The Database of Integer Sequences, Part 9 

     Part of the On-Line Encyclopedia of Integer Sequences


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

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

    Seis.html:          Welcome
    index.html:         Lookup 
    indexfr.html:       Francais
    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 
    JIS/index.html:     Journal of Integer Sequences
    pages.html:         More pages 

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


(start)



%I A006704 M0119
%S A006704 1,1,2,1,2,5,8,2,1,3,10,7,3,15,4,1,4,17,170,4,5,197,24,5,1,5,26,16,70,11,
%T A006704 1520,6,23,35,6,1,6,37,25,6,32,13,3482,20,4,24335,6,7,1,7,50,36,182,485,89
%N A006704 Solution to Pellian: x such that x^2 - n y^2 = +- 1, +- 4.
%D A006704 A. Cayley, Report of a committee appointed for the purpose of carrying on the tables connected with the Pellian equation ..., Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 13, pp. 430-443.
%D A006704 C. F. Degen, Canon Pellianus. Hafniae, Copenhagen, 1817.
%D A006704 D. H. Lehmer, Guide to Tables in the Theory of Numbers. Bulletin No. 105, National Research Council, Washington, DC, 1941, p. 55.
%Y A006704 Cf. A006705.
%Y A006704 Sequence in context: A047991 A120898 A052532 this_sequence A006702 A129394 A049901
%Y A006704 Adjacent sequences: A006701 A006702 A006703 this_sequence A006705 A006706 A006707
%K A006704 nonn
%O A006704 1,3
%A A006704 njas

%I A006702 M0120
%S A006702 1,1,2,1,2,5,8,3,1,3,10,7,18,15,4,1,4,17,170,9,55,197,24,5,1,5,26,
%T A006702 127,70,11,1520,17,23,35,6,1,6,37,25,19,32,13,3482,199,161,24335,48,
%U A006702 7,1,7,50,649,182,485,89,15,151,99,530,31,29718,63,8,1,8,65,48842
%N A006702 Solution to a Pellian equation: least x such that x^2 - n y^2 = +- 1.
%C A006702 When n is a square, the trivial solution x=1, y=1 is taken; otherwise we take the least x that satisfies either the +1 or -1 equation. - T. D. Noe, May 19 2007
%D A006702 A. Cayley, Report of a committee appointed for the purpose of carrying on the tables connected with the Pellian equation ..., Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 13, pp. 430-443.
%D A006702 C. F. Degen, Canon Pellianus. Hafniae, Copenhagen, 1817.
%D A006702 D. H. Lehmer, Guide to Tables in the Theory of Numbers. Bulletin No. 105, National Research Council, Washington, DC, 1941, p. 55.
%H A006702 T. D. Noe, <a href="http://www.research.att.com/~njas/sequences/b006702.txt">Table of n, a(n) for n=1..10000</a>
%H A006702 M. Zuker, <a href="http://www.bioinfo.rpi.edu/~zukerm/cgi-bin/dq.html">Fundamental solution to Pell's Equation x^2 - d*y^2 = +-1</a>
%Y A006702 Cf. A006703, A077232
%Y A006702 Sequence in context: A120898 A052532 A006704 this_sequence A129394 A049901 A117715
%Y A006702 Adjacent sequences: A006699 A006700 A006701 this_sequence A006703 A006704 A006705
%K A006702 nonn,nice,easy
%O A006702 1,3
%A A006702 njas
%E A006702 Corrected and extended by T. D. Noe, May 19 2007

