="http://dx.doi.org/10.1016/j.chaos.2005.08.158">The continuous functions for the Fibonacci and Lucas p-numbers</a>, Chaos, Solit. and Fractals 26 (4) (2006) 1014-1025.
%e A171909 F(1.67668837258...)=0.896946387424606172912600371068765... = A172081
%o A171909 (Other) Gerd Lamprecht online Iterationsrechner: #@P@Q5)*0.5+0.5,x)/@Q5)+@P@Q5)*0.5-0.5, x)*sin(PI*(x-0.5))/@Q5)@Na=0.19; b=1.6; @B2]=2; @N@B0]=Fx(b); @B1]=Fx(b-a); @B2]=Fx(b+a); if(@B0]%3C@B1]&&@B0]%3C@B2])a/=10; @Eif(@B1]%3C@B2])b-=a; @Eb+=a; @N@A@B1]-@B2])%3C1e-17@N1@N1@Nc=Fx(b);
%Y A171909 Cf. A001622, A089260
%K A171909 cons,nonn,new
%O A171909 1,2
%A A171909 Gerd Lamprecht (gerdlamprecht(AT)googlemail.com), Dec 31 2009
%E A171909 Description edited, JavaScript calculations embedded in URL's removed, Weisstein and Stakhov-Rozin ref added by R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Feb 02 2010

%I A171908
%S A171908 20,30,0,10,0,0
%N A171908 a(n)= Number of 5 X 5 permutation matrices with trace n and determinant -1
%e A171908 a(0)=20 because we have 20 different permutation matrices 5 x 5 with trace 0 and determinant -1.
%Y A171908 A171806, A171808, A171809, A171906, A171907
%K A171908 fini,full,nonn
%O A171908 0,1
%A A171908 Artur Jasinski (grafix(AT)csl.pl), Dec 31 2009

%I A171907
%S A171907 24,15,20,0,0,1
%N A171907 a(n)= Number of 5 X 5 permutation matrices with trace n and determinant +1
%e A171907 a(0)=24 because we have 24 different permutation matrices 5 x 5 with trace 0 and determinant +1.
%Y A171907 A171806, A171808, A171809, A171906, A171908
%K A171907 fini,full,nonn
%O A171907 0,1
%A A171907 Artur Jasinski (grafix(AT)csl.pl), Dec 31 2009

%I A171906
%S A171906 44,45,20,10,0,1
%N A171906 a(n)= Number of 5 X 5 permutation matrices with trace n
%e A171906 a(0)=44 because we have 44 different permutation matrices 5 x 5 with trace 0.
%Y A171906 A171806, A171808, A171809, A171907, A171908
%K A171906 nonn
%O A171906 0,1
%A A171906 Artur Jasinski (grafix(AT)csl.pl), Dec 31 2009

%I A171870
%S A171870 0,1,0,4,5,3,1,4,2,5,0,3,6,40,4,38,7,2,5,10,39,8,3,37,6,6,1,40,9,9,4,38,
%T A171870 7,7,2,36,41,2,5,10,5,39,0,8,8,32,3,37,42,6,6,30,11,35,40,23,1,9,4,9,33,
%U A171870 14,38,14,43,7,7,12,31,12,2,36,41,41,5,2,10,29,10,17,34,5,39,22,15,44,8
%N A171870 a(x) : associated numbers of 3 in the Collatz 3x + 1 problem.
%C A171870 In the Collatze problem 3x + 1, it's possible to associate every number x with a polynomial f(z). Exemple with x = 17 : if we keep only odd number, we obtain the trajectory (17, 13, 5, 1). 13 = (3.17 + 1) / 4 5 = (3.13 + 1) / 8 = (3(3.17 + 1) / 4 ) + 1 ) / 8 1 = (3.5 + 1) / 16 = (3(3(3.17 + 1) / 4 ) + 1 ) / 8) / 16 If we substitute the number 3 by z, we obtain : 17z^3 + z^2 + 4z - 480 = 0. This equation has 3 roots : the real trivial solution z0 = 3, and two complex roots z1 and z2. The associated numbers of 3 are z1 and z2. In this way, a(17) = 2. For the ten first odd numbers, we obtain : a(1) = 0, a(3) = 1, a(5) = 0, a(7) = 4, a(9) = 5, a(11) = 3, a(13) = 1, a(15) = 4, a(17) = 2, a(21) = 5 Remark : a(2x) = a(x)
%D A171870 3x+1 problem, J. C. Lagarias's 3x+1 Problem Annotated Bibliography
%F A171870 In the classical 3x + 1 algorithm, compute N = M - 1 where M is the number of multiplications by 3 (M is the degree of the associated polynomial).
%K A171870 nonn,uned
%O A171870 0,4
%A A171870 Michel Lagneau (mn.lagneau2(AT)orange.fr), Dec 30 2009

%I A171869
%S A171869 1,2,2,2,2,4,8,16,32,64,128,256,512,1024,2048,4096,8192,16384,32768,
%T A171869 65536,131072
%N A171869 The triangle of partial sums of A170827 show periodics sequences s(i) with periods p(n) : s(1) = (5,5, 5,5,...,5, 5, ...) , p(1) = 1 s(2) = (2,7,2,7, ...,2,7, ...) , p(2) = 2 s(3) = (3,6,3,6, ...,3,6, ...), p(3) = 2 s(4) = (0,9,0,9, ...,0,9, ...), p(4) = 2 s(5) = (5,9,5,9, ...,5,9, ...) , p(5) = 2 s(6) = (3,8,8,3,8,8, ...,3,8,8, ...) , p(6) = 4 s(7) = (0,2,3,0,5,7,8,5, 0,2,3,0,5,7,8,5, ...) , p(7) = 8 s(8) = (6,4,5,5,3,6,2,8,1,9,0,0,8,1,7,3, 6,4,5,5,3,6,2,8,1,9,0,0,8,1,7,3, ...) , p(8) = 16 ... ... ... p(n) = 2^(n-4) Proof : (3/2)^n and (3/2)^(n + q), q = 2^(n-4)k with k = 0,1,2,.. have the same sequence of n last digits in the decimal expansion.
%Y A171869 Cf. A002379, A170828 , A170827.
%K A171869 nonn,uned
%O A171869 1,2
%A A171869 Michel Lagneau (mn.lagneau2(AT)orange.fr), Dec 30 2009

%I A171929
%S A171929 1,3,9,15,45,105,315,1155,7425,8415,8925,31815,32445,351351,442365,
%T A171929 13800465,14571585,16286445,20355825,20487159,78524145,132701205,
%U A171929 159030135,815634435,2586415095,29169504045,40833636525,125208115065
%N A171929 Odd numbers whose abundancy is closer to 2 than any smaller odd number.
%C A171929 So far all known perfect numbers (abundancy = 2) are even. Some examples are: 6, 28, 496, 8128. It has been conjectured but not proved that there are no odd perfect numbers. This sequence provides the list of odd numbers that approach perfection (odd numbers which abundancy is closer to two than the abundancy of any smaller odd number)
%C A171929 Odd numbers n such that abs(sigma(n)/n-2) < abs(sigma(m)/m-2) for all m < n. That is, each n is closer to being an odd perfect number than the preceding n. Interestingly, if abs(sigma(n)/n-2) is expressed as a reduced fraction, the numerator of the fraction is 2 for 25 out of the first 30 terms. Terms a(29) and a(30) are 127595519865 and 154063853475. [From T. D. Noe (noe(AT)sspectra.com), Jan 28 2010]
%C A171929 Contribution from Max Alekseyev (maxale(AT)gmail.com), Jan 26 2010: (Start)
%C A171929 Indices of successive minima in the sequence |A000203(n)/n - 2| for odd n.
%C A171929 The sequence would terminate at the smallest odd perfect number (if it exists). (End)
%H A171929 Mathworld, <a href="http://mathworld.wolfram.com/Abundancy.html">Abundancy</a>
%H A171929 T. D. Noe, <a href="http://www.research.att.com/~njas/sequences/b171929.txt">Table of n, a(n) for n=1..36</a> (terms < 10^12)
%e A171929 Example: a(8) = 1155 since S(1155)/1155 = 1.9948 which is closer to 2 than any smaller a(n)
%t A171929 minDiff=Infinity; k=-1; Table[k=k+2; While[abun=DivisorSigma[1,k]/k; Abs[2-abun] > minDiff, k=k+2]; minDiff=Abs[2-abun]; k, {15}] [From T. D. Noe (noe(AT)sspectra.com), Jan 28 2010]
%o A171929 (PARI) { m=2; forstep(n=1,10^10,2, t=abs(sigma(n)/n - 2); if(t<m,m=t;print1(n,", ");); ); } [From Max Alekseyev (maxale(AT)gmail.com), Jan 26 2010]
%Y A171929 Cf. A000396, A053624, A119239
%Y A171929 Cf. A088012, A117349 [From T. D. Noe (noe(AT)sspectra.com), Jan 28 2010]
%K A171929 hard,nonn,new
%O A171929 1,2
%A A171929 Sergio Pimentel (ferdiego(AT)suddenlink.net), Jan 05 2010
%E A171929 Name improved by T. D. Noe (noe(AT)sspectra.com), Jan 28 2010
%E A171929 More terms from Max Alekseyev (maxale(AT)gmail.com), T. D. Noe (noe(AT)sspectra.com) and J. Mulder (jasper.mulder(AT)planet.nl), Jan 26 2010

%I A171928
%S A171928 3,6,487,1461,4383,13149
%N A171928 Numbers n which can divide the periodic part of the decimal expansion of 1/n.
%e A171928 a(3)=487 means: 1/487=0.(002053388090349075975359342915811088295687885010266940451745379876796714579055441478439425051334702258726899383983572895277207392197125256673511293634496919917864476386036960985626283367556468172484599589322381930184804928131416837782340862422997946611909650924024640657084188911704312114989733059548254620123203285420944558521560574948665297741273100616016427104722792607802874743326488706365503080082135523613963039014373716632443531827515400410677618069815195071868583162217659137577),
%e A171928 and the periodic part of it: 2053388090349075975359342915811088295687885010266940451745379876796714579055441478439425051334702258726899383983572895277207392197125256673511293634496919917864476386036960985626283367556468172484599589322381930184804928131416837782340862422997946611909650924024640657084188911704312114989733059548254620123203285420944558521560574948665297741273100616016427104722792607802874743326488706365503080082135523613963039014373716632443531827
%K A171928 base,easy,nonn
%O A171928 1,1
%A A171928 Zhining Yang (northwolves(AT)163.com), Jan 05 2010

%I A171995
%S A171995 1,625,1715
%N A171995 Twin natural nonprimes with nonprime number of prime factors.
%e A171995 a(1)=1(without prime factors), a(2)=625(=5*5*5*5), a(3)=1715(=5*7*7*7).
%Y A171995 Cf. A171697.
%K A171995 nonn,new,bref
%O A171995 1,2
%A A171995 Juri-Stepan Gerasimov (2stepan(AT)rambler.ru), Jan 21 2010

%I A171994
%S A171994 245,325,343,435,475,637,665,715,805,833,845,847,925,1001,1025,1045,
%T A171994 1075,1175,1265,1331,1463,1475,1505,1645,1675,1705,1729,1775,1805,1855
%N A171994 Twin natural nonprimes are products of free primes.
%e A171994 a(1)=245(=5*7*7), a(2)=325(=5*5*13).
%Y A171994 Cf. A171897.
%K A171994 nonn,new
%O A171994 1,1
%A A171994 Juri-Stepan Gerasimov (2stepan(AT)rambler.ru), Jan 21 2010

%I A171993
%S A171993 1,4,8,10,14,16,20,22,25,26,28,32,34,35,38,40,44,46,49,50,52,55,56,58,
%T A171993 62,64,65,68,70,74,76,77,80,82,85,86,88,91,92,94,95,98,100,104,106,110,
%U A171993 112,115,116,118,119,121,122,124,125,128,130,133,134,136,140,142,143
%N A171993 Nonprimes of the form 3*n+-1.
%C A171993 Unit together with composite numbers that are not multiples of free.
%F A171993 a(n) = A157695(n-1), n>1. [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Feb 07 2010]
%Y A171993 Cf. A018252, A157695.
%K A171993 nonn,new
%O A171993 1,2
%A A171993 Juri-Stepan Gerasimov (2stepan(AT)rambler.ru), Jan 21 2010

%I A171992
%S A171992 0,1,1,2,1,2,1,2,3,1,3,2,1,2,3,3,1,3,2,1,3,2,3,4,2,1,2,1,2,7,2,3,1,5,1,
%T A171992 3,3,2,3,3,1,5,1,2,1,6,6,2,1,2,3,1,5,3,3,3,1,3,2,1,5,7,2,1,2,7,3,5,1,2,
%U A171992 3,4,3,3,2,3,4,2,4,5,1,5,1,3,2,3,4,2,1,2,6,4,2,4,2,3,6
%N A171992 a(n)=((nth prime of the form 3*k+-1)-nth prime)/2.
%F A171992 a(n)=(A045344(n)-A000040(n))/2; a(n-1)=(A001223(n-1))/2.
%F A171992 a(n)=A028334(n), n>0. [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jan 23 2010]
%e A171992 a(1)=((3*1-1)-2)/2=0, a(2)=((3*2-1)-3)/2=1, a(3)=((3*2+1)-5)/2=1.
%Y A171992 Cf. A000040, A001223, A045344.
%K A171992 nonn,new
%O A171992 1,4
%A A171992 Juri-Stepan Gerasimov (2stepan(AT)rambler.ru), Jan 21 2010

%I A171991
%S A171991 0,2,2,4,2,4,2,4,6,2,6,4,2,4,6,6,2,6,4,2,6,4,6,8,4,2,4,2,4,14,4,6,2,10,
%T A171991 2,6,6,4,6,6,2,10,2,4,2,12,12,4,2,4,6,2,10,6,6,6,2,6,4,2,10,14,4,2,4,14,
%U A171991 6,10,2,4,6,8,6,6,4,6,8,4,8,10,2,10,2,6,4,6,8,4,2,4,12,8,4,8,4,6,12
%N A171991 nth prime of the form 3*k+-1 minus nth prime.
%C A171991 a(n-1)=differences between consecutive odd primes.
%F A171991 a(n)=A045344(n)-A000040(n).
%e A171991 a(1)=(3*1-1)-2=0, a(2)=(3*2-1)-3=2, a(3)=(3*2+1)-5=2.
%Y A171991 Cf. A000040, A001223, A045344.
%K A171991 nonn,new
%O A171991 1,2
%A A171991 Juri-Stepan Gerasimov (2stepan(AT)rambler.ru), Jan 21 2010

%I A171845
%S A171845 0,4,6,9,12,15,18,21,25,26,27,30,33,34,35,39,42,45,49,50,51,55,56,57,60,
%T A171845 63,64,65,69,72,75,76,77,81,85,86,87,91,92,93,94,95,99,102,105,106,108,
%U A171845 111,115,116,117,118,119,120,121,122,123,124,125,129,133,134,135,138
%N A171845 The nonfrontier or nontrivial nonprimes: nonprimes n such that neither n+-1 is prime or nonprime.
%C A171845 Or, numbers n such that n+1 and n-1 are both prime or both nonprime. A168546 U A171845 = A171690, where A171690 U A100317 = A001477.
%Y A171845 Cf. A001477(the nonnegative numbers), A100317(the frontier or trivial numbers), A169546(the nonfrontier or nontrivial primes), A171690(the nonfrontier or nontrivial numbers).
%K A171845 nonn,new
%O A171845 1,2
%A A171845 Juri-Stepan Gerasimov (2stepan(AT)rambler.ru), Jan 21 2010

%I A171990
%S A171990 1,2,3,16,3814280
%N A171990 Lowest integer a(n) for which the iterated function ln, iterated n times, is defined.
%C A171990 ln(a(1)) is defined if a(1)>0 => a(1) = 1. ln(ln(a(2))) is defined if ln(a(2))>0 => a(2)>1 => a(2)=2. The sequence grows rapidly and a(6) = 2.33206...10^1656520
%F A171990 a(n) = ceil(e^(e^...))), n times.
%e A171990 a(2) = 2 because ln(ln(2)) is defined and ln(ln(1)) is not, a(3) = 3 because ln(ln(ln(3))) is defined, a(4) = 16 because ln(ln(ln(ln(16)))) is defined.
%K A171990 nonn,new
%O A171990 1,2
%A A171990 Alexis Monnerot-Dumaine (alexis.monnerotdumaine(AT)gmail.com), Jan 21 2010