%I A129394
%S A129394 0,1,2,1,2,5,8,5,2,3,9,4,0,4,9,3,4,19,12,16,32,16,12,19,4,5,107,116,28,
%T A129394 76,120,76,28,116,107,5,6,287,276,96,64,132,144,132,64,96,276,287,6,7,
%U A129394 367,380,300,196,8,268,392,268,8,196,300,380,367,7,8,245,260,352,992
%V A129394 0,1,2,1,2,-5,-8,-5,2,3,9,4,0,4,9,3,4,19,12,-16,-32,-16,12,19,4,5,-107,-116,-28,76,120,
%W A129394 76,-28,-116,-107,5,6,287,276,96,-64,-132,-144,-132,-64,96,276,287,6,7,-367,-380,-300,
%X A129394 -196,8,268,392,268,8,-196,-300,-380,-367,7,8,-245,-260,352,992,940,144,-804,-1216
%N A129394 Triangle read by rows: T(r,c)=T(r,c-1)+T(r,c+1)+T(r-1,c-1).
%C A129394 Like A129392, except edge elements=0,1,2,3,4,5,...
%H A129394 R. J. Mathar, <a href="http://www.research.att.com/~njas/sequences/a129392.pdf">Comments on A129392, A129394 and A129396</a>
%Y A129394 Cf. A129392, A129395.
%Y A129394 Sequence in context: A052532 A006704 A006702 this_sequence A049901 A117715 A107087
%Y A129394 Adjacent sequences: A129391 A129392 A129393 this_sequence A129395 A129396 A129397
%K A129394 sign,tabf
%O A129394 0,3
%A A129394 Jonas Wallgren (jonwa(AT)ida.liu.se), Apr 13 2007
%E A129394 Corrected and extended by R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jan 17 2008

%I A049901
%S A049901 1,2,1,2,5,9,19,37,75,114,246,502,1008,2019,4039,8077,16155,24234,
%T A049901 52506,107032,215075,430656,861568,1723268,3446575,6893188,13786394,
%U A049901 27572798,55145600,110291203,220582407,441164813,882329627,1323494442
%N A049901 a(n)=a(1)+a(2)+...+a(n-1)-a(m), where m=2^(p+1)+2-n, and 2^p<n-1<=2^(p+1).
%Y A049901 Sequence in context: A006704 A006702 A129394 this_sequence A117715 A107087 A115141
%Y A049901 Adjacent sequences: A049898 A049899 A049900 this_sequence A049902 A049903 A049904
%K A049901 nonn
%O A049901 1,2
%A A049901 Clark Kimberling (ck6(AT)evansville.edu)

%I A117715
%S A117715 0,1,1,0,1,2,1,2,5,10,0,3,12,33,72,1,5,29,109,305,701,0,8,70,360,1292,
%T A117715 3640,8658,1,13,169,1189,5473,18901,53353,129949,0,21,408,3927,23184,
%U A117715 98145,328776,927843,2298912,1,34,985,12970,98209,509626,2026009
%N A117715 Triangle based on expansion x/(1-m*x-x^2)=Sum[Fibonacci[n,m]*x^n/n!,{n,0,Infinity}].
%D A117715 Steven Wolfram, The Mathematica Book,Cambridge University Press, 3rd ed. 1996, page 728
%F A117715 a(n,m) = Fibonacci[n,m]
%e A117715 0
%e A117715 1, 1
%e A117715 0, 1, 2
%e A117715 1, 2, 5, 10
%e A117715 0, 3, 12, 33, 72
%e A117715 1, 5, 29, 109, 305, 701
%e A117715 0, 8, 70, 360, 1292, 3640, 8658
%e A117715 1, 13, 169, 1189, 5473, 18901, 53353, 129949
%p A117715 with(combinat):for n from 0 to 9 do seq(fibonacci(n,m), m = 0 .. n) od; - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 09 2008
%t A117715 a = Table[Table[Fibonacci[n, m], {m, 0, n}], {n, 0, 10}] Flatten[a]
%Y A117715 Cf. A000045, A117716.
%Y A117715 Cf. A073133.
%Y A117715 Sequence in context: A006702 A129394 A049901 this_sequence A107087 A115141 A031148
%Y A117715 Adjacent sequences: A117712 A117713 A117714 this_sequence A117716 A117717 A117718
%K A117715 nonn,uned,tabl
%O A117715 0,6
%A A117715 Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Apr 13 2006

%I A107087
%S A107087 1,2,1,2,5,12,30,82,233,668,1949,5802,17503,53302,163783,507418,1582869,
%T A107087 4966790,15667573,49658264,158059506,505013014,1619144976,5207596574,16797286048,
%U A107087 54323516786,176113786590,572236347200,1863213388504,6078422747600,19865661574555
%V A107087 1,2,-1,2,-5,12,-30,82,-233,668,-1949,5802,-17503,53302,-163783,507418,-1582869,
%W A107087 4966790,-15667573,49658264,-158059506,505013014,-1619144976,5207596574,-16797286048,
%X A107087 54323516786,-176113786590,572236347200,-1863213388504,6078422747600,-19865661574555
%N A107087 G.f. A(x) satisfies: A(x)^2 = A(x^2) + 4*x.
%C A107087 Self-convolution of A107086. Self-convolution yields A107088.
%F A107087 G.f. A(x) satisfies 0=f(A(x), A(x^2), A(x^4)) where f(u, v, w)=u^4+4*w-2*u^2*v-3*v^2 . - Michael Somos May 15 2005
%e A107087 A(x)^2 = 1 + 4*x + 2*x^2 - x^4 + 2*x^6 - 5*x^8 + 12*x^10 - 30*x^12 +...
%e A107087 A(x^2) = 1 + 2*x^2 - x^4 + 2*x^6 - 5*x^8 + 12*x^10 - 30*x^12 +...
%o A107087 (PARI) {a(n)=local(A=1+x);for(i=1,n,A=(subst(A,x,x^2)+4*x+x*O(x^n))^(1/2)); polcoeff(A,n,x)}
%o A107087 (PARI) {a(n)=local(m,A); if(n<0,0, A=1+O(x); m=1; while(m<=n, A=sqrt(4*x+subst(A,x,x^2));m*=2); polcoeff(A,n))} /* Michael Somos May 15 2005 */
%Y A107087 Cf. A107086, A107088.
%Y A107087 Sequence in context: A129394 A049901 A117715 this_sequence A115141 A031148 A032238
%Y A107087 Adjacent sequences: A107084 A107085 A107086 this_sequence A107088 A107089 A107090
%K A107087 sign
%O A107087 0,2
%A A107087 Paul D. Hanna (pauldhanna(AT)juno.com), May 11 2005

%I A115141
%S A115141 1,2,1,2,5,14,42,132,429,1430,4862,16796,58786,208012,742900,2674440,
%T A115141 9694845,35357670,129644790,477638700,1767263190,6564120420,24466267020,
%U A115141 91482563640,343059613650,1289904147324,4861946401452,18367353072152
%V A115141 1,-2,-1,-2,-5,-14,-42,-132,-429,-1430,-4862,-16796,-58786,-208012,-742900,-2674440,
%W A115141 -9694845,-35357670,-129644790,-477638700,-1767263190,-6564120420,-24466267020,
%X A115141 -91482563640,-343059613650,-1289904147324,-4861946401452,-18367353072152
%N A115141 Convolution of A115140 with itself.
%F A115141 O.g.f.: 1/c(x)^2 = (1-x) - x*c(x) with the o.g.f. c(x):=(1-sqrt(1-4*x))/(2*x) of A000108 (Catalan numbers).
%F A115141 a(0)=1, a(1)=-2, a(n)=-C(n-1), n>=2, with C(n):=A000108(n) (Catalan). The start [1, -2] is row n=2 of signed A034807 (signed Lucas polynomials). See A115149 and A034807 for comments.
%Y A115141 Sequence in context: A049901 A117715 A107087 this_sequence A031148 A032238 A000619
%Y A115141 Adjacent sequences: A115138 A115139 A115140 this_sequence A115142 A115143 A115144
%K A115141 sign,easy
%O A115141 0,2
%A A115141 Wolfdieter Lang (wolfdieter.lang_AT_physik_DOT_uni-karlsruhe_DOT_de), Jan 13 2006