%I A171989
%S A171989 2,6,30,210,2310,29464,476928,9671392,222388792,6438663000,200560490130,
%T A171989 7379606916000,299261862900000,13004421443456272,614231422273479360,
%U A171989 31727029501157817600,1915248189055217892480,116762424492324428512272
%N A171989 A000010((A006862(n))
%p A171989 A006862 := proc(n) if n = 0 then 2; else 1+mul(ithprime(j),j=1..n) ; end if: end proc: A171989 := proc(n) numtheory[phi](A006862(n)) ; end proc: seq(A171989(n),n=1..18) ; [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jan 30 2010]
%Y A171989 Cf. A000010, A006862.
%K A171989 nonn,new
%O A171989 1,1
%A A171989 Giovanni Teofilatto (g.teofilatto(AT)tiscalinet.it), Jan 21 2010
%E A171989 a(1) inserted and extended beyond a(5) by R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jan 30 2010

%I A171988
%S A171988 1,2,101,146,424,848,1370,2404,3049,3250,3257,3700,4709,5954,9805,11237,
%T A171988 11889,14885,19465,20738,22261,22736,26216,28564,29042,35113,38900,
%U A171988 44433,44433,46660,57128
%N A171988 Appearance radii of visible vectors in the medial axis test mask for the Euclidean distance in Z^2
%D A171988 J. Hulin and E. Thiel. Visible vectors and discrete Euclidean medial axis. Discrete and Computational Geometry, 42(4):759-773, 2009.
%D A171988 J. Hulin and E. Thiel. Farey Sequences and the Planar Euclidean Medial Axis Test Mask. In 13th IWCIA, LNCS 5852, pages 82-95, Cancun, Mexico, Nov 2009.
%D A171988 E. Remy and E. Thiel. Exact Medial Axis with Euclidean Distance. Image and Vision Computing, 23(2):167-175, 2005.
%H A171988 E. Thiel, <a href="b171988.txt">Table of n, R(n) for n=1..350</a>
%H A171988 E. Thiel, the <a href="http://pageperso.lif.univ-mrs.fr/~edouard.thiel/npic/index.html">Npic</a> library and tools.
%H A171988 E. Thiel and E. Remy, Medial Axis for Euclidean Distance in nD : <a href="http://pageperso.lif.univ-mrs.fr/~edouard.thiel/IVC2004/index.html">examples and sources</a>.
%K A171988 nonn,new
%O A171988 1,2
%A A171988 Edouard THIEL (Edouard.Thiel(AT)lif.univ-mrs.fr), Jan 21 2010

%I A171987
%S A171987 1,2,3,4,5,6,7,8,9,10,12,13,14,15,16,17,18,20,24,25,26,28,29,30,31,32,
%T A171987 33,34,36,40,48,49,50,52,56,57,58,60,61,62,63,64
%N A171987 Best explained by example: in the binary representation, start with 10000, then add 1 and push the 1 to the left: 10001, 10010,10100,11000, then add another one, 11001,11010,11100,etc: 11101,11110,11111. Then proceed with the next length of numbers: 100000 etc.
%K A171987 nonn,new
%O A171987 1,2
%A A171987 Stefan Maubach (s.maubach(AT)science.ru.nl), Jan 21 2010

%I A171967
%S A171967 1,2,2,5,5,10,12,20,25,37,49,68,90,119,158,206,269,344,446,565,722,908,
%T A171967 1148,1435,1795,2229,2765,3416,4204,5164,6315,7717,9380,11406,13793,
%U A171967 16692,20093,24203,29012,34799,41552,49636,59059,70279,83341,98822
%N A171967 Number of partitions of n with distinct numbers of odd and even parts.
%C A171967 a(n) = A000041(n) - A045931(n) = A108949(n) + A108950(n).
%Y A171967 Cf. A130780, A171966.
%K A171967 nonn,new
%O A171967 0,2
%A A171967 Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Jan 21 2010

%I A171966
%S A171966 1,0,1,1,2,3,4,6,8,12,15,21,28,37,49,63,83,105,138,171,223,275,353,433,
%T A171966 551,673,846,1031,1282,1558,1922,2327,2848,3440,4179,5032,6078,7293,
%U A171966 8763,10482,12534,14943,17797,21146,25090,29719,35138,41493,48908,57578
%N A171966 Number of partitions of n having not more odd than even parts.
%C A171966 a(n) = A108949(n) + A045931(n) = A000041(n) - A108950(n).
%Y A171966 Cf. A130780, A171967.
%K A171966 nonn,new
%O A171966 0,5
%A A171966 Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Jan 21 2010

%I A171986
%S A171986 1,1,12,11,22,11,24,13,22,31,42,11,24,13,22,31,42,11,24,13,22,31,46,15,
%T A171986 24,33,42,51,62,11,26,15,24,33,42,51,64,13,22,31,42,11,24,13,22,31,46,
%U A171986 15,24,33,42,51,66,15,24,33,42,51,62,11,26,15,24,33,42,51,64,13,22,31
%N A171986 Concatenate distance to previous prime & distance to next prime. If no previous prime, use 0.
%e A171986 a(7)=24 as 5 is largest previous prime to 7, distance is 2, & 11 is smallest next prime to 7, distance 4. Concatenate 2 & 4 gives 24.
%K A171986 nonn,new
%O A171986 1,3
%A A171986 Gerald Hillier (adr.rabbicat(AT)gmail.com), Jan 21 2010

%I A171984
%S A171984 4,8,12,16,20,24,28,32,37,41,45,49,53,57,61,65,70,74,78,82,86,90,94,98,
%T A171984 103,107,111,115,119,123,127,131,136,140,144,148,152,156,160,164,169,
%U A171984 173,177,181,185,189,193,197,202,206,210,214,218,222,226,230,235,239
%N A171984 Beatty sequence for sqrt(17)
%K A171984 nonn,new
%O A171984 1,1
%A A171984 Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Jan 21 2010

%I A171985
%S A171985 1,2,5,11,23,44,82,146,252,423,695,1116,1763,2738,4192,6334,9459,13968,
%T A171985 20425,29588,42496,60547,85628,120246,167762,232605,320635,439544,
%U A171985 599412,813360,1098480,1476870,1977087,2635869,3500382,4630932,6104533
%N A171985 Number of partitions of 2*n-1 into parts not greater than n.
%C A171985 Central terms of the triangle in A026820: a(n)=A026820(2*n-1,n).
%e A171985 a(4) = (partitions of 7 into parts <= 4) = #{4+3, 4+2+1, 4+1+1+1, 3+3+1, 3+2+2, 3+2+1+1, 3+1+1+1+1, 2+2+2+1, 2+2+1+1+1, 2+1+1+1+1+1, 1+1+1+1+1+1+1} = 11.
%K A171985 nonn,new
%O A171985 1,2
%A A171985 Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Jan 21 2010

%I A171983
%S A171983 3,7,10,14,18,21,25,28,32,36,39,43,46,50,54,57,61,64,68,72,75,79,82,86,
%T A171983 90,93,97,100,104,108,111,115,118,122,126,129,133,137,140,144,147,151,
%U A171983 155,158,162,165,169,173,176,180,183,187,191,194,198,201,205
%N A171983 Beatty sequence for sqrt(13)
%K A171983 nonn,new
%O A171983 1,1
%A A171983 Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Jan 21 2010

%I A171982
%S A171982 3,6,9,13,16,19,23,26,29,33,36,39,43,46,49,53,56,59,63,66,69,72,76,79,
%T A171982 82,86,89,92,96,99,102,106,109,112,116,119,122,126,129,132,135,139,142,
%U A171982 145,149,152,155,159,162,165,169,172,175,179,182,185,189,192,195,198
%N A171982 Beatty sequence for sqrt(11)
%K A171982 nonn,new
%O A171982 1,1
%A A171982 Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Jan 21 2010

%I A171874
%S A171874 0,0,0,1,1,2,4,7,16,46,174,3311,268446771,401906756202069927727330981
%N A171874 a(n) = a(n-1) + a(n-2)*a(n-3) + a(n-4)^a(n-5)
%C A171874 First 5 terms are {0,0,0,1,1}; thereafter apply the recurrence. Note that 0^0=1.
%H A171874 R. Munafo, <a href="http://mrob.com/pub/math/seq-accelerate.html">Accelerating Sequences</a>
%K A171874 nonn,new
%O A171874 0,6
%A A171874 Robert Munafo (mrob27(AT)gmail.com), Jan 21 2010

%I A171873
%S A171873 1,1,2,10,280,1173468
%N A171873 Column sums of triangle A171871
%C A171873 Next term is known to be greater than 220146725295227, based on link between A171871 and A039754.
%H A171873 R. Munafo, <a href="http://mrob.com/pub/math/seq-a005646.html">Classifications of N Elements</a>
%K A171873 nonn,new
%O A171873 0,3
%A A171873 Robert Munafo (mrob27(AT)gmail.com), Jan 21 2010

%I A171872
%S A171872 1,0,1,0,1,1,0,2,3,3,1,1,0,3,17,36,60,56,50,27,19,6,4,1,1,0,6,74,573,
%T A171872 2802,10087,26512,55088,91984,130267,157662,168890,158658,133576,98804,
%U A171872 65664,38073,19963,9013,3779
%N A171872 Triangle read by columns: Distinct classifications of N elements containing exactly R binary partitions.
%C A171872 The bottom of each column is marked by a single 0 in this sequence. Value is 0 for all (N,R) for which N is greater than 2^R.
%C A171872 See note on efficient computation in A171871.
%H A171872 R. Munafo, <a href="http://mrob.com/pub/math/seq-a005646.html">Classifications of N Elements</a>
%Y A171872 First term in each column is A000055(R+1).
%Y A171872 Column 4 shares terms with A034189, column 5 with A034190.
%K A171872 nonn,new
%O A171872 0,8
%A A171872 Robert Munafo (mrob27(AT)gmail.com), Jan 21 2010

%I A171871
%S A171871 1,0,1,0,0,1,0,0,1,2,0,0,0,3,3,0,0,0,3,17,6,0,0,0,1,36,74,11,0,0,0,1,60,
%T A171871 573,358,23,0,0,0,0,56,2802,7311,1631,47,0,0,0,0,50,10087,107938,83170,
%U A171871 7563,106,0,0,0,0,27,26512,1186969,3121840,866657,34751,235,0,0,0,0,19
%N A171871 Triangle read by rows: Distinct classifications of N elements containing exactly R binary partitions
%C A171871 Significance of triangle suggested by Franklin T. Adams-Watters on 19 Dec 2009 Row N has N terms in this sequence. The triangle starts:
%C A171871 1
%C A171871 0,1
%C A171871 0,0,1
%C A171871 0,0,1,2
%C A171871 0,0,0,3,3
%C A171871 0,0,0,0,3,17,6
%C A171871 0,0,0,0,1,36,74,11
%C A171871 Value is A000055(N) when R=N-1 (last term in each row). (Conjectured by R. Munafo Dec 28 2009, then proved by A. Weimholt and F. T. Adams-Watters on Dec 29 2009)
%C A171871 Value is 1 when N=2^R.
%C A171871 Value is 1 when N=(2^R)-1.
%C A171871 Value is R when R>2 and N=(2^R)-2.
%C A171871 Value is A034198(R) when R>2 and N=(2^R)-3.
%C A171871 Conjecture by R. Munafo: In general, in each column, the last 2^(R-1) values are the same as the first 2^(N-1) values from the corresponding row of A039754. - R. Munafo, (mrob(AT)mrob.com) Dec 30 2009.
%C A171871 Value is 0 for all (N,R) for which N is greater than 2^R.
%C A171871 Each term A(N,R) can be computed most efficiently by first enumerating all classifications in A(N-1,R) plus those in A(N-1,R-1), and then adding an additional type and/or partition to each.
%H A171871 R. Munafo, <a href="http://mrob.com/pub/math/seq-a005646.html">Classifications of N Elements</a>
%Y A171871 Cf. Row sums are A005646, Column sums are A171832.
%Y A171871 Cf. A039754.
%Y A171871 Last term in each row is A000055(N).
%Y A171871 Same triangle read by columns is A171872.
%K A171871 nonn,new
%O A171871 0,10
%A A171871 Robert Munafo (mrob27(AT)gmail.com), Jan 21 2010

%I A171963
%S A171963 0,0,0,1,1,1,1,0,2,1,0,1,3,1,3,1,3,1,5,2,2,1,3,3,3,4,2,2,4,4,8,3,3,8,4,
%T A171963 5,8,3,6,7,3,5,7,9,5,5,7,10,7,6,11,5,8,7,5,9,8,8,9,6,10,8,8,7,11,9,9,10,
%U A171963 9,7,15,12,10,11,9,10,15,9,12,10,12,12,13,11,11,11,15,12,17,12,13,16,14
%N A171963 Number of partitions of the n-th semiprime into two semiprimes.
%C A171963 a(n) = A072931(A001358(n)).
%H A171963 R. Zumkeller, <a href="b171963.txt">Table of n, a(n) for n = 1..1000</a>
%e A171963 a(13) = A072931(A001358(13)) = A072931(35) = #{26+9,25+10,21+14} = #{2*13+3*3,5*5+2*5,3*7+2*7} = 3.
%K A171963 nonn,new
%O A171963 1,9
%A A171963 Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Jan 21 2010

%I A171776
%S A171776 1,1,5,141,25161,25295385,129002055885,3167498196303525,
%T A171776 363195624958803434385,190409085693362565632615985,
%U A171776 449225585595812339036501379506325
%N A171776 E.g.f.: A(x) = exp( Sum_{n>=1} 2^(n(n-1)) * x^n/n ).
%F A171776 a(n) = A155200(n)*n!/2^n and is odd for n>=0.
%e A171776 E.g.f.: A(x) = 1 + x + 5*x^2/2! + 141*x^3/3! + 25161*x^4/4! +...
%e A171776 log(A(x)) = x + 4*x^2/2 + 64*x^3/3 + 4096*x^4/4 + 1048576*x^5/5 +..
%o A171776 (PARI) {a(n)=n!*polcoeff(exp(sum(m=1, n+1, 2^(m*(m-1))*x^m/m)+x*O(x^n)), n)}
%Y A171776 Cf. A155200.
%K A171776 nonn,new
%O A171776 0,3
%A A171776 Paul D. Hanna (pauldhanna(AT)juno.com), Jan 20 2010

%I A171799
%S A171799 1,2,20,648,78608,37949472,74258600000,589859028828288,
%T A171799 18957096840069579008,2455889836782322072945152,
%U A171799 1278835681226410156250000000000,2671465293024628033252951422140418048
%N A171799 O.g.f.: Sum_{n>=0} 2^(n^2)*x^n/(1 - 2^n*x)^n.
%F A171799 a(n) = 2^n*(2^n + 1)^(n-1) for n>0 with a(0)=1.
%e A171799 G.f.: A(x) = 1 + 2*x + 20*x^2 + 648*x^3 + 78608*x^4 +...
%e A171799 A(x) = 1 + 2*x/(1-2*x) + 2^4*x^2/(1-2^2*x)^2 + 2^9*x^3/(1-2^3*x)^3 +...
%o A171799 (PARI) {a(n)=polcoeff(sum(m=0,n,2^(m^2)*x^m/(1-2^m*x+x*O(x^n))^m),n)}
%o A171799 (PARI) {a(n)=if(n==0,1,2^n*(2^n+1)^(n-1))}
%Y A171799 Cf. A171801, A171800, A136516.
%K A171799 nonn,new
%O A171799 0,2
%A A171799 Paul D. Hanna (pauldhanna(AT)juno.com), Jan 20 2010

%I A171801
%S A171801 1,4,56,2448,379168,223096896,514098000000,4691436926959872,
%T A171801 170097530401558168064,24520599890836361905701888,
%U A171801 14055963692387060312500000000000
%N A171801 O.g.f.: Sum_{n>=0} (n+1)*2^(n^2)*x^n/(1 - 2^n*x)^n.
%F A171801 a(n) = 2^n * ((n+1)*2^n + 2) * (2^n + 1)^(n-2) for n>0 with a(0)=1.
%e A171801 G.f.: A(x) = 1 + 4*x + 56*x^2 + 2448*x^3 + 379168*x^4 +...
%e A171801 A(x) = 1 + 2*2*x/(1-2*x) + 3*2^4*x^2/(1-2^2*x)^2 + 4*2^9*x^3/(1-2^3*x)^3 +...
%o A171801 (PARI) {a(n)=polcoeff(sum(m=0,n,(m+1)*2^(m^2)*x^m/(1-2^m*x+x*O(x^n))^m),n)}
%o A171801 (PARI) {a(n)=if(n==0,1,2^n*((n+1)*2^n + 2)*(2^n + 1)^(n-2))}
%Y A171801 Cf. A171799, A171800.
%K A171801 nonn,new
%O A171801 0,2
%A A171801 Paul D. Hanna (pauldhanna(AT)juno.com), Jan 20 2010

%I A171800
%S A171800 1,5,65,2673,397953,228882753,520970490625,4723480504289025,
%T A171800 170687922720157732865,24563695027660686202250241,
%U A171800 14068441356460459384918212890625
%N A171800 a(n) = ((n+1)*2^n + 1)*(2^n + 1)^(n-1).
%F A171800 O.G.f.: Sum_{n>=0} (n+1)*2^(n^2) * x^n/(1 - 2^n*x)^(n+1).
%F A171800 E.g.f.: Sum_{n>=0} (n+1)*2^(n^2) * exp(2^n*x) * x^n/n!.
%e A171800 G.f.: A(x) = 1 + 5*x + 65*x^2 + 2673*x^3 + 397953*x^4 +...
%e A171800 A(x) = 1/(1-x) + 2*2*x/(1-2*x)^2 + 3*2^4*x^2/(1-2^2*x)^3 + 4*2^9*x^3/(1-2^3*x)^4 +...
%o A171800 (PARI) {a(n)=polcoeff(sum(m=0,n,(m+1)*2^(m^2)*x^m/(1-2^m*x+x*O(x^n))^(m+1)),n)}
%o A171800 (PARI) {a(n)=n!*polcoeff(sum(k=0, n, (k+1)*2^(k^2)*exp(2^k*x)*x^k/k!), n)}
%o A171800 (PARI) {a(n)=((n+1)*2^n+1)*(2^n+1)^(n-1)}
%Y A171800 Cf. A136516, A171801, A171799.
%K A171800 nonn,new
%O A171800 0,2
%A A171800 Paul D. Hanna (pauldhanna(AT)juno.com), Jan 20 2010

%I A171981
%S A171981 5,75025,9006076025,332813125,54036081025,162108093025,12304690625,
%T A171981 3662109765625,1238325212525,225150026875625,8562180281412026525,
%U A171981 309581286250625,15197626762525,4520507828125,2059936966552758203125
%N A171981 Smallest multiples from A129066 of primes from A171980.
%C A171981 A129066 lists integers n such that n divides n-th Fibonacci number A000045(n) with multiples of 12 excluded, while A171980 lists possible prime divisors of elements of A129066 in the increasing order. This sequence lists smallest multiples from A129066 of primes from A171980.
%F A171981 a(n) = min { A129066(m) : A171980(n)|A129066(m) }
%Y A171981 Cf. A114207, A140258
%K A171981 nonn,new
%O A171981 1,1
%A A171981 Max Alekseyev (maxale(AT)gmail.com), Jan 20 2010

%I A171980
%S A171980 5,3001,120041,532501,720241,2160721,3937501,9375001,16505501,120040001,
%T A171980 158453021,165055001,202567501,289312501,562500061,900307501,985937501,
%U A171980 1500512501,1512504701,3169060421,3301100021,3908604433,3993757501
%N A171980 Prime divisors of elements of A129066.
%C A171980 Corresponding smallest multiples from A129066 are given in A171981.
%C A171980 Prime p>5 is in this sequence if the multiplicative order of (sqrt(5)-3)/2 modulo p is the product of smaller terms of this sequence.
%H A171980 Max Alekseyev, <a href="b171980.txt">Table of n, a(n) for n = 1..25</a> (complete up to 10^10)
%Y A171980 Cf. A057719, A066364, A129729, A171981
%K A171980 nonn,new
%O A171980 1,1
%A A171980 Max Alekseyev (maxale(AT)gmail.com), Jan 20 2010

%I A171975
%S A171975 0,1,1,2,3,3,4,4,5,6,6,7,7,8,9,9,10,11,11,12,12,13,14,14,15,15,16,17,17,
%T A171975 18,18,19,20,20,21,22,22,23,23,24,25,25,26,26,27,28,28,29,30,30,31,31,
%U A171975 32,33,33,34,34,35,36,36,37,37,38,39,39,40,41,41,42,42,43,44,44,45,45
%N A171975 Integer part of the circumsphere radius of a regular tetrahedron with edge length n.
%C A171975 -3 <= 4*a(n) - 3*A171974(n) < 3;
%C A171975 a(n)*A171974(n) <= A007590(n).
%H A171975 Wikipedia, <a href="http://en.wikipedia.org/wiki/Tetrahedron">Tetrahedron</a>
%H A171975 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Tetrahedron.html">Tetrahedron</a>
%F A171975 a(n) = floor(n*sqrt(6)/4).
%Y A171975 Cf. A171973, A171972, A022840.
%K A171975 nonn,new
%O A171975 1,4
%A A171975 Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Jan 20 2010

%I A171974
%S A171974 0,1,2,3,4,4,5,6,7,8,8,9,10,11,12,13,13,14,15,16,17,17,18,19,20,21,22,
%T A171974 22,23,24,25,26,26,27,28,29,30,31,31,32,33,34,35,35,36,37,38,39,40,40,
%U A171974 41,42,43,44,44,45,46,47,48,48,49,50,51,52,53,53,54,55,56,57,57,58,59
%N A171974 Integer part of the height of a regular tetrahedron with edge length n.
%C A171974 -3 <= 4*A171975(n) - 3*a(n) < 3;
%C A171974 a(n)*A171975(n) <= A007590(n);
%C A171974 floor(a(n)*A171971(n)/3) <= A171973(n).
%H A171974 Wikipedia, <a href="http://en.wikipedia.org/wiki/Tetrahedron">Tetrahedron</a>
%H A171974 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Tetrahedron.html">Tetrahedron</a>
%F A171974 a(n) = floor(n*sqrt(6)/3).
%Y A171974 Cf. A171972, A022840.
%K A171974 nonn,new
%O A171974 1,3
%A A171974 Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Jan 20 2010

%I A171973
%S A171973 0,0,3,7,14,25,40,60,85,117,156,203,258,323,397,482,579,687,808,942,
%T A171973 1091,1254,1433,1629,1841,2071,2319,2587,2874,3181,3510,3861,4235,4632,
%U A171973 5052,5498,5969,6466,6990,7542,8122,8731,9369,10039,10739,11471,12235
%N A171973 Integer part of the volume of a regular tetrahedron with edge length n.
%C A171973 Lim{n->oo} a(n)/A000292(n) = sqrt(2)/2;
%C A171973 floor(A171971(n)*A171974(n)/3) <= a(n).
%H A171973 Wikipedia, <a href="http://en.wikipedia.org/wiki/Tetrahedron">Tetrahedron</a>
%H A171973 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Tetrahedron.html">Tetrahedron</a>
%F A171973 a(n) = floor(n^3 * sqrt(2) / 12).
%Y A171973 Cf. A001951, A171972, A171975.
%K A171973 nonn,new
%O A171973 1,3
%A A171973 Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Jan 20 2010

%I A171972
%S A171972 1,6,15,27,43,62,84,110,140,173,209,249,292,339,389,443,500,561,625,692,
%T A171972 763,838,916,997,1082,1170,1262,1357,1456,1558,1664,1773,1886,2002,2121,
%U A171972 2244,2371,2501,2634,2771,2911,3055,3202,3353,3507,3665,3826,3990,4158
%N A171972 Integer part of the surface area of a regular tetrahedron with edge length n.
%C A171972 a(n) = A022838(A000290(n));
%C A171972 A171970(n)*A005843(n) <= a(n);
%C A171972 a(n) <= 4*A171971(n); 0 <= a(n) - 4*A171971(n) < 4.
%H A171972 Wikipedia, <a href="http://en.wikipedia.org/wiki/Tetrahedron">Tetrahedron</a>
%H A171972 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Tetrahedron.html">Tetrahedron</a>
%F A171972 a(n) = floor(n^2 * sqrt(3)).
%Y A171972 Cf. A171974, A171973, A171975.
%K A171972 nonn,new
%O A171972 1,2
%A A171972 Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Jan 20 2010

%I A171971
%S A171971 0,1,3,6,10,15,21,27,35,43,52,62,73,84,97,110,125,140,156,173,190,209,
%T A171971 229,249,270,292,315,339,364,389,416,443,471,500,530,561,592,625,658,
%U A171971 692,727,763,800,838,876,916,956,997,1039,1082,1126,1170,1216,1262,1309
%N A171971 Integer part of the area of an equilateral triangle with side length n.
%C A171971 a(n)*A171974(n)/3 <= A171973(n);
%C A171971 A171970(n)*A004526(n) <= a(n).
%H A171971 Wikipedia, <a href="http://en.wikipedia.org/wiki/Equilateral_triangle">Equilateral triangle</a>
%H A171971 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/EquilateralTriangle.html">Equilateral Triangle</a>
%F A171971 a(n) = floor(n^2 * sqrt(3) / 4).
%Y A171971 Cf. A171972, A022838.
%K A171971 nonn,new
%O A171971 1,3
%A A171971 Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Jan 20 2010

%I A171970
%S A171970 0,1,2,3,4,5,6,6,7,8,9,10,11,12,12,13,14,15,16,17,18,19,19,20,21,22,23,
%T A171970 24,25,25,26,27,28,29,30,31,32,32,33,34,35,36,37,38,38,39,40,41,42,43,
%U A171970 44,45,45,46,47,48,49,50,51,51,52,53,54,55,56,57,58,58,59,60,61,62,63
%N A171970 Integer part of the height of an equilateral triangle with side length n.
%C A171970 a(n) = floor(A022838(n)/2);
%C A171970 a(n)*A004526(n) <= A171971(n);
%C A171970 a(n)*A005843(n) <= A171972(n).
%H A171970 Wikipedia, <a href="http://en.wikipedia.org/wiki/Equilateral_triangle">Equilateral triangle</a>
%H A171970 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/EquilateralTriangle.html">Equilateral Triangle</a>
%F A171970 a(n) = floor(n*sqrt(3)/2).
%K A171970 nonn,new
%O A171970 1,3
%A A171970 Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Jan 20 2010

%I A171960
%S A171960 2,1,0,5,4,3,2,1,0,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,53,52,51,
%T A171960 50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,
%U A171960 27,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2
%N A171960 Values of the 2-complement of n in ternary representation.
%C A171960 a(A134026(n)) < A134026(n);
%C A171960 a(A003462(n)) = A003462(n);
%C A171960 a(A134025(n)) >= A134025(n).
%H A171960 R. Zumkeller, <a href="b171960.txt">Table of n, a(n) for n = 0..10000</a>
%F A171960 a(n) = if n<3 then 2 - n else 3*a(floor(n/3)) + 2 - n mod 3.
%Y A171960 Cf. A007089, A035327, A061601.
%K A171960 nonn,new
%O A171960 0,1
%A A171960 Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Jan 20 2010

%I A171979
%S A171979 1,1,2,3,4,5,8,8,12,14,19,21,30,31,42,50,62,69,91,99,126,144,175,198,
%T A171979 246,275,331,379,452,509,612,686,811,922,1076,1219,1428,1604,1863,2108,
%U A171979 2434,2739,3162,3551,4075,4593,5240,5885,6721,7527,8556,9597,10870
%N A171979 Number of partitions of n such that smaller parts dont't occur more frequently than greater parts.
%C A171979 A000009(n) <= a(n) <= A000041(n).
%H A171979 <a href="http://www.research.att.com/~njas/sequences/Sindx_Par.html#partN">Index entries for sequences related to partitions</a>
%F A171979 a(n) = p(n,0,1,1) with p(n,i,j,k) = if k<=n then p(n-k,i,j+1,k)+p(n,max(i,j),1,k+1) else (if j<i or n>0 then 0 else 1).
%e A171979 a(5) = #{5, 4+1, 3+2, 2+2+1, 5x1} = 5;
%e A171979 a(6) = #{6, 5+1, 4+2, 3+3, 3+2+1, 2+2+2, 2+2+1+1, 6x1} = 8;
%e A171979 a(7) = #{7, 6+1, 5+2, 4+3, 4+2+1, 3+3+1, 2+2+2+1, 7x1} = 8;
%e A171979 a(8) = #{8, 7+1, 6+2, 5+3, 5+2+1, 4+4, 4+3+1, 3+3+2, 3+3+1+1, 2+2+2+2, 2+2+2+1+1, 8x1} = 12.
%K A171979 nonn,new
%O A171979 0,3
%A A171979 Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Jan 20 2010

%I A171978
%S A171978 1,1,2,4,7,22,37,84,172,454,745,2904,4722,10464,38546,88769
%N A171978 Number of partitions of n into fractions k/(k+1), 0<k<=n.
%H A171978 <a href="http://www.research.att.com/~njas/sequences/Sindx_Par.html#partN">Index entries for sequences related to partitions</a>
%F A171978 a(n) = q(n,1) with q(x,k) = if x < k/(k+1) then 0^x else if k>n then 0 else q(x-k/(k+1),k) + q(x,k+1).
%e A171978 a(3) = 4 partitions into parts 1/2, 2/3, or 3/4:
%e A171978 #1: 3/4 + 3/4 + 3/4 + 3/4 = 3,
%e A171978 #2: (3/4 + 3/4) + (1/2 + 1/2 + 1/2) = 3,
%e A171978 #3: (2/3 + 2/3 + 2/3) + (1/2 + 1/2) = 3,
%e A171978 #4: 1/2 + 1/2 + 1/2 + 1/2 + 1/2 + 1/2 = 3;
%e A171978 a(4) = 7 partitions into parts 1/2, 2/3, 3/4, or 4/5:
%e A171978 #1: 4/5 + 4/5 + 4/5 + 4/5 + 4/5 = 4,
%e A171978 #2: (3/4 + 3/4 + 3/4 + 3/4) + (1/2 + 1/2) = 4,
%e A171978 #3: (3/4 + 3/4) + (2/3 + 2/3 + 2/3) + 1/2 = 4,
%e A171978 #4: (3/4 + 3/4) + (1/2 + 1/2 + 1/2 + 1/2 + 1/2) = 4,
%e A171978 #5: 2/3 + 2/3 + 2/3 + 2/3 + 2/3 + 2/3 = 4,
%e A171978 #6: 2/3 + 2/3 + 2/3 + 1/2 + 1/2 + 1/2 + 1/2 = 4,
%e A171978 #7: 1/2 + 1/2 + 1/2 + 1/2 + 1/2 + 1/2 + 1/2 + 1/2 = 4.
%K A171978 more,nonn,new
%O A171978 1,3
%A A171978 Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Jan 20 2010

%I A170904
%S A170904 1,0,0,2,24,572,21280,1074390,70299264,5792903144,587159944704,71822748886440,
%T A170904 10435273503677440,1776780701352504408,350461958856515690496,79284041282799128098778,
%U A170904 20392765404792755583221760,5917934230798152486136427600,1924427226324694427836833857536
%N A170904 a(n) = K_n as defined by Riordan (Eq. (30a), p. 206 and in Theorem 3, p. 209).
%C A170904 The formula (see the Maple code here) is in fact incorrect. The correct formula is given in A000186. Thanks to Neven Juric for alerting me to this error of Riordan's.
%D A170904 J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, pp. 206, 209.
%p A170904 # A000166
%p A170904 unprotect(D);
%p A170904 D := proc(n) option remember; if n<=1 then 1-n else (n-1)*(D(n-1)+D(n-2)); fi; end;
%p A170904 [seq(D(n),n=0..30)];
%p A170904 # A000179
%p A170904 U := proc(n) if n<=1 then 1-n else add ((-1)^k*(2*n)*binomial(2*n-k, k)*(n-k)!/(2*n-k), k=0..n); fi; end;
%p A170904 [seq(U(n),n=0..30)];
%p A170904 # bad A000186 (A170904)
%p A170904 Kbad:=proc(n) local k; global D, U; add( binomial(n,k)*D(n-k)*D(k)*U(n-2*k), k=0..floor(n/2) ); end;
%p A170904 [seq(Kbad(n),n=0..30)];
%K A170904 nonn,new
%O A170904 0,4
%A A170904 N. J. A. Sloane (njas(AT)research.att.com), Jan 21 2010