%I A031148
%S A031148 2,1,2,5,14,43,138,455,1540,5305,18546,65616,234546,845683,
%T A031148 3072350,11235393,41326470,152793376,567518950,2116666670,
%U A031148 7924062430,29765741831,112157686170,423809991041,1605622028100
%N A031148 Number of series-reduced planted trees with n leaves of 2 colors and no symmetries.
%H A031148 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/sequences/transforms.txt">Transforms</a>
%H A031148 <a href="http://www.research.att.com/~njas/sequences/Sindx_Ro.html#rooted">Index entries for sequences related to rooted trees</a>
%F A031148 Doubles (index 2+) under WEIGH transform.
%Y A031148 Essentially the same as A052301. Cf. A000669, A001678, A038075, A050381.
%Y A031148 Sequence in context: A117715 A107087 A115141 this_sequence A032238 A000619 A006602
%Y A031148 Adjacent sequences: A031145 A031146 A031147 this_sequence A031149 A031150 A031151
%K A031148 nonn,eigen
%O A031148 1,1
%A A031148 Christian G. Bower (bowerc(AT)usa.net)

%I A032238
%S A032238 2,1,2,5,14,43,142,475,1640,5774,20682,74911,274392,1014155,
%T A032238 3778158,14170988,53471322,202835554,773073462,2958951714,
%U A032238 11368816862,43832643969,169531531422,657591743743,2557480816940
%N A032238 Number of series-reduced dyslexic planted compound windmills with n leaves of 2 colors where any 2 submills extending from the same node are different.
%H A032238 <a href="http://www.research.att.com/~njas/sequences/Sindx_Mo.html#mobiles">Index entries for sequences related to mobiles</a>
%F A032238 Doubles (index 2+) under "DGK" (bracelet, element, unlabeled) transform.
%Y A032238 Sequence in context: A107087 A115141 A031148 this_sequence A000619 A006602 A049404
%Y A032238 Adjacent sequences: A032235 A032236 A032237 this_sequence A032239 A032240 A032241
%K A032238 nonn
%O A032238 1,1
%A A032238 Christian G. Bower (bowerc(AT)usa.net)

%I A000619 M0121 N0048
%S A000619 2,1,2,5,17,92,994,28262,2700791
%N A000619 NP-equivalence classes of threshold functions of exactly n variable.
%D A000619 S. Muroga, Threshold Logic and Its Applications. Wiley, NY, 1971, p. 38, Table 2.3.2. - Row 15.
%D A000619 S. Muroga, T. Tsuboi and C. R. Baugh, Enumeration of threshold functions of eight variables, IEEE Trans. Computers, 19 (1970), 818-825.
%Y A000619 Sequence in context: A115141 A031148 A032238 this_sequence A006602 A049404 A083773
%Y A000619 Adjacent sequences: A000616 A000617 A000618 this_sequence A000620 A000621 A000622
%K A000619 nonn
%O A000619 0,1
%A A000619 njas

%I A006602 M1532
%S A006602 2,1,2,5,20,180,16143
%N A006602 Number of hierarchical models with linear terms forced.
%C A006602 Also number of pure (= irreducible) group-testing histories of n items - A. Boneh, Mar 31, 2000
%C A006602 Also number of antichain covers of an unlabeled n-set, so a(n) equals first differences of A003182 - Vladeta Jovovic, Goran Kilibarda (vladeta(AT)Eunet.yu), Aug 18 2000
%C A006602 Also number of inequivalent (under permutation of variables) nondegenerate monotone Boolean functions of n variables. We say h and g (functions of n variables) are equivalent if there exists a permutation p of Sn such that hp=g. E.g. a(3)=5 because xyz, xy+xz+yz, x+yz+xyz, xy+xz+xyz, x+y+z+xy+xz+yz+xyz are 5 inequivalent nondegenerate monotone Boolean functions that generate (by permutation of variables) the other 4. For example, y+xz+xyz can be obtained from x+yz+xyz by permuting x with y. - Alan Veliz-Cuba (alanavc(AT)vt.edu), Jun 16 2006
%D A006602 Y. M. M. Bishop, S. E. Fienberg and P. W. Holland, Discrete Multivariate Analysis. MIT Press, 1975, p. 34.
%D A006602 V. Jovovic and G. Kilibarda, On the number of Boolean functions in the Post classes F^{mu}_8, Diskretnaya Matematika, 11 (1999), no. 4, 127-138 (translated in Discrete Mathematics and Applications, 9, (1999), no. 6)
%D A006602 V. Jovovic and G. Kilibarda, On enumeration of the class of all monotone Boolean functions, in preparation.
%D A006602 A. A. Mcintosh, personal communication.
%Y A006602 a(n)=A007411(n)+1. Cf. A006126 (labeled case).
%Y A006602 Sequence in context: A031148 A032238 A000619 this_sequence A049404 A083773 A096179
%Y A006602 Adjacent sequences: A006599 A006600 A006601 this_sequence A006603 A006604 A006605
%K A006602 nonn,nice,hard
%O A006602 0,1
%A A006602 C. L. Mallows
%E A006602 a(6) from A. Boneh, 32 Hantkeh St., Haifa 34608, Israel, Mar 31, 2000.
%E A006602 Entry revised by njas, Jul 23 2006