%I A170903
%S A170903 1,1,5,1,5,9,13,1,5,9,13,9,21,33,29,1,5,9,13,9,21,33,29,9,21,33,37,41,
%T A170903 77,97,61,1,5,9,13,9,21,33,29,9,21,33,37,41,77,97,61,9,21,33,37,41,77,97,
%U A170903 69,41,77,105,117,161,253,257,125,1,5,9,13,9,21,33,29,9,21,33,37,41,77
%N A170903 a(n) = 2*A160552(n)-1.
%H A170903 David Applegate, Omar E. Pol and N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/tooth.pdf">The Toothpick Sequence and Other Sequences from Cellular Automata</a>
%H A170903 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/toothlist.html">Catalog of Toothpick and Cellular Automata Sequences in the OEIS</a>
%e A170903 When written as a triangle:
%e A170903 1
%e A170903 1, 5;
%e A170903 1, 5, 9, 13;
%e A170903 1, 5, 9, 13, 9, 21, 33, 29;
%e A170903 ...
%e A170903 Rows sums are A006516 (this is immediate from the definition).
%K A170903 nonn,new
%O A170903 1,3
%A A170903 Gary W. Adamson, Jan 21 2010

%I A170902
%S A170902 1,3,4,8,14,23,44,86,161,281,486,908,1966,3744,7574
%N A170902 Number of 2-generator Schur towers of order 2^n that have trivial Schur multiplier.
%D A170902 B. Eick, Computing p-groups with trivial Schur multiplier, J. Algebra, 322 (2009), 741-751.
%K A170902 nonn,new
%O A170902 2,2
%A A170902 N. J. A. Sloane (njas(AT)research.att.com), Jan 19 2010

%I A170901
%S A170901 3,8,11,26,43,91,173,388,798,1586,3502,7693,16447,34410,94960
%N A170901 Number of 2-generator Schur towers of order 2^n.
%D A170901 B. Eick, Computing p-groups with trivial Schur multiplier, J. Algebra, 322 (2009), 741-751.
%K A170901 nonn,new
%O A170901 3,1
%A A170901 N. J. A. Sloane (njas(AT)research.att.com), Jan 19 2010

%I A170900
%S A170900 3,8,19,53,162,540,2043
%N A170900 Number of 2-generator groups of order 2^n.
%D A170900 B. Eick, Computing p-groups with trivial Schur multiplier, J. Algebra, 322 (2009), 741-751.
%D A170900 B. Eick and E. O'Brien, Enumerating p-groups, J. Australian Math. Soc., 67 (1999), 191-205.
%Y A170900 Cf. A170901, A170902.
%K A170900 nonn,new
%O A170900 3,1
%A A170900 N. J. A. Sloane (njas(AT)research.att.com), Jan 19 2010

%I A170899
%S A170899 0,0,1,0,1,2,3,0,1,2,4,4,3,6,7,0,1,2,4,4,4,8,12,8,3,6,11,13,9,15,15,
%T A170899 0,1,2,4,4,4,8,12,8,4,8,14,18,16,20,28,16,3,6,11,13,13,21,33,29,13,
%U A170899 15,27,34,24,34,31,0,1,2,4,4,4,8,12,8,4,8,14,18,16,20,28,16,4,8,14
%N A170899 Triangle read by rows, obtained by subtracting 1 from the terms of A170898 and dividing by 2.
%C A170899 Row k has 2^k terms.
%C A170899 I was expecting this to have a simple formula, analogous to that for A147582!
%e A170899 Triangle begins:
%e A170899 .0,
%e A170899 .0, 1,
%e A170899 .0, 1, 2, 3,
%e A170899 .0, 1, 2, 4, 4, 3, 6, 7,
%e A170899 .0, 1, 2, 4, 4, 4, 8, 12, 8, 3, 6, 11, 13, 9, 15, 15,
%e A170899 .0, 1, 2, 4, 4, 4, 8, 12, 8, 4, 8, 14, 18, 16, 20, 28, 16, 3, 6, 11, 13, 13, 21, 33, 29, 13, 15, 27, 34, 24, 34, 31,
%e A170899 .0, 1, 2, 4, 4, 4, 8, 12, 8, 4, 8, 14, 18, 16, 20, 28, 16, 4, 8, 14, 18, 18, 26, 42, 42, 24, 20, 36, 50, 46, 50, 62, 32, 3, 6, 11, 13, 13, 21, 33, 29, 17, 21, 37, 51, 51, 57, 77, 61, 21, 15, 27, 34, 36, 52, 80, 80, 44, 38, 62, 81, 58, 73, 63,
%e A170899 .0, 1, 2, 4, 4, 4, 8, 12, 8, 4, 8, 14, 18, 16, 20, 28, 16, 4, 8, 14, 18, 18, 26, 42, 42, 24, 20, 36, 50, 46, 50, 62, 32, 4, 8, 14, 18, 18, 26, 42, 42, 26, 26, 46, 66, 70, 74, 98, 90, 40, 20, 36, 50, 54, 70, 110, 126, 86, 58, 86, 124, 118, 118, 132, 64, 3, 6, 11, 13, 13, 21, 33, 29, 17, 21, 37, 51, 51, 57, 77, 61, 25, 21, 37, 51, 55, 71, 111, 127, 91, 65, 93, 137, 143, 147, 175, 127, 37, 15, 27, 34, 36, 52, 80, 80, 56, 56, 88, 126, 136, 150, 192, 172, 84, 46, 62, 81, 90, 124, 184, 196, 124, 96, 139, 183, 131, 152, 127
%e A170899 ...
%Y A170899 Cf. A139250, A151723, A151724, A170898.
%K A170899 nonn,tabf
%O A170899 0,6
%A A170899 N. J. A. Sloane (njas(AT)research.att.com), Jan 10 2010

%I A170898
%S A170898 1,1,3,1,3,5,7,1,3,5,9,9,7,13,15,1,3,5,9,9,9,17,25,17,7,13,23,27,19,
%T A170898 31,31,1,3,5,9,9,9,17,25,17,9,17,29,37,33,41,57,33,7,13,23,27,27,43,
%U A170898 67,59,27,31,55,69,49,69,63,1,3,5,9,9,9,17,25,17,9,17,29,37,33,41
%N A170898 Triangle read by rows, obtained by dividing A151724 by 6.
%C A170898 Row k has 2^k terms.
%e A170898 Triangle begins:
%e A170898 .1,
%e A170898 .1,3,
%e A170898 .1,3,5,7,
%e A170898 .1,3,5,9,9,7,13,15,
%e A170898 .1,3,5,9,9,9,17,25,17,7,13,23,27,19,31,31,
%e A170898 .1,3,5,9,9,9,17,25,17,9,17,29,37,33,41,57,33,7,13,23,27,27,43,67,59,27,31,55,69,49,69,63,
%e A170898 ....
%Y A170898 Cf. A139250, A151723, A151724, A170899.
%K A170898 nonn,tabf
%O A170898 0,3
%A A170898 N. J. A. Sloane (njas(AT)research.att.com), Jan 10 2010

%I A170897
%S A170897 0,1,4,4,4,12,4,12,12,12,20,12,20,28,12
%N A170897 First differences of A170896.
%C A170897 First differs from A151896 at n=14.
%K A170897 nonn,more
%O A170897 0,3
%A A170897 N. J. A. Sloane (njas(AT)research.att.com), Jan 09 2010

%I A170896
%S A170896 0,1,5,9,13,25,29,41,53,65,85,97,117,145,157
%N A170896 Number of ON cells after n generations of the Schrandt-Ulam cellular automaton on the square grid that is described in the Comments.
%C A170896 The cells are the squares of the standard square grid.
%C A170896 Once a cell is ON it stays ON.
%C A170896 At generation 1 the cell at the center is turned ON.
%C A170896 Candidates for cells to be turned ON at generation n are those cells, not presently ON, which share exactly one edge with an ON cell of generation n-1 but do not share an outer vertex (a vertex not adjacent to a generation n-1 ON cell) with an ON cell of generation n-3 or earlier. That is, they may share a vertex with a grandparent but not with any earlier ancestor.
%C A170896 However, if two such candidates share an outer vertex, then neither is turned ON. Otherwise, candidates are turned ON.
%C A170896 First differs from A151895 at n=14.
%D A170896 R. G. Schrandt and S. M. Ulam, ``On recursively defined geometric objects and patterns of growth,'' Los Alamos Scientific Laboratory, Report LA-3762, Aug 16 1967; published in A. W. Burks, editor, Essays on Cellular Automata. Univ. Ill. Press, 1970, pp. 238ff.
%D A170896 S. Ulam, On some mathematical problems connected with patterns of growth of figures, pp. 215-224 of R. E. Bellman, ed., Mathematical Problems in the Biological Sciences, Proc. Sympos. Applied Math., Vol. 14, Amer. Math. Soc., 1962.
%H A170896 David Applegate, Omar E. Pol and N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/tooth.pdf">The Toothpick Sequence and Other Sequences from Cellular Automata</a>
%H A170896 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/toothlist.html">Catalog of Toothpick and Cellular Automata Sequences in the OEIS</a>
%H A170896 R. G. Schrandt and S. M. Ulam, <a href="http://library.lanl.gov/cgi-bin/getfile?00359037.pdf">On recursively defined geometric objects and patterns of growth</a> [Link supplied by Laurinda J. Alcorn, Jan 09 2010.]
%Y A170896 Cf. A139250, A170897, A151895, A151896.
%K A170896 nonn,more
%O A170896 0,3
%A A170896 N. J. A. Sloane (njas(AT)research.att.com), Jan 09 2010

%I A170895
%S A170895 0,1,1,2,3,3,3,6,8,6,4
%N A170895 First differences of A170894.
%H A170895 David Applegate, Omar E. Pol and N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/tooth.pdf">The Toothpick Sequence and Other Sequences from Cellular Automata</a>
%H A170895 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/toothlist.html">Catalog of Toothpick and Cellular Automata Sequences in the OEIS</a>
%K A170895 nonn,more
%O A170895 0,4
%A A170895 Omar E. Pol (info(AT)polprimos.com), Jan 09 2010

%I A170894
%S A170894 0,1,2,4,7,10,13,19,27,33,37
%N A170894 Similar to A160406, always staying outside the wedge, but starting with a horizontal toothpick whose endpoint touches the vertex of the wedge.
%H A170894 David Applegate, Omar E. Pol and N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/tooth.pdf">The Toothpick Sequence and Other Sequences from Cellular Automata</a>
%H A170894 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/toothlist.html">Catalog of Toothpick and Cellular Automata Sequences in the OEIS</a>
%K A170894 nonn,more
%O A170894 0,3
%A A170894 Omar E. Pol (info(AT)polprimos.com), Jan 09 2010

%I A170893
%S A170893 0,1,1,2,4,4,4,8,10,10,4
%N A170893 First differences of A170892.
%H A170893 David Applegate, Omar E. Pol and N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/tooth.pdf">The Toothpick Sequence and Other Sequences from Cellular Automata</a>
%H A170893 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/toothlist.html">Catalog of Toothpick and Cellular Automata Sequences in the OEIS</a>
%K A170893 nonn,more
%O A170893 0,4
%A A170893 Omar E. Pol (info(AT)polprimos.com), Jan 09 2010

%I A170892
%S A170892 0,1,2,4,8,12,16,24,34,44
%N A170892 Similar to A160406, always staying outside the wedge, but starting with a vertical toothpick whose endpoint touches the vertex of the wedge.
%H A170892 David Applegate, Omar E. Pol and N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/tooth.pdf">The Toothpick Sequence and Other Sequences from Cellular Automata</a>
%H A170892 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/toothlist.html">Catalog of Toothpick and Cellular Automata Sequences in the OEIS</a>
%K A170892 nonn,more
%O A170892 0,3
%A A170892 Omar E. Pol (info(AT)polprimos.com), Jan 09 2010

%I A170891
%S A170891 0,1,1,2,3,3,4,7,8,7
%N A170891 First differences of A170890.
%H A170891 David Applegate, Omar E. Pol and N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/tooth.pdf">The Toothpick Sequence and Other Sequences from Cellular Automata</a>
%H A170891 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/toothlist.html">Catalog of Toothpick and Cellular Automata Sequences in the OEIS</a>
%K A170891 nonn,more
%O A170891 0,4
%A A170891 Omar E. Pol (info(AT)polprimos.com), Jan 09 2010

%I A170890
%S A170890 0,1,2,4,7,10,14,21,29,36
%N A170890 Similar to A160406, always staying outside the wedge, but starting with a horizontal half-toothpick which protrudes from the vertex of the wedge.
%H A170890 David Applegate, Omar E. Pol and N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/tooth.pdf">The Toothpick Sequence and Other Sequences from Cellular Automata</a>
%H A170890 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/toothlist.html">Catalog of Toothpick and Cellular Automata Sequences in the OEIS</a>
%K A170890 nonn,more
%O A170890 0,3
%A A170890 Omar E. Pol (info(AT)polprimos.com), Jan 09 2010

%I A170889
%S A170889 0,1,2,4,4,4,6,10,8,4,6,10,10,12,20,26,16,4,6,10,10,12,20,26,18,12,20,
%T A170889 28,30,42,64,66,32,4,6,10,10,12,20,26,18,12,20,28,30,42,64,66,34,12,20,
%U A170889 28,30,42,64,68,46,42,66,84,100,146,192,162,64,4,6,10,10,12,20,26,18,12
%N A170889 First differences of A170888.
%H A170889 David Applegate, Omar E. Pol and N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/tooth.pdf">The Toothpick Sequence and Other Sequences from Cellular Automata</a>
%H A170889 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/toothlist.html">Catalog of Toothpick and Cellular Automata Sequences in the OEIS</a>
%K A170889 nonn,new
%O A170889 0,3
%A A170889 Omar E. Pol (info(AT)polprimos.com), Jan 09 2010
%E A170889 Terms beyond a(10) from R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jan 25 2010

%I A170888
%S A170888 0,1,3,7,11,15,21,31,39,43,49,59,69,81,101,127,143,147,153,163,173,185,
%T A170888 205,231,249,261,281,309,339,381,445,511,543,547,553,563,573,585,605,
%U A170888 631,649,661,681,709,739,781,845,911,945,957,977,1005,1035,1077,1141
%N A170888 Similar to A160406, always staying outside the wedge, but starting with a vertical half-toothpick which protrudes from the vertex of the wedge.
%H A170888 David Applegate, Omar E. Pol and N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/tooth.pdf">The Toothpick Sequence and Other Sequences from Cellular Automata</a>
%H A170888 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/toothlist.html">Catalog of Toothpick and Cellular Automata Sequences in the OEIS</a>
%K A170888 nonn,new
%O A170888 0,3
%A A170888 Omar E. Pol (info(AT)polprimos.com), Jan 09 2010
%E A170888 Terms beyond a(10) from R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jan 25 2010

%I A170887
%S A170887 0,1,2,2,2,4,6,6,6,8,12,6,8,12,18,14,14,20,20,6,8,12,18,14,16,24,26,16,
%T A170887 24,38,46,38,42,52,36,6,8,12,18,14,16,24,26,16,24,38,46,38,44,56,42,16,
%U A170887 24,38,46,40,52,70,64,52,82,118,126,114,130,132,68,6,8,12,18,14,16,24
%N A170887 First differences of A170886.
%H A170887 David Applegate, Omar E. Pol and N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/tooth.pdf">The Toothpick Sequence and Other Sequences from Cellular Automata</a>
%H A170887 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/toothlist.html">Catalog of Toothpick and Cellular Automata Sequences in the OEIS</a>
%K A170887 nonn,new
%O A170887 0,3
%A A170887 Omar E. Pol (info(AT)polprimos.com), Jan 09 2010
%E A170887 More terms from R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jan 25 2010

%I A170886
%S A170886 0,1,3,5,7,11,17,23,29,37,49,55,63,75,93,107,121,141,161,167,175,187,
%T A170886 205,219,235,259,285,301,325,363,409,447,489,541,577,583,591,603,621,
%U A170886 635,651,675,701,717,741,779,825,863,907,963,1005,1021,1045,1083,1129
%N A170886 Similar to A160406, always staying outside the wedge, but starting with a toothpick whose midpoint touches the vertex of the wedge.
%H A170886 David Applegate, Omar E. Pol and N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/tooth.pdf">The Toothpick Sequence and Other Sequences from Cellular Automata</a>
%H A170886 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/toothlist.html">Catalog of Toothpick and Cellular Automata Sequences in the OEIS</a>
%K A170886 nonn,new
%O A170886 0,3
%A A170886 Omar E. Pol (info(AT)polprimos.com), Jan 09 2010
%E A170886 Terms beyond a(9) from R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jan 25 2010

%I A170885
%S A170885 0,3,4,8,12,12,12
%N A170885 First differences of A170884.
%H A170885 David Applegate, Omar E. Pol and N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/tooth.pdf">The Toothpick Sequence and Other Sequences from Cellular Automata</a>
%H A170885 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/toothlist.html">Catalog of Toothpick and Cellular Automata Sequences in the OEIS</a>
%K A170885 nonn,more
%O A170885 0,2
%A A170885 David Applegate, Omar E. Pol and N. J. A. Sloane, Jan 09 2010

%I A170884
%S A170884 0,3,7,15,27,39,51
%N A170884 In the toothpick structure of A160160, the number of nodes occupied after n steps, assuming that the toothpicks have length 2.
%H A170884 David Applegate, Omar E. Pol and N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/tooth.pdf">The Toothpick Sequence and Other Sequences from Cellular Automata</a>
%H A170884 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/toothlist.html">Catalog of Toothpick and Cellular Automata Sequences in the OEIS</a>
%K A170884 nonn,more
%O A170884 0,2
%A A170884 David Applegate, Omar E. Pol and N. J. A. Sloane, Jan 09 2010

%I A170883
%S A170883 0,1,3,7,12,17,24,35,46,53,60,73,92,111,130,155,178,189,196,209,228,
%T A170883 249,274,311,356,391,414,445,494,547,598,657,706,725,732,745,764,
%U A170883 785,810,847,892,929,958,999,1064,1141,1218,1307,1404,1471,1502
%N A170883 "Corner" sequence associated with A161644 and A161645.
%H A170883 David Applegate and N. J. A. Sloane, <a href="b170883.txt">Table of n, a(n) for n = 0..1000</a>
%H A170883 David Applegate, Omar E. Pol and N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/tooth.pdf">The Toothpick Sequence and Other Sequences from Cellular Automata</a>
%H A170883 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/toothlist.html">Catalog of Toothpick and Cellular Automata Sequences in the OEIS</a>
%H A170883 David Applegate, <a href="http://www.research.att.com/~david/oeis/toothpick.html">The movie version of A139250, etc.</a> [See A161644, corner version.]
%Y A170883 Cf. A139250, A161644, A161645, A170882.
%K A170883 nonn
%O A170883 0,3
%A A170883 David Applegate (david(AT)research.att.com) and N. J. A. Sloane (njas(AT)research.att.com), Jan 08 2010

%I A170882
%S A170882 0,1,2,4,5,5,7,11,11,7,7,13,19,19,19,25,23,11,7,13,19,21,25,37,45,
%T A170882 35,23,31,49,53,51,59,49,19,7,13,19,21,25,37,45,37,29,41,65,77,77,
%U A170882 89,97,67,31,31,49,59,69,97,125,111,75,79,119,135,129,135,103,35,7
%N A170882 First differences of "corner" sequence associated with A161644 and A161645.
%H A170882 David Applegate and N. J. A. Sloane, <a href="b170882.txt">Table of n, a(n) for n = 0..1000</a>
%H A170882 David Applegate, Omar E. Pol and N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/tooth.pdf">The Toothpick Sequence and Other Sequences from Cellular Automata</a>
%H A170882 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/toothlist.html">Catalog of Toothpick and Cellular Automata Sequences in the OEIS</a>
%H A170882 David Applegate, <a href="http://www.research.att.com/~david/oeis/toothpick.html">The movie version of A139250, etc.</a> [See A161644, corner version.]
%Y A170882 Cf. A139250, A161644, A161645, A170883.
%K A170882 nonn
%O A170882 0,3
%A A170882 David Applegate (david(AT)research.att.com) and N. J. A. Sloane (njas(AT)research.att.com), Jan 08 2010

%I A170881
%S A170881 1,3,8,21,53,129,305,705,1601,3585,7937,17409,37889,81921,176129,376833,802817,
%T A170881 1703937,3604481,7602177,15990785,33554433,70254593,146800641,306184193,
%U A170881 637534209,1325400065,2751463425,5704253441,11811160065,24427626497,50465865729
%N A170881 a(0)=0; thereafter a(n) = (3*n+1)*2^(n-2)+1.
%Y A170881 Essentially the first column of the triangular array in A151747.
%K A170881 nonn
%O A170881 0,2
%A A170881 N. J. A. Sloane (njas(AT)research.att.com), Jan 07 2010

%I A170880
%S A170880 0,1,6,11,22,29,44,63,86,93,108,129,158,187,236,295,342,349,364,385,414,
%T A170880 443,492,553,606,635,686,757,844,951,1108,1271,1366,1373,1388,1409,1438,
%U A170880 1467,1516,1577,1630,1659,1710,1781,1868,1975,2132,2297,2398,2427,2478
%N A170880 Partial sums of A151728.
%H A170880 David Applegate, Omar E. Pol and N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/tooth.pdf">The Toothpick Sequence and Other Sequences from Cellular Automata</a>
%H A170880 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/toothlist.html">Catalog of Toothpick and Cellular Automata Sequences in the OEIS</a>
%K A170880 nonn
%O A170880 0,3
%A A170880 N. J. A. Sloane (njas(AT)research.att.com), Jan 07 2010

%I A170879
%S A170879 0,1,4,9,17,26,37,54,75,90,101,119,144,173,212,266,319,346,357,375,400,
%T A170879 429,468,523,580,621,661,722,801,898,1030,1190,1319,1370,1381,1399,1424,
%U A170879 1453,1492,1547,1604,1645,1685,1746,1825,1922,2054,2215,2348,2413,2453
%N A170879 Partial sums of A151747.
%H A170879 David Applegate, Omar E. Pol and N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/tooth.pdf">The Toothpick Sequence and Other Sequences from Cellular Automata</a>
%H A170879 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/toothlist.html">Catalog of Toothpick and Cellular Automata Sequences in the OEIS</a>
%K A170879 nonn
%O A170879 0,3
%A A170879 N. J. A. Sloane (njas(AT)research.att.com), Jan 07 2010

%I A170878
%S A170878 4,0,12,12,20,8,44,56,20,0,60,68,68,24,156,212,20,0,60,60,100,40,220,288,
%T A170878 68,0,204,228,244,88,556,768,20,0,60,60,100,40,220,280,100,0,300,340,340,
%U A170878 120,780,1068,68,0,204,204,340
%V A170878 4,0,12,-12,20,-8,44,-56,20,0,60,-68,68,-24,156,-212,20,0,60,-60,100,-40,220,-288,
%W A170878 68,0,204,-228,244,-88,556,-768,20,0,60,-60,100,-40,220,-280,100,0,300,-340,340,
%X A170878 -120,780,-1068,68,0,204,-204,340
%N A170878 First differences of A072272.
%H A170878 David Applegate, Omar E. Pol and N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/tooth.pdf">The Toothpick Sequence and Other Sequences from Cellular Automata</a>
%H A170878 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/toothlist.html">Catalog of Toothpick and Cellular Automata Sequences in the OEIS</a>
%K A170878 sign
%O A170878 0,1
%A A170878 N. J. A. Sloane (njas(AT)research.att.com), Jan 07 2010

%I A170877
%S A170877 1,2,3,5,7,10,15,22,30,43,61,88,123,173,246,348,487,688,972,1371,1928,
%T A170877 2714,3822,5387
%N A170877 Number of binary words of length n with properties that there is no pair of adjacent 1's and no subword of the form X^4.
%e A170877 a(3) = 5: 000, 001, 010, 100, 101.
%e A170877 a(4) = 7: 0001, 0010, 0100, 1000, 0101, 1010, 1001.
%Y A170877 Cf. A003410, A028445, A135491.
%K A170877 nonn,more
%O A170877 0,2
%A A170877 Benjamin Chaffin (chaffin(AT)gmail.com) and N. J. A. Sloane (njas(AT)research.att.com), Jan 07 2010

%I A170875
%S A170875 0,1,4,16,16,16,64,80,64,144,160,224,176,256,320,400,512,480,688,768,704,
%T A170875 816,896,1120,1168,1536,1568,1936,1600
%N A170875 First differences of A170876.
%H A170875 David Applegate, Omar E. Pol and N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/tooth.pdf">The Toothpick Sequence and Other Sequences from Cellular Automata</a>
%H A170875 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/toothlist.html">Catalog of Toothpick and Cellular Automata Sequences in the OEIS</a>
%K A170875 nonn,more
%O A170875 0,3
%A A170875 N. J. A. Sloane, Jan 05 2010, based on email from R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jun 02 2009; revised by R. J. Mathar, Jan 08 2010

%I A170876
%S A170876 0,1,5,21,37,53,117,197,261,405,565,789,965,1221,1541,1941,2453,
%T A170876 2933,3621,4389,5093,5909,6805,7925,9093,10629,12197,14133,15733,
%U A170876 17717,19493,21605,23909,26453,29109,32117,35013,38085,41285
%N A170876 Number of toothpicks after n stages of 3-D toothpick structure defined in Comments.
%C A170876 We are in 3-D, and we are placing ordinary toothpicks, as in A139250.
%C A170876 We start with one toothpick in the z direction
%C A170876 We place toothpicks at any free end, as in A139250.
%C A170876 We always place new toothpicks in pairs, two perpendicular toothpicks that are perpendicular to the original toothpick
%C A170876 The toothpicks are always in 2 out of the 3 (x, y or z) directions.
%C A170876 The initial values are as follows (this should be checked!):
%C A170876 n:.0..1..2..3..4..5..6
%C A170876 ----------------------------
%C A170876 x..0..0..2..4..8..4.24 (Number added in x direction)
%C A170876 y..0..0..2..4..8..4.24 (Number added in y direction)
%C A170876 z..0..1..0..8..0..8.16 (Number added in z direction)
%C A170876 ----------------------------
%C A170876 ...0..1..4.16.16.16.64 (Total number added at nth stage, A170876)
%C A170876 ----------------------------
%C A170876 a..0..1..5.21.37.53.117 (Total so far, this sequence)
%C A170876 ----------------------------
%H A170876 David Applegate, Omar E. Pol and N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/tooth.pdf">The Toothpick Sequence and Other Sequences from Cellular Automata</a>
%H A170876 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/toothlist.html">Catalog of Toothpick and Cellular Automata Sequences in the OEIS</a>
%H A170876 R. J. Mathar, <a href="a170876.cc">C++ program</a>
%H A170876 R. J. Mathar, <a href="a170876_2.eps">View after stage 1</a>
%H A170876 R. J. Mathar, <a href="a170876_3.eps">View after stage 2</a>
%H A170876 R. J. Mathar, <a href="a170876_4.eps">View after stage 3</a>
%H A170876 R. J. Mathar, <a href="a170876_5.eps">View after stage 4</a>
%H A170876 R. J. Mathar, <a href="a170876_6.eps">View after stage 5</a>
%H A170876 R. J. Mathar, <a href="a170876_7.eps">View after stage 6</a>
%H A170876 R. J. Mathar, <a href="a170876_8.eps">View after stage 7</a>
%H A170876 R. J. Mathar, <a href="a170876_9.eps">View after stage 8</a>
%H A170876 R. J. Mathar, <a href="a170876_10.eps">View after stage 9</a>
%H A170876 R. J. Mathar, <a href="a170876_11.eps">View after stage 10</a>
%H A170876 O. E. Pol, <a href="http://www.polprimos.com/imagenespub/poltp053.jpg">Illustration of initial terms</a>
%e A170876 At stage 2 we have a horizontal cross, a vertical toothpick then another horizontal cross, for a total of 5 toothpicks.
%e A170876 Then we add 8 vertical toothpicks at the ends of the crosses and 8 horizontal toothpicks in the same planes as the crosses, fore a total of 13 toothpicks.
%Y A170876 Cf. A139250, A170875 (first differences), A160160, A160170. For another version see A170837.
%K A170876 nonn
%O A170876 0,3
%A A170876 N. J. A. Sloane, Jan 05 2010, based on email from R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jun 02 2009; revised by R. J. Mathar, Jan 08 2010, Jan 09 2010

%I A170836
%S A170836 0,1,4,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,
%T A170836 16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,
%U A170836 16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16
%N A170836 First differences of A170837.
%F A170836 a(n)=16 for n >= 3.
%K A170836 nonn
%O A170836 0,3
%A A170836 N. J. A. Sloane, Jan 05 2010, based on email from R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jun 02 2009

%I A170837
%S A170837 0,1,5,21,37,53,69,85,101,117,133,149,165,181,197,213,229,245,261,277,293,309,
%T A170837 325,341,357,373,389,405,421,437,453,469,485,501,517,533,549,565,581,597,
%U A170837 613,629,645,661,677,693,709,725,741,757,773,789,805
%N A170837 a(0)=0, a(1)=1 and a(n) = 16n-27 for n >= 2.
%F A170837 G.f.: x*(3*x+12*x^2+1)/(x-1)^2. a(n)= 2*a(n-1) -a(n-2), n>=4.
%Y A170837 Cf. A170836 (first differences), A170876.
%K A170837 nonn,easy
%O A170837 0,3
%A A170837 N. J. A. Sloane, Jan 05 2010, based on email from R. J. Mathar (mathar(AT)strw.leidenuniv.nl) and Benout Jubin, Jun 02 2009; revised Jan 09 2010

%I A162795
%S A162795 1,5,9,21,25,37,53,85,89,101,117,149,165,201,261,341,345,357,373,405,421,
%T A162795 457,517,597,613,649,709,793,853,965,1173,1365,1369,1381,1397,1429,1445,
%U A162795 1481,1541,1621,1637,1673,1733,1817,1877,1989,2197,2389,2405,2441,2501
%N A162795 Partial sums of A162793.
%C A162795 Number of toothpicks in the toothpick structure A139250 that are parallel to the initial toothpick, after n odd rounds.
%H A162795 David Applegate, Omar E. Pol and N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/tooth.pdf">The Toothpick Sequence and Other Sequences from Cellular Automata</a>
%H A162795 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/toothlist.html">Catalog of Toothpick and Cellular Automata Sequences in the OEIS</a>
%Y A162795 Cf. A139250, A139251, A159791, A159792, A162793, A162794, A162796, A162797.
%K A162795 nonn
%O A162795 1,2
%A A162795 Omar E. Pol (info(AT)polprimos.com), Jul 14 2009
%E A162795 More terms from N. J. A. Sloane (njas(AT)research.att.com), Dec 28 2009

%I A170872
%S A170872 1,2,9,33,9,33,126,126,9,33,126,126,126,477,756,405,9,33,126,126,126,477,
%T A170872 756,405,126,477,756,756,1809,3699,3483,1242,9,33,126,126,126,477,756,405,
%U A170872 126,477,756,756,1809,3699,3483,1242,126,477,756,756,1809,3699,3483,1593
%N A170872 a(0) = 1, a(1) = 2; a(2^i + j) = 3a(j) + 3a(j + 1) for 0 <= j < 2^i.
%H A170872 David Applegate, Omar E. Pol and N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/tooth.pdf">The Toothpick Sequence and Other Sequences from Cellular Automata</a>
%H A170872 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/toothlist.html">Catalog of Toothpick and Cellular Automata Sequences in the OEIS</a>
%K A170872 nonn
%O A170872 0,2
%A A170872 N. J. A. Sloane (njas(AT)research.att.com), Jan 02 2010

%I A170871
%S A170871 1,2,8,28,8,28,100,80,8,28,100,80,100,356,440,184,8,28,100,80,100,356,440,
%T A170871 184,100,356,440,460,1268,2032,1432,392,8,28,100,80,100,356,440,184,100,
%U A170871 356,440,460,1268,2032,1432,392,100,356,440,460,1268,2032,1432,668,1268
%N A170871 a(0) = 1, a(1) = 2; a(2^i + j) = 2a(j) + 3a(j + 1) for 0 <= j < 2^i.
%H A170871 David Applegate, Omar E. Pol and N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/tooth.pdf">The Toothpick Sequence and Other Sequences from Cellular Automata</a>
%H A170871 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/toothlist.html">Catalog of Toothpick and Cellular Automata Sequences in the OEIS</a>
%K A170871 nonn
%O A170871 0,2
%A A170871 N. J. A. Sloane (njas(AT)research.att.com), Jan 02 2010

%I A170870
%S A170870 1,2,7,23,7,23,76,44,7,23,76,44,76,251,208,65,7,23,76,44,76,251,208,65,
%T A170870 76,251,208,272,829,875,403,86,7,23,76,44,76,251,208,65,76,251,208,272,
%U A170870 829,875,403,86,76,251,208,272,829,875,403,293,829,875,1024,2759,3454,2084
%N A170870 a(0) = 1, a(1) = 2; a(2^i + j) = a(j) + 3a(j + 1) for 0 <= j < 2^i.
%H A170870 David Applegate, Omar E. Pol and N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/tooth.pdf">The Toothpick Sequence and Other Sequences from Cellular Automata</a>
%H A170870 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/toothlist.html">Catalog of Toothpick and Cellular Automata Sequences in the OEIS</a>
%K A170870 nonn
%O A170870 0,2
%A A170870 N. J. A. Sloane (njas(AT)research.att.com), Jan 02 2010