%I A049404
%S A049404 1,2,1,2,6,1,0,20,12,1,0,40,80,20,1,0,40,360,220,30,1,0,0,1120,1680,
%T A049404 490,42,1,0,0,2240,9520,5600,952,56,1,0,0,2240,40320,48720,15120,1680,
%U A049404 72,1,0,0,0,123200,332640,184800,35280,2760,90,1,0,0,0,246400,1786400
%N A049404 A triangle of numbers related to triangle A049324.
%C A049404 a(n,1)= A008279(2,n-1). a(n,m)=: S1(-2; n,m), a member of a sequence of lower triangular Jabotinsky matrices, including S1(1; n,m)= A008275 (signed Stirling first kind), S1(2; n,m)= A008297(n,m) (signed Lah numbers).
%C A049404 a(n,m) matrix is inverse to signed matrix ((-1)^(n-m))*A004747(n,m). The monic row polynomials E(n,x) := sum(a(n,m)*x^m,m=1..n), E(0,x) := 1 are exponential convolution polynomials (see A039692 for the definition and a Knuth reference).
%H A049404 W. Lang, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">On generalizations of Stirling number triangles</a>, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
%F A049404 a(n, m) = n!*A049324(n, m)/(m!*3^(n-m)); a(n, m) = (3*m-n+1)*a(n-1, m) + a(n-1, m-1), n >= m >= 1; a(n, m)=0, n<m; a(n, 0) := 0; a(1, 1)=1. E.g.f. for m-th column: ((x+x^2+(x^3)/3)^m)/m!.
%e A049404 {1}; {2,1}; {2,6,1}; {0,20,12,1}; ... E.g. row polynomial E(3,x)= 2*x+6*x^2+x^3.
%Y A049404 Row sums give A049425.
%Y A049404 Sequence in context: A032238 A000619 A006602 this_sequence A083773 A096179 A133643
%Y A049404 Adjacent sequences: A049401 A049402 A049403 this_sequence A049405 A049406 A049407
%K A049404 easy,nonn,tabl
%O A049404 1,2
%A A049404 Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de)

%I A083773
%S A083773 1,1,2,1,2,6,1,2,3,6,1,2,3,5,18,1,2,3,4,8,2275,1,2,3,4,5,10,222,1,2,3,4,
%T A083773 5,6,9,762986,1,2,3,4,5,6,7,9,2203418,1,2,3,4,5,6,7,8,15
%N A083773 The n-th row of the following triangle contains n distinct numbers such that the product of (n-1) of them + 1 is always a prime. The first (n-1) numbers are the smallest set whose product +1 is a prime, and the n-th term is chosen to satisfy the requirement. a(1) = 1 by convention. Sequence contains the triangle by rows.
%e A083773 1
%e A083773 1 2
%e A083773 1 2 6
%e A083773 1 2 3 6
%e A083773 1 2 3 5 18
%e A083773 ...
%o A083773 (PARI) fac = 1; for (n = 1, 40, fac *= n; for (i = 1, n, print1(i); print1(" ")); i = n + 1; while (!isprime(fac*i + 1), i++); print1(i); print1(" "); fails = 1; j = i; while (fails, j++; fails = !isprime(j*fac + 1); k = 1; while (!fails && k <= n, if (isprime(j*i*fac/k + 1), k++, fails = 1))); print(j));
%Y A083773 Cf. A083774, A083775.
%Y A083773 Sequence in context: A000619 A006602 A049404 this_sequence A096179 A133643 A008305
%Y A083773 Adjacent sequences: A083770 A083771 A083772 this_sequence A083774 A083775 A083776
%K A083773 more,nonn,tabl
%O A083773 1,3
%A A083773 Amarnath Murthy and Meenakshi Srikanth (amarnath_murthy(AT)yahoo.com), May 07 2003
%E A083773 More terms from David Wasserman (wasserma(AT)spawar.navy.mil), Nov 29 2004