%I A170869
%S A170869 1,2,7,20,7,20,61,74,7,20,61,74,61,182,331,236,7,20,61,74,61,182,331,236,
%T A170869 61,182,331,344,547,1208,1465,722,7,20,61,74,61,182,331,236,61,182,331,
%U A170869 344,547,1208,1465,722,61,182,331,344,547,1208,1465,830,547,1208,1681,2126
%N A170869 a(0) = 1, a(1) = 2; a(2^i + j) = 3a(j) + 2a(j + 1) for 0 <= j < 2^i.
%H A170869 David Applegate, Omar E. Pol and N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/tooth.pdf">The Toothpick Sequence and Other Sequences from Cellular Automata</a>
%H A170869 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/toothlist.html">Catalog of Toothpick and Cellular Automata Sequences in the OEIS</a>
%K A170869 nonn
%O A170869 0,2
%A A170869 N. J. A. Sloane (njas(AT)research.att.com), Jan 02 2010

%I A170868
%S A170868 1,2,5,11,5,11,26,38,5,11,26,38,26,59,116,119,5,11,26,38,26,59,116,119,
%T A170868 26,59,116,140,137,293,467,362,5,11,26,38,26,59,116,119,26,59,116,140,137,
%U A170868 293,467,362,26,59,116,140,137,293,467,383,137,293,488,557,704,1346,1763
%N A170868 a(0) = 1, a(1) = 2; a(2^i + j) = 3a(j) + a(j + 1) for 0 <= j < 2^i.
%H A170868 David Applegate, Omar E. Pol and N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/tooth.pdf">The Toothpick Sequence and Other Sequences from Cellular Automata</a>
%H A170868 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/toothlist.html">Catalog of Toothpick and Cellular Automata Sequences in the OEIS</a>
%K A170868 nonn
%O A170868 0,2
%A A170868 N. J. A. Sloane (njas(AT)research.att.com), Jan 02 2010

%I A170867
%S A170867 1,1,6,21,6,21,81,81,6,21,81,81,81,306,486,261,6,21,81,81,81,306,486,261,
%T A170867 81,306,486,486,1161,2376,2241,801,6,21,81,81,81,306,486,261,81,306,486,
%U A170867 486,1161,2376,2241,801,81,306,486,486,1161,2376,2241,1026,1161,2376,2916
%N A170867 a(0) = 1, a(1) = 1; a(2^i + j) = 3a(j) + 3a(j + 1) for 0 <= j < 2^i.
%H A170867 David Applegate, Omar E. Pol and N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/tooth.pdf">The Toothpick Sequence and Other Sequences from Cellular Automata</a>
%H A170867 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/toothlist.html">Catalog of Toothpick and Cellular Automata Sequences in the OEIS</a>
%K A170867 nonn
%O A170867 0,3
%A A170867 N. J. A. Sloane (njas(AT)research.att.com), Jan 02 2010

%I A170866
%S A170866 1,1,5,17,5,17,61,49,5,17,61,49,61,217,269,113,5,17,61,49,61,217,269,113,
%T A170866 61,217,269,281,773,1241,877,241,5,17,61,49,61,217,269,113,61,217,269,281,
%U A170866 773,1241,877,241,61,217,269,281,773,1241,877,409,773,1241,1381,2881,5269
%N A170866 a(0) = 1, a(1) = 1; a(2^i + j) = 2a(j) + 3a(j + 1) for 0 <= j < 2^i.
%H A170866 David Applegate, Omar E. Pol and N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/tooth.pdf">The Toothpick Sequence and Other Sequences from Cellular Automata</a>
%H A170866 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/toothlist.html">Catalog of Toothpick and Cellular Automata Sequences in the OEIS</a>
%K A170866 nonn
%O A170866 0,3
%A A170866 N. J. A. Sloane (njas(AT)research.att.com), Jan 02 2010

%I A170865
%S A170865 1,1,4,13,4,13,43,25,4,13,43,25,43,142,118,37,4,13,43,25,43,142,118,37,
%T A170865 43,142,118,154,469,496,229,49,4,13,43,25,43,142,118,37,43,142,118,154,
%U A170865 469,496,229,49,43,142,118,154,469,496,229,166,469,496,580,1561,1957,1183
%N A170865 a(0) = 1, a(1) = 1; a(2^i + j) = a(j) + 3a(j + 1) for 0 <= j < 2^i.
%H A170865 David Applegate, Omar E. Pol and N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/tooth.pdf">The Toothpick Sequence and Other Sequences from Cellular Automata</a>
%H A170865 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/toothlist.html">Catalog of Toothpick and Cellular Automata Sequences in the OEIS</a>
%K A170865 nonn
%O A170865 0,3
%A A170865 N. J. A. Sloane (njas(AT)research.att.com), Jan 02 2010

%I A170864
%S A170864 1,1,5,13,5,13,41,49,5,13,41,49,41,121,221,157,5,13,41,49,41,121,221,157,
%T A170864 41,121,221,229,365,805,977,481,5,13,41,49,41,121,221,157,41,121,221,229,
%U A170864 365,805,977,481,41,121,221,229,365,805,977,553,365,805,1121,1417,2705
%N A170864 a(0) = 1, a(1) = 1; a(2^i + j) = 3a(j) + 2a(j + 1) for 0 <= j < 2^i.
%H A170864 David Applegate, Omar E. Pol and N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/tooth.pdf">The Toothpick Sequence and Other Sequences from Cellular Automata</a>
%H A170864 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/toothlist.html">Catalog of Toothpick and Cellular Automata Sequences in the OEIS</a>
%K A170864 nonn
%O A170864 0,3
%A A170864 N. J. A. Sloane (njas(AT)research.att.com), Jan 02 2010

%I A170863
%S A170863 1,1,4,7,4,7,19,25,4,7,19,25,19,40,82,79,4,7,19,25,19,40,82,79,19,40,82,
%T A170863 94,97,202,325,241,4,7,19,25,19,40,82,79,19,40,82,94,97,202,325,241,19,
%U A170863 40,82,94,97,202,325,256,97,202,340,379,493,931,1216,727,4,7,19,25,19,40
%N A170863 a(0) = 1, a(1) = 1; a(2^i + j) = 3a(j) + a(j + 1) for 0 <= j < 2^i.
%H A170863 David Applegate, Omar E. Pol and N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/tooth.pdf">The Toothpick Sequence and Other Sequences from Cellular Automata</a>
%H A170863 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/toothlist.html">Catalog of Toothpick and Cellular Automata Sequences in the OEIS</a>
%K A170863 nonn
%O A170863 0,3
%A A170863 N. J. A. Sloane (njas(AT)research.att.com), Jan 02 2010

%I A170862
%S A170862 1,0,3,9,3,9,36,36,3,9,36,36,36,135,216,117,3,9,36,36,36,135,216,117,36,
%T A170862 135,216,216,513,1053,999,360,3,9,36,36,36,135,216,117,36,135,216,216,513,
%U A170862 1053,999,360,36,135,216,216,513,1053,999,459,513,1053,1296,2187,4698,6156
%N A170862 a(0) = 1, a(1) = 0; a(2^i + j) = 3a(j) + 3a(j + 1) for 0 <= j < 2^i.
%H A170862 David Applegate, Omar E. Pol and N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/tooth.pdf">The Toothpick Sequence and Other Sequences from Cellular Automata</a>
%H A170862 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/toothlist.html">Catalog of Toothpick and Cellular Automata Sequences in the OEIS</a>
%K A170862 nonn
%O A170862 0,3
%A A170862 N. J. A. Sloane (njas(AT)research.att.com), Jan 02 2010

%I A170861
%S A170861 1,0,2,6,2,6,22,18,2,6,22,18,22,78,98,42,2,6,22,18,22,78,98,42,22,78,98,
%T A170861 102,278,450,322,90,2,6,22,18,22,78,98,42,22,78,98,102,278,450,322,90,22,
%U A170861 78,98,102,278,450,322,150,278,450,502,1038,1906,1866,914,186,2,6,22,18
%N A170861 a(0) = 1, a(1) = 0; a(2^i + j) = 2a(j) + 3a(j + 1) for 0 <= j < 2^i.
%H A170861 David Applegate, Omar E. Pol and N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/tooth.pdf">The Toothpick Sequence and Other Sequences from Cellular Automata</a>
%H A170861 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/toothlist.html">Catalog of Toothpick and Cellular Automata Sequences in the OEIS</a>
%K A170861 nonn
%O A170861 0,3
%A A170861 N. J. A. Sloane (njas(AT)research.att.com), Jan 02 2010

%I A170860
%S A170860 1,0,1,3,1,3,10,6,1,3,10,6,10,33,28,9,1,3,10,6,10,33,28,9,10,33,28,36,109,
%T A170860 117,55,12,1,3,10,6,10,33,28,9,10,33,28,36,109,117,55,12,10,33,28,36,109,
%U A170860 117,55,39,109,117,136,363,460,282,91,15,1,3,10,6,10,33,28,9,10,33,28,36
%N A170860 a(0) = 1, a(1) = 0; a(2^i + j) = a(j) + 3a(j + 1) for 0 <= j < 2^i.
%H A170860 David Applegate, Omar E. Pol and N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/tooth.pdf">The Toothpick Sequence and Other Sequences from Cellular Automata</a>
%H A170860 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/toothlist.html">Catalog of Toothpick and Cellular Automata Sequences in the OEIS</a>
%K A170860 nonn
%O A170860 0,4
%A A170860 N. J. A. Sloane (njas(AT)research.att.com), Jan 02 2010

%I A170859
%S A170859 1,0,3,6,3,6,21,24,3,6,21,24,21,60,111,78,3,6,21,24,21,60,111,78,21,60,
%T A170859 111,114,183,402,489,240,3,6,21,24,21,60,111,78,21,60,111,114,183,402,489,
%U A170859 240,21,60,111,114,183,402,489,276,183,402,561,708,1353,2184,1947,726,3
%N A170859 a(0) = 1, a(1) = 0; a(2^i + j) = 3a(j) + 2a(j + 1) for 0 <= j < 2^i.
%H A170859 David Applegate, Omar E. Pol and N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/tooth.pdf">The Toothpick Sequence and Other Sequences from Cellular Automata</a>
%H A170859 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/toothlist.html">Catalog of Toothpick and Cellular Automata Sequences in the OEIS</a>
%K A170859 nonn
%O A170859 0,3
%A A170859 N. J. A. Sloane (njas(AT)research.att.com), Jan 02 2010

%I A170858
%S A170858 1,0,3,3,3,3,12,12,3,3,12,12,12,21,48,39,3,3,12,12,12,21,48,39,12,21,48,
%T A170858 48,57,111,183,120,3,3,12,12,12,21,48,39,12,21,48,48,57,111,183,120,12,
%U A170858 21,48,48,57,111,183,129,57,111,192,201,282,516,669,363,3,3,12,12,12,21
%N A170858 a(0) = 1, a(1) = 0; a(2^i + j) = 3a(j) + a(j + 1) for 0 <= j < 2^i.
%H A170858 David Applegate, Omar E. Pol and N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/tooth.pdf">The Toothpick Sequence and Other Sequences from Cellular Automata</a>
%H A170858 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/toothlist.html">Catalog of Toothpick and Cellular Automata Sequences in the OEIS</a>
%K A170858 nonn
%O A170858 0,3
%A A170858 N. J. A. Sloane (njas(AT)research.att.com), Jan 02 2010

%I A170857
%S A170857 0,1,3,12,3,12,45,45,3,12,45,45,45,171,270,144,3,12,45,45,45,171,270,144,
%T A170857 45,171,270,270,648,1323,1242,441,3,12,45,45,45,171,270,144,45,171,270,
%U A170857 270,648,1323,1242,441,45,171,270,270,648,1323,1242,567,648,1323,1620,2754
%N A170857 a(0) = 0, a(1) = 1; a(2^i + j) = 3a(j) + 3a(j + 1) for 0 <= j < 2^i.
%H A170857 David Applegate, Omar E. Pol and N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/tooth.pdf">The Toothpick Sequence and Other Sequences from Cellular Automata</a>
%H A170857 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/toothlist.html">Catalog of Toothpick and Cellular Automata Sequences in the OEIS</a>
%K A170857 nonn
%O A170857 0,3
%A A170857 N. J. A. Sloane (njas(AT)research.att.com), Jan 02 2010

%I A170856
%S A170856 0,1,3,11,3,11,39,31,3,11,39,31,39,139,171,71,3,11,39,31,39,139,171,71,
%T A170856 39,139,171,179,495,791,555,151,3,11,39,31,39,139,171,71,39,139,171,179,
%U A170856 495,791,555,151,39,139,171,179,495,791,555,259,495,791,879,1843,3363,3247
%N A170856 a(0) = 0, a(1) = 1; a(2^i + j) = 2a(j) + 3a(j + 1) for 0 <= j < 2^i.
%H A170856 David Applegate, Omar E. Pol and N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/tooth.pdf">The Toothpick Sequence and Other Sequences from Cellular Automata</a>
%H A170856 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/toothlist.html">Catalog of Toothpick and Cellular Automata Sequences in the OEIS</a>
%K A170856 nonn
%O A170856 0,3
%A A170856 N. J. A. Sloane (njas(AT)research.att.com), Jan 02 2010

%I A170855
%S A170855 0,1,3,10,3,10,33,19,3,10,33,19,33,109,90,28,3,10,33,19,33,109,90,28,33,
%T A170855 109,90,118,360,379,174,37,3,10,33,19,33,109,90,28,33,109,90,118,360,379,
%U A170855 174,37,33,109,90,118,360,379,174,127,360,379,444,1198,1497,901,285,46
%N A170855 a(0) = 0, a(1) = 1; a(2^i + j) = a(j) + 3a(j + 1) for 0 <= j < 2^i.
%H A170855 David Applegate, Omar E. Pol and N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/tooth.pdf">The Toothpick Sequence and Other Sequences from Cellular Automata</a>
%H A170855 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/toothlist.html">Catalog of Toothpick and Cellular Automata Sequences in the OEIS</a>
%K A170855 nonn
%O A170855 0,3
%A A170855 N. J. A. Sloane (njas(AT)research.att.com), Jan 02 2010

%I A170854
%S A170854 0,1,2,7,2,7,20,25,2,7,20,25,20,61,110,79,2,7,20,25,20,61,110,79,20,61,
%T A170854 110,115,182,403,488,241,2,7,20,25,20,61,110,79,20,61,110,115,182,403,488,
%U A170854 241,20,61,110,115,182,403,488,277,182,403,560,709,1352,2185,1946,727,2
%N A170854 a(0) = 0, a(1) = 1; a(2^i + j) = 3a(j) + 2a(j + 1) for 0 <= j < 2^i.
%H A170854 David Applegate, Omar E. Pol and N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/tooth.pdf">The Toothpick Sequence and Other Sequences from Cellular Automata</a>
%H A170854 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/toothlist.html">Catalog of Toothpick and Cellular Automata Sequences in the OEIS</a>
%K A170854 nonn
%O A170854 0,3
%A A170854 N. J. A. Sloane (njas(AT)research.att.com), Jan 02 2010

%I A170852
%S A170852 1,0,0,3,3,0,0,3,3,0,9,18,9,0,0,3,3,0,9,18,9,0,9,18,9,27,81,81,27,0,0,3,
%T A170852 3,0,9,18,9,0,9,18,9,27,81,81,27,0,9,18,9,27,81,81,27,27,81,81,108,324,
%U A170852 486,324,81,0,0,3,3,0,9,18,9,0,9,18,9,27,81,81,27,0,9,18,9,27,81,81,27
%N A170852 G.f.: Prod_{k >= 2} (1 + 3x^(2^k-1) + 3x^(2^k)).
%H A170852 David Applegate, Omar E. Pol and N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/tooth.pdf">The Toothpick Sequence and Other Sequences from Cellular Automata</a>
%H A170852 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/toothlist.html">Catalog of Toothpick and Cellular Automata Sequences in the OEIS</a>
%K A170852 nonn
%O A170852 0,4
%A A170852 N. J. A. Sloane (njas(AT)research.att.com), Jan 02 2010

%I A170851
%S A170851 1,3,3,3,12,18,9,3,12,18,18,45,90,81,27,3,12,18,18,45,90,81,36,45,90,108,
%T A170851 189,405,513,324,81,3,12,18,18,45,90,81,36,45,90,108,189,405,513,324,90,
%U A170851 45,90,108,189,405,513,351,243,405,594,891,1782,2754,2511,1215,243,3,12
%N A170851 G.f.: Prod_{k >= 1} (1 + 3x^(2^k-1) + 3x^(2^k)).
%H A170851 David Applegate, Omar E. Pol and N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/tooth.pdf">The Toothpick Sequence and Other Sequences from Cellular Automata</a>
%H A170851 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/toothlist.html">Catalog of Toothpick and Cellular Automata Sequences in the OEIS</a>
%K A170851 nonn
%O A170851 0,2
%A A170851 N. J. A. Sloane (njas(AT)research.att.com), Jan 02 2010

%I A170850
%S A170850 4,15,21,21,57,108,90,39,57,108,126,234,495,594,351,93,57,108,126,234,495,
%T A170850 594,387,288,495,702,1080,2187,3267,2835,1296,255,57,108,126,234,495,594,
%U A170850 387,288,495,702,1080,2187,3267,2835,1332,450,495,702,1080,2187,3267,2943
%N A170850 G.f.: Prod_{k >= 0} (1 + 3x^(2^k-1) + 3x^(2^k)).
%H A170850 David Applegate, Omar E. Pol and N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/tooth.pdf">The Toothpick Sequence and Other Sequences from Cellular Automata</a>
%H A170850 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/toothlist.html">Catalog of Toothpick and Cellular Automata Sequences in the OEIS</a>
%K A170850 nonn
%O A170850 0,1
%A A170850 N. J. A. Sloane (njas(AT)research.att.com), Jan 02 2010

%I A170849
%S A170849 1,0,0,3,2,0,0,3,2,0,9,12,4,0,0,3,2,0,9,12,4,0,9,12,4,27,54,36,8,0,0,3,
%T A170849 2,0,9,12,4,0,9,12,4,27,54,36,8,0,9,12,4,27,54,36,8,27,54,36,89,216,216,
%U A170849 96,16,0,0,3,2,0,9,12,4,0,9,12,4,27,54,36,8,0,9,12,4,27,54,36,8,27,54,36
%N A170849 G.f.: Prod_{k >= 2} (1 + 3x^(2^k-1) + 2x^(2^k)).
%H A170849 David Applegate, Omar E. Pol and N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/tooth.pdf">The Toothpick Sequence and Other Sequences from Cellular Automata</a>
%H A170849 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/toothlist.html">Catalog of Toothpick and Cellular Automata Sequences in the OEIS</a>
%K A170849 nonn
%O A170849 0,4
%A A170849 N. J. A. Sloane (njas(AT)research.att.com), Jan 02 2010

%I A170848
%S A170848 1,3,2,3,11,12,4,3,11,12,13,39,58,36,8,3,11,12,13,39,58,36,17,39,58,63,
%T A170848 143,252,224,96,16,3,11,12,13,39,58,36,17,39,58,63,143,252,224,96,25,39,
%U A170848 58,63,143,252,224,123,151,252,305,555,1042,1176,736,240,32,3,11,12,13
%N A170848 G.f.: Prod_{k >= 1} (1 + 3x^(2^k-1) + 2x^(2^k)).
%H A170848 David Applegate, Omar E. Pol and N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/tooth.pdf">The Toothpick Sequence and Other Sequences from Cellular Automata</a>
%H A170848 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/toothlist.html">Catalog of Toothpick and Cellular Automata Sequences in the OEIS</a>
%K A170848 nonn
%O A170848 0,2
%A A170848 N. J. A. Sloane (njas(AT)research.att.com), Jan 02 2010

%I A170847
%S A170847 4,14,14,16,50,70,40,20,50,70,76,182,310,260,104,28,50,70,76,182,310,260,
%T A170847 140,190,310,368,698,1294,1400,832,256,44,50,70,76,182,310,260,140,190,
%U A170847 310,368,698,1294,1400,832,292,206,310,368,698,1294,1400,940,850,1310,1724
%N A170847 G.f.: Prod_{k >= 0} (1 + 3x^(2^k-1) + 2x^(2^k)).
%H A170847 David Applegate, Omar E. Pol and N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/tooth.pdf">The Toothpick Sequence and Other Sequences from Cellular Automata</a>
%H A170847 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/toothlist.html">Catalog of Toothpick and Cellular Automata Sequences in the OEIS</a>
%K A170847 nonn
%O A170847 0,1
%A A170847 N. J. A. Sloane (njas(AT)research.att.com), Jan 02 2010

%I A170846
%S A170846 1,0,0,3,1,0,0,3,1,0,9,6,1,0,0,3,1,0,9,6,1,0,9,6,1,27,27,9,1,0,0,3,1,0,
%T A170846 9,6,1,0,9,6,1,27,27,9,1,0,9,6,1,27,27,9,1,27,27,9,82,108,54,12,1,0,0,3,
%U A170846 1,0,9,6,1,0,9,6,1,27,27,9,1,0,9,6,1,27,27,9,1,27,27,9,82,108,54,12,1,0
%N A170846 G.f.: Prod_{k >= 2} (1 + 3x^(2^k-1) + x^(2^k)).
%H A170846 David Applegate, Omar E. Pol and N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/tooth.pdf">The Toothpick Sequence and Other Sequences from Cellular Automata</a>
%H A170846 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/toothlist.html">Catalog of Toothpick and Cellular Automata Sequences in the OEIS</a>
%K A170846 nonn
%O A170846 0,4
%A A170846 N. J. A. Sloane (njas(AT)research.att.com), Jan 02 2010

%I A170845
%S A170845 1,3,1,3,10,6,1,3,10,6,10,33,28,9,1,3,10,6,10,33,28,9,10,33,28,36,109,117,
%T A170845 55,12,1,3,10,6,10,33,28,9,10,33,28,36,109,117,55,12,10,33,28,36,109,117,
%U A170845 55,39,109,117,136,363,460,282,91,15,1,3,10,6,10,33,28,9,10,33,28,36,109
%N A170845 G.f.: Prod_{k >= 1} (1 + 3x^(2^k-1) + x^(2^k)).
%H A170845 David Applegate, Omar E. Pol and N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/tooth.pdf">The Toothpick Sequence and Other Sequences from Cellular Automata</a>
%H A170845 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/toothlist.html">Catalog of Toothpick and Cellular Automata Sequences in the OEIS</a>
%K A170845 nonn
%O A170845 0,2
%A A170845 N. J. A. Sloane (njas(AT)research.att.com), Jan 02 2010

%I A170844
%S A170844 4,13,7,13,43,34,10,13,43,34,46,142,145,64,13,13,43,34,46,142,145,64,49,
%T A170844 142,145,172,472,577,337,103,16,13,43,34,46,142,145,64,49,142,145,172,472,
%U A170844 577,337,103,52,142,145,172,472,577,337,211,475,577,661,1588,2203,1588
%N A170844 G.f.: Prod_{k >= 0} (1 + 3x^(2^k-1) + x^(2^k)).
%H A170844 David Applegate, Omar E. Pol and N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/tooth.pdf">The Toothpick Sequence and Other Sequences from Cellular Automata</a>
%H A170844 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/toothlist.html">Catalog of Toothpick and Cellular Automata Sequences in the OEIS</a>
%K A170844 nonn
%O A170844 0,1
%A A170844 N. J. A. Sloane (njas(AT)research.att.com), Jan 02 2010

%I A170843
%S A170843 1,0,0,2,3,0,0,2,3,0,4,12,9,0,0,2,3,0,4,12,9,0,4,12,9,8,36,54,27,0,0,2,
%T A170843 3,0,4,12,9,0,4,12,9,8,36,54,27,0,4,12,9,8,36,54,27,8,36,54,43,96,216,216,
%U A170843 81,0,0,2,3,0,4,12,9,0,4,12,9,8,36,54,27,0,4,12,9,8,36,54,27,8,36,54,43
%N A170843 G.f.: Prod_{k >= 2} (1 + 2x^(2^k-1) + 3x^(2^k)).
%H A170843 David Applegate, Omar E. Pol and N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/tooth.pdf">The Toothpick Sequence and Other Sequences from Cellular Automata</a>
%H A170843 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/toothlist.html">Catalog of Toothpick and Cellular Automata Sequences in the OEIS</a>
%K A170843 nonn
%O A170843 0,4
%A A170843 N. J. A. Sloane (njas(AT)research.att.com), Jan 02 2010

%I A170842
%S A170842 1,2,3,2,7,12,9,2,7,12,13,20,45,54,27,2,7,12,13,20,45,54,31,20,45,62,79,
%T A170842 150,243,216,81,2,7,12,13,20,45,54,31,20,45,62,79,150,243,216,85,20,45,
%U A170842 62,79,150,243,224,133,150,259,344,537,936,1161,810,243,2,7,12,13,20,45
%N A170842 G.f.: Prod_{k >= 1} (1 + 2x^(2^k-1) + 3x^(2^k)).
%H A170842 David Applegate, Omar E. Pol and N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/tooth.pdf">The Toothpick Sequence and Other Sequences from Cellular Automata</a>
%H A170842 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/toothlist.html">Catalog of Toothpick and Cellular Automata Sequences in the OEIS</a>
%K A170842 nonn
%O A170842 0,2
%A A170842 N. J. A. Sloane (njas(AT)research.att.com), Jan 02 2010

%I A170841
%S A170841 3,9,15,15,27,57,63,33,27,57,75,99,195,297,243,87,27,57,75,99,195,297,255,
%T A170841 153,195,321,423,687,1179,1377,891,249,27,57,75,99,195,297,255,153,195,
%U A170841 321,423,687,1179,1377,903,315,195,321,423,687,1179,1401,1071,849,1227
%N A170841 G.f.: Prod_{k >= 0} (1 + 2x^(2^k-1) + 3x^(2^k)).
%H A170841 David Applegate, Omar E. Pol and N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/tooth.pdf">The Toothpick Sequence and Other Sequences from Cellular Automata</a>
%H A170841 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/toothlist.html">Catalog of Toothpick and Cellular Automata Sequences in the OEIS</a>
%K A170841 nonn
%O A170841 0,1
%A A170841 N. J. A. Sloane (njas(AT)research.att.com), Jan 02 2010

%I A170840
%S A170840 1,0,0,1,3,0,0,1,3,0,1,6,9,0,0,1,3,0,1,6,9,0,1,6,9,1,9,27,27,0,0,1,3,0,
%T A170840 1,6,9,0,1,6,9,1,9,27,27,0,1,6,9,1,9,27,27,1,9,27,28,12,54,108,81,0,0,1,
%U A170840 3,0,1,6,9,0,1,6,9,1,9,27,27,0,1,6,9,1,9,27,27,1,9,27,28,12,54,108,81,0
%N A170840 G.f.: Prod_{k >= 2} (1 + x^(2^k-1) + 3x^(2^k)).
%H A170840 David Applegate, Omar E. Pol and N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/tooth.pdf">The Toothpick Sequence and Other Sequences from Cellular Automata</a>
%H A170840 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/toothlist.html">Catalog of Toothpick and Cellular Automata Sequences in the OEIS</a>
%K A170840 nonn
%O A170840 0,5
%A A170840 N. J. A. Sloane (njas(AT)research.att.com), Jan 02 2010

%I A170839
%S A170839 1,1,3,1,4,6,9,1,4,6,10,7,18,27,27,1,4,6,10,7,18,27,28,7,18,28,37,39,81,
%T A170839 108,81,1,4,6,10,7,18,27,28,7,18,28,37,39,81,108,82,7,18,28,37,39,81,109,
%U A170839 91,39,82,121,150,198,351,405,243,1,4,6,10,7,18,27,28,7,18,28,37,39,81
%N A170839 G.f.: Prod_{k >= 1} (1 + x^(2^k-1) + 3x^(2^k)).
%H A170839 David Applegate, Omar E. Pol and N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/tooth.pdf">The Toothpick Sequence and Other Sequences from Cellular Automata</a>
%H A170839 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/toothlist.html">Catalog of Toothpick and Cellular Automata Sequences in the OEIS</a>
%K A170839 nonn
%O A170839 0,3
%A A170839 N. J. A. Sloane (njas(AT)research.att.com), Jan 02 2010

%I A170838
%S A170838 2,5,9,11,11,24,36,29,11,24,38,44,57,108,135,83,11,24,38,44,57,108,137,
%T A170838 98,57,110,158,189,279,459,486,245,11,24,38,44,57,108,137,98,57,110,158,
%U A170838 189,279,459,488,260,57,110,158,189,279,461,509,351,281,488,663,846,1296
%N A170838 G.f.: Prod_{k >= 0} (1 + x^(2^k-1) + 3x^(2^k)).
%C A170838 A170838-A170852, A170854-A170872 were added to supplement Gary Adamson's A162956.
%H A170838 David Applegate, Omar E. Pol and N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/tooth.pdf">The Toothpick Sequence and Other Sequences from Cellular Automata</a>
%H A170838 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/toothlist.html">Catalog of Toothpick and Cellular Automata Sequences in the OEIS</a>
%p A170838 Maple program for A170838-A170852, A162956, A170854-A170872.
%p A170838 read format;
%p A170838 G := proc(a,b,c); mul( 1 + a*x^(2^n-1) + b*x^(2^n), n=c..20); end;
%p A170838 f := proc(a,b,c) seriestolist(series(G(a,b,c),x,120)); end;
%p A170838 at:=170838:
%p A170838 for a from 1 to 2 do for c from 0 to 2 do
%p A170838 b:=3;
%p A170838 t1:=f(a,b,c);
%p A170838 lprint( format(t1,at) );
%p A170838 lprint("G.f.: Prod_{k >= ", c, "} (1 +",a,"* x^(2^k-1) +",b,"* x^(2^k)).");
%p A170838 at:=at+1; od: od:
%p A170838 for b from 1 to 3 do for c from 0 to 2 do
%p A170838 a:=3;
%p A170838 t1:=f(a,b,c);
%p A170838 lprint( format(t1,at) );
%p A170838 lprint("G.f.: Prod_{k >= ", c, "} (1 +",a,"* x^(2^k-1) +",b,"* x^(2^k)).");
%p A170838 at:=at+1; od: od:
%p A170838 h:=proc(r,s,a,b) local s1,n,i,j;
%p A170838 s1:=array(0..120);
%p A170838 s1[0]:=r; s1[1]:=s;
%p A170838 for n from 2 to 120 do i:=floor(log(n)/log(2));
%p A170838 j:=n-2^i; s1[n]:=a*s1[j]+b*s1[j+1]; od:
%p A170838 [seq(s1[n],n=0..120)];
%p A170838 end;
%p A170838 l1:=[[0,1],[1,0],[1,1],[1,2]];
%p A170838 l2:=[[3,1],[3,2],[1,3],[2,3],[3,3]];
%p A170838 for i from 1 to 4 do for j from 1 to 5 do
%p A170838 r:=l1[i][1];
%p A170838 s:=l1[i][2];
%p A170838 a:=l2[j][1];
%p A170838 b:=l2[j][2];
%p A170838 t1:=h(r,s,a,b);
%p A170838 lprint(format(t1,at)); at:=at+1;
%p A170838 lprint("a(0)=",r,", a(1)=", s, "; a(2^i+j)=",a,"*a(j)+",b,"a(j+1) for 0 <= j < 2^i.");
%p A170838 od: od:
%K A170838 nonn
%O A170838 0,1
%A A170838 N. J. A. Sloane (njas(AT)research.att.com), Jan 02 2010

%I A170834
%S A170834 5,5,25,25,145,125,725,725,3625,3625,18625,18125,93125,93125,465625,465625,
%T A170834 2340625,2328125,11703125,11703125,58515625,58515625,292890625,292578125,
%U A170834 1464453125,1464453125,7322265625,7322265625,36619140625,36611328125,183095703125
%N A170834 5^(floor(n/2))+5^(floor(n/2)-1)-5^(floor((n-1)/3)).
%D A170834 R. P. Stanley, Problem 11348, Amer. Math. Monthly, 117 (2010), 87-88.
%K A170834 nonn
%O A170834 2,1
%A A170834 N. J. A. Sloane (njas(AT)research.att.com), Dec 30 2009

%I A170833
%S A170833 4,4,16,16,76,64,304,304,1216,1216,5056,4864,20224,20224,80896,80896,326656,
%T A170833 323584,1306624,1306624,5226496,5226496,20955136,20905984,83820544,83820544,
%U A170833 335282176,335282176,1341915136,1341128704,5367660544,5367660544,21470642176
%N A170833 4^(floor(n/2))+4^(floor(n/2)-1)-4^(floor((n-1)/3)).
%D A170833 R. P. Stanley, Problem 11348, Amer. Math. Monthly, 117 (2010), 87-88.
%K A170833 nonn
%O A170833 2,1
%A A170833 N. J. A. Sloane (njas(AT)research.att.com), Dec 30 2009