%I A096179
%S A096179 1,1,2,1,2,6,1,2,4,12,1,2,4,12,60,1,2,4,6,12,60,1,2,4,6,12,60,420,1,2,4,
%T A096179 6,12,24,120,840,1,2,4,6,12,24,72,360,2520,1,2,4,6,12,24,60,120,360,
%U A096179 2520,1,2,4,6,12,24,60,120,360,2520,27720,1,2,4,6,12,12,24,60,120,360
%N A096179 Triangle read by rows: T(n,k) is the smallest positive integer having k of the first n positive integers as divisors.
%H A096179 Wikipedia, <a href="http://en.wikipedia.org/wiki/Table_of_divisors">Table of divisors</a>.
%e A096179 Triangle begins:
%e A096179 1
%e A096179 1 2
%e A096179 1 2 6
%e A096179 1 2 4 12
%e A096179 1 2 4 12 60
%e A096179 1 2 4 6 12 60
%Y A096179 Main diagonal is A003418. See also A094348, A096180.
%Y A096179 Sequence in context: A006602 A049404 A083773 this_sequence A133643 A008305 A133644
%Y A096179 Adjacent sequences: A096176 A096177 A096178 this_sequence A096180 A096181 A096182
%K A096179 nonn,tabl
%O A096179 1,3
%A A096179 Matthew Vandermast (ghodges14(AT)comcast.net), Jun 19, 2004

%I A133643
%S A133643 1,1,2,1,2,6,1,2,9,24,1,2,12,44,120,1,2,17,80,265,720,1,2,24,144,
%T A133643 578,1854,5040,1,2,33,248,1249,4738,14833,40320,1,2,42,440,2681,
%U A133643 12000,43386,133496,362880,1,2,60,764,5713,30240,126117,439792,1334961,3628800,1,2,83,1316,12105,75510,364503,1441788,4890740,14684570,39916800
%N A133643 Triangle read by rows: T(n,k) (n >= 1, 1 <= k <= n) = smallest permanent of any n X n (0,1) matrix with k 1's in each row and column.
%D A133643 G.-S. Cheon and I. M. Wanless, An update on Minc's survey on open problems involving permanents, Lin. Alg. Applic., 403 (2005), 314-342.
%D A133643 L. Hogben, ed., Handbook of Linear Algebra, pp. 31-6, 31-7.
%e A133643 Triangle begins:
%e A133643 1
%e A133643 1,2
%e A133643 1,2,6
%e A133643 1,2,9,24
%e A133643 1,2,12,44,120
%e A133643 1,2,17,80,265,720
%e A133643 1,2,24,144,578,1854,5040
%Y A133643 Sequence in context: A049404 A083773 A096179 this_sequence A008305 A133644 A098361
%Y A133643 Adjacent sequences: A133640 A133641 A133642 this_sequence A133644 A133645 A133646
%K A133643 nonn,tabl
%O A133643 1,3
%A A133643 njas, Dec 17 2007