%I A170832
%S A170832 3,3,9,9,33,27,99,99,297,297,945,891,2835,2835,8505,8505,26001,25515,78003,
%T A170832 78003,234009,234009,706401,702027,2119203,2119203,6357609,6357609,19112193,
%U A170832 19072827,57336579,57336579,172009737,172009737,516383505,516029211,1549150515
%N A170832 3^(floor(n/2))+3^(floor(n/2)-1)-3^(floor((n-1)/3)).
%D A170832 R. P. Stanley, Problem 11348, Amer. Math. Monthly, 117 (2010), 87-88.
%K A170832 nonn
%O A170832 2,1
%A A170832 N. J. A. Sloane (njas(AT)research.att.com), Dec 30 2009

%I A170831
%S A170831 2,2,4,4,10,8,20,20,40,40,88,80,176,176,352,352,736,704,1472,1472,2944,
%T A170831 2944,6016,5888,12032,12032,24064,24064,48640,48128,97280,97280,194560,
%U A170831 194560,391168,389120,782336,782336,1564672,1564672,3137536,3129344,6275072
%N A170831 2^(floor(n/2))+2^(floor(n/2)-1)-2^(floor((n-1)/3)).
%D A170831 R. P. Stanley, Problem 11348, Amer. Math. Monthly, 117 (2010), 87-88.
%K A170831 nonn
%O A170831 2,1
%A A170831 N. J. A. Sloane (njas(AT)research.att.com), Dec 30 2009

%I A170826
%S A170826 1,2,3,8,5,36,7,64,81,100,11,144,13,196,225,256,17,324,19,400,441,484,
%T A170826 23,576,625,676,729,784,29,900,31,1024,1089,1156,1225,1296,37,1444,1521,
%U A170826 1600,41,1764,43,1936,2025,2116,47,2304,2401,2500,2601,2704,53,2916
%N A170826 a(n) = GCD(n^2, n!).
%F A170826 If n is prime then a(n)=n else if n<>4 a(n)=n^2. - Zak Seidov, Dec 28 2009
%p A170826 GCDWITHFACTORIAL:=proc(a) local b,i,k:
%p A170826 if whattype(a) <> list then RETURN([]); fi:
%p A170826 b:=[]:
%p A170826 for i to nops(a) do b:=[op(b), gcd(a[i],i!)]: od;
%p A170826 RETURN(b);
%p A170826 end:
%K A170826 nonn
%O A170826 1,2
%A A170826 N. J. A. Sloane (njas(AT)research.att.com), Dec 27 2009

%I A170825
%S A170825 1,1,1,1,5,1,1,1,1,5,11,1,1,1,5,1,17,1,1,5,1,11,23,1,5,1,1,1,29,5,1,1,
%T A170825 11,17,5,1,1,1,1,5,41,1,1,11,5,23,47,1,1,5,17,1,53,1,55,1,1,29,59,5,1,1,
%U A170825 1,1,5,11,1,17,23,5,71,1,1,1,5,1,11,1,1,5,1,41,83,1,85,1,29,11,89,5,1
%N A170825 a(n) = product of distinct primes of form 6k-1 that divide n.
%p A170825 A170825 := proc(n) a := 1 ; for p in numtheory[factorset](n) do if p mod 6 = 5 then a := a*p ; end if ; end do ; a ; end proc: seq(A170825(n),n=1..120) ; [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jan 21 2010]
%Y A170825 Cf. A007528, A170819.
%Y A170825 Cf. A140214. [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jan 21 2010]
%K A170825 nonn,new
%O A170825 1,5
%A A170825 N. J. A. Sloane (njas(AT)research.att.com), Dec 25 2009, following a suggestion from Jonathan Vos Post.
%E A170825 More terms from R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jan 21 2010

%I A170824
%S A170824 1,1,1,1,1,1,7,1,1,1,1,1,13,7,1,1,1,1,19,1,7,1,1,1,1,13,1,7,1,1,31,1,1,
%T A170824 1,7,1,37,19,13,1,1,7,43,1,1,1,1,1,7,1,1,13,1,1,1,7,19,1,1,1,61,31,7,1,
%U A170824 13,1,67,1,1,7,1,1,73,37,1,19,7,13,79,1,1,1,1,7,1,43,1,1,1,1,91,1,31,1
%N A170824 a(n) = product of distinct primes of form 6k+1 that divide n.
%p A170824 A170824 := proc(n) a := 1 ; for p in numtheory[factorset](n) do if p mod 6 = 1 then a := a*p ; end if ; end do ; a ; end proc: A140213 := proc(n) a := 1 ; for p in numtheory[divisors](n) do if p mod 6 = 1 then a := a*p ; end if ; end do ; a ; end proc: seq(A170824(n),n=1..120) ; [From R.J. Mathar (mathar(AT)strw.leidenuniv.nl), Jan 21 2010]
%Y A170824 Cf. A002476, A170817.
%Y A170824 Cf. A140213. [From R.J. Mathar (mathar(AT)strw.leidenuniv.nl), Jan 21 2010]
%K A170824 nonn,new
%O A170824 1,7
%A A170824 N. J. A. Sloane (njas(AT)research.att.com), Dec 25 2009, following a suggestion from Jonathan Vos Post.
%E A170824 More terms from R.J. Mathar (mathar(AT)strw.leidenuniv.nl), Jan 21 2010

%I A170823
%S A170823 1,2,3,2,1,2,3,1,3,2,3,1,2,1,3,2,3,1,3,2,1,2,3,2,1,2,3,1,3,2,3,1,2,1,3,
%T A170823 1,2,3,2,1,3,1,2,1,3,2,3,1,3,2,3,1,2,1,3,1,2,3,2,1,2,3,1,3,2,1,2,3,2,1,
%U A170823 3,1,2,1,3,2,3,1,3,2,3,1,2,1,3,1,2,3,2,1,3,1,2,1,3,2,3,1,3,2,1,2,3,2,1
%N A170823 An infinite word on the alphabet 1, 2, 3 in which no block appears twice in succession.
%C A170823 A concatenation of blocks u_k, k >= 0, where u_k has length 5^k. The sequence is defined recursively - see the Maple code.
%D A170823 B. Bollobas, The Art of Mathematics: Coffee Time in Memphis, Cambridge, 2006, p. 226.
%p A170823 a:=[1,2,3,2,1]; b:=[2,3,1,3,2]; c:=[3,1,2,1,3]; S:=[1];
%p A170823 for m from 1 to 6 do S:=subs({1=a[],2=b[],3=c[]},S); od: S;
%Y A170823 Cf. A010060, A005678-A005681, A006156, A007413.
%K A170823 nonn
%O A170823 0,2
%A A170823 N. J. A. Sloane (njas(AT)research.att.com), Dec 25 2009

%I A170822
%S A170822 1,3,2,2,1,1,2,1,1,12,1,1,2,1,2,4,1,14,2,1,2,1,2,1,1,2,2,1,1,10,1,3,1,1,
%T A170822 4,9,2,1,2,18,2,16,1,1,1,1,2,2,1,2,6,2,1,2,1,1,2,1,1,1,3,10,12,1,1,42,2,
%U A170822 12,1,2,1,4,27,2,1,4,1,6,2,6,10,4,3,2,1,2,1,1,2,2,1,2,3,2,1,5
%N A170822 Let p = n-th prime; a(n) = (p-1)/(order of A170821(n) mod p).
%D A170822 I. Anderson and D. A. Preece, Combinatorially fruitful properies of ..., Discr. Math., 310 (2010), 312-324.
%e A170822 n=3: p=5, A170821(n)=2, order of 2 mod 5 = 4, (5-1)/4 = 1 = a(3).
%Y A170822 Cf. A001917, A170820, A170821.
%K A170822 nonn
%O A170822 3,2
%A A170822 N. J. A. Sloane (njas(AT)research.att.com), Dec 24 2009

%I A170821
%S A170821 0,2,6,9,4,5,15,18,8,24,10,11,33,36,14,45,16,51,54,19,60,63,23,25,26,78,
%T A170821 81,28,29,96,99,35,105,38,114,40,123,126,44,135,46,144,49,50,150,159,168,
%U A170821 171,58,59,180,61,189,65,198,68,204,70,71,213,74,231,234,79,80,249,85,261
%N A170821 Let p = n-th prime; a(n) = smallest b >= 0 such that 4b == 3 mod p.
%D A170821 I. Anderson and D. A. Preece, Combinatorially fruitful properies of ..., Discr. Math., 310 (2010), 312-324.
%p A170821 f:=proc(n) local b; for b from 0 to n-1 do if 4*b mod n = 3 then RETURN(b); fi; od: -1; end; [seq(f(ithprime(n)),n=2..100)]; # Gives wrong answer for n=2.
%K A170821 sign
%O A170821 2,2
%A A170821 N. J. A. Sloane (njas(AT)research.att.com), Dec 24 2009

%I A170820
%S A170820 2,1,1,3,1,6,2,4,1,1,1,2,2,4,1,5,2,10,2,3,1,1,12,4,1,1,1,1,1,1,1,2,2,1,
%T A170820 1,2,2,2,1,5,2,2,4,3,42,1,1,1,1,2,8,1,1,2,4,1,1,7,2,4,6,2,2,4,30,2,1,1,
%U A170820 1,2,1,3,2,2,2,1,25,4,11,1,10,2,3,1,1,8,10,33,1,2,3,1,6,2,4,1,2,1,2,2,1
%N A170820 Let p = n-th prime; a(n) = (p-1)/(order of (p+3)/2 mod p).
%D A170820 I. Anderson and D. A. Preece, Combinatorially fruitful properies of ..., Discr. Math., 310 (2010), 312-324.
%p A170820 with(numtheory); [seq((ithprime(n)-1)/order((ithprime(n)+3)/2,ithprime(n)),n=3..130)];
%Y A170820 Cf. A014664, A001917, A170821, A170822.
%K A170820 nonn
%O A170820 3,1
%A A170820 N. J. A. Sloane (njas(AT)research.att.com), Dec 24 2009

%I A170819
%S A170819 1,1,3,1,1,3,7,1,3,1,11,3,1,7,3,1,1,3,19,1,21,11,23,3,1,1,3,7,1,3,
%T A170819 31,1,33,1,7,3,1,19,3,1,1,21,43,11,3,23,47,3,7,1,3,1,1,3,11,7,57,1,
%U A170819 59,3,1,31,21,1,1,33,67,1,69,7,71,3,1,1,3,19,77,3,79,1,3,1,83,21,1
%N A170819 a(n) = product of distinct primes of form 4k-1 that divide n.
%o A170819 (PARI) for(n=1,99, t=select(x->x%4==3, factor(n)[,1]); print1(prod(i=1,#t,t[i])","))
%Y A170819 Cf. A170817-A170818, A097706, A083025.
%K A170819 nonn
%O A170819 1,3
%A A170819 N. J. A. Sloane (njas(AT)research.att.com), Dec 23 2009
%E A170819 Extended with PARI program by M. F. Hasler, Dec 23 2009

%I A170818
%S A170818 1,1,1,1,5,1,1,1,1,5,1,1,13,1,5,1,17,1,1,5,1,1,1,1,25,13,1,1,29,5,1,
%T A170818 1,1,17,5,1,37,1,13,5,41,1,1,1,5,1,1,1,1,25,17,13,53,1,5,1,1,29,1,5,
%U A170818 61,1,1,1,65,1,1,17,1,5,1,1,73,37,25,1,1,13,1,5,1,41,1,1,85,1,29,1
%N A170818 a(n) = product of primes (with multiplicity) of form 4k+1 that divide n.
%D A170818 A. Tripathi, On Pythagorean triples containing a fixed integer, Fib. Q., 46/47 (2008/2009), 331-340.
%p A170818 a:= n-> mul (i, i=map (x-> x[1]^x[2], select (x-> isprime (x[1]) and irem (x[1], 4)=1, ifactors(n)[2]))): seq (a(n), n=1..120);
%Y A170818 Cf. A170817-A170819, A097706, A083025.
%K A170818 nonn
%O A170818 1,5
%A A170818 N. J. A. Sloane (njas(AT)research.att.com), Dec 22 2009
%E A170818 Corrected and extended with Maple program by Alois Heinz, Dec 23 2009

%I A170817
%S A170817 1,1,1,1,5,1,1,1,1,5,1,1,13,1,5,1,17,1,1,5,1,1,1,1,5,13,1,1,29,5,1,
%T A170817 1,1,17,5,1,37,1,13,5,41,1,1,1,5,1,1,1,1,5,17,13,53,1,5,1,1,29,1,5,
%U A170817 61,1,1,1,65,1,1,17,1,5,1,1,73,37,5,1,1,13,1,5,1,41,1,1,85,1,29,1
%N A170817 a(n) = product of distinct primes of form 4k+1 that divide n.
%p A170817 a:= n-> mul (i, i=map (x-> x[1], select (x-> isprime (x[1]) and irem (x[1], 4)=1, ifactors(n)[2]))): seq (a(n), n=1..120);
%Y A170817 Cf. A170818-A170819, A097706, A083025, A170824, A170825.
%K A170817 nonn
%O A170817 1,5
%A A170817 N. J. A. Sloane (njas(AT)research.att.com), Dec 22 2009
%E A170817 Corrected and extended with Maple program by Alois Heinz, Dec 23 2009

%I A170874
%S A170874 9,3,12,4,6,7,14,3,7,13,11,0,12,7,10,4,13,1,11,14,3,15,8,1,0,1,5,2,12,
%T A170874 11,5,6,10,1,12,14,12,12,3,10,15,6,5,12,12,0,1,9,0,12,0,3,13,15,3,4,7,0,
%U A170874 9,10,15,15,11,13,8,14,4,11,5,9,15,10,0,3,10,9,15,0,14,14,13,0,6,4,9,12
%N A170874 Hexadecimal expansion of Euler's constant (or Euler-Mascheroni constant) gamma.
%F A170874 a(n) = 8*A104015(4n)+4*A104015(4n+1)+2*A104015(4n+2)+1*A104015(4n+3).
%e A170874 0x0.93C467E37DB0C7A4D1BE3F810152CB56A1CECC3AF65CC0190C03DF34709AFFBD8E4B5...
%t A170874 RealDigits[EulerGamma~N~200, 16][[1]]
%Y A170874 Cf. A104015, A001620.
%K A170874 cons,base,easy,nonn
%O A170874 0,1
%A A170874 Andrew Robbins (and_j_rob(AT)yahoo.com), Jan 03 2010, at the request of N. J. A. Sloane

%I A170873
%S A170873 2,11,7,14,1,5,1,6,2,8,10,14,13,2,10,6,10,11,15,7,1,5,8,8,0,9,12,15,4,
%T A170873 15,3,12,7,6,2,14,7,1,6,0,15,3,8,11,4,13,10,5,6,10,7,8,4,13,9,0,4,5,1,9,
%U A170873 0,12,15,14,15,3,2,4,14,7,7,3,8,9,2,6,12,15,11,14,5,15,4,11,15,8,13,8
%N A170873 Hexadecimal expansion of e.
%F A170873 a(n) = 8*A004593(4n)+4*A004593(4n+1)+2*A004593(4n+2)+1*A004593(4n+3).
%e A170873 0x2.B7E151628AED2A6ABF7158809CF4F3C762E7160F38B4DA56A784D9045190CFEF324E...
%t A170873 RealDigits[E~N~200, 16][[1]]
%Y A170873 Cf. A004593, A001113.
%K A170873 cons,base,easy,nonn
%O A170873 1,1
%A A170873 Andrew Robbins (and_j_rob(AT)yahoo.com), Jan 03 2010, at the request of N. J. A. Sloane

(end)

        The On-Line Encyclopedia of Integer Sequences, Recent Additions

This is a section of the main database for the On-Line Encyclopedia of Integer Sequences.

It shows the most recently added sequences in reverse chronological order.

For more information see the following pages:
( www.research.att.com/~njas/sequences/ then )

    Seis.html:          Welcome
    index.html:         Lookup 
    demo1.html:         Demos 
    Sindx.html:         Index 
    WebCam.html:        WebCam
    Submit.html:        Contribute new sequence or comment 
    eishelp1.html:      Internal format 
    eishelp2.html:      Beautified format
    transforms.html:    Transforms 
    Spuzzle.html:       Puzzles 
    Shot.html:          Hot 
    classic.html:       Classics
    ol.html:            Superseeker 
    pages.html:         More pages 

Maintained  by:         N. J. A. Sloane (njas@research.att.com), 
    home page:          www.research.att.com/~njas/