%I A008305
%S A008305 1,1,2,1,2,6,1,2,9,24,1,2,13,44,120,1,2,20,80,265,720,1,2,31,144,579,
%T A008305 1854,5040,1,2,49,264,1265,4738,14833,40320,1,2,78,484,2783,12072,43387,
%U A008305 133496,362880,1,2,125,888,6208,30818,126565,439792,1334961,3628800,1,2
%N A008305 Triangle read by rows: a(n,k) = number of permutations of [ n ] allowing i->i+j (mod n),j=0..k-1.
%D A008305 N. S. Mendelsohn, Permutations with restricted displacement, Canad. Math. Bull., 4 (1961), 29-38.
%D A008305 H. Minc, Permanents, Encyc. Math. #6, 1978, p. 48
%F A008305 a(n, k)=per(sum(P^j, j=0..k-1)), where P is n by n, P[ i, i+1 (mod n) ]=1, 0's otherwise.
%e A008305 1; 1,2; 1,2,6; 1,2,9,24; etc. (so a(4,3)=9)
%Y A008305 A000142, A000166, A000179, A000183, A004307, A000211, A000496, A000803, A004306.
%Y A008305 Cf. A000804, A000805.
%Y A008305 Sequence in context: A083773 A096179 A133643 this_sequence A133644 A098361 A050977
%Y A008305 Adjacent sequences: A008302 A008303 A008304 this_sequence A008306 A008307 A008308
%K A008305 tabl,nonn
%O A008305 1,3
%A A008305 njas
%E A008305 Comments and more terms from Len Smiley (smiley(AT)math.uaa.alaska.edu)
%E A008305 More terms from Vladeta Jovovic (vladeta(AT)Eunet.yu), Oct 02 2003

%I A133644
%S A133644 1,1,2,1,2,6,1,4,9,24,1,4,13,44,120,1,8,36,82,265,720,1,8,54,148,580,
%T A133644 1854,5040,1,16,81,576,1313,4752,14833,40320,1,16,216,1056,2916,
%U A133644 12108,43424,133496,362880,1,32,324,1968,14400,32826,127044,440192,1334961,3628800,1,32,486,3608,31800,86400,373208,1448640,4893072,14684570,39916800
%N A133644 Triangle read by rows: T(n,k) (n >= 1, 1 <= k <= n) = largest permanent of any n X n (0,1) matrix with k 1's in each row and column.
%D A133644 G.-S. Cheon and I. M. Wanless, An update on Minc's survey on open problems involving permanents, Lin. Alg. Applic., 403 (2005), 314-342.
%D A133644 L. Hogben, ed., Handbook of Linear Algebra, pp. 31-6, 31-7.
%e A133644 Triangle begins:
%e A133644 1
%e A133644 1,2
%e A133644 1,2,6
%e A133644 1,4,9,24
%e A133644 1,4,13,44,120
%e A133644 1,8,36,82,265,720
%e A133644 1,8,54,148,580,1854,5040
%Y A133644 Sequence in context: A096179 A133643 A008305 this_sequence A098361 A050977 A053448
%Y A133644 Adjacent sequences: A133641 A133642 A133643 this_sequence A133645 A133646 A133647
%K A133644 nonn,tabl
%O A133644 1,3
%A A133644 njas, Dec 17 2007

%I A098361
%S A098361 1,1,1,2,1,2,6,2,2,6,24,6,4,6,24,120,24,12,12,24,120,720,120,48,36,48,
%T A098361 120,720,5040,720,240,144,144,240,720,5040,40320,5040,1440,720,576,720,
%U A098361 1440,5040,40320,362880,40320,10080,4320,2880,2880,4320,10080,40320
%N A098361 Multiplication table of the factorial numbers read by antidiagonals.
%C A098361 This sequence gives the variance of the 2-dimensional Polynomial Chaoses (See the Stochastic Finite Elements reference) - Stephen Crowley (crow(AT)crowlogic.net), Mar 28 2007
%D A098361 R. Ghanem and P. Spanos, Stochastic Finite Elements: A Spectral Approach (Revised Edition), 2003, Ch 2.4 Table 2-2.
%e A098361 1; 1,1; 2,1,2; 6,2,2,6; ...
%Y A098361 Cf. A003991, A098358, A098359, A098360.
%Y A098361 Sequence in context: A133643 A008305 A133644 this_sequence A050977 A053448 A060550
%Y A098361 Adjacent sequences: A098358 A098359 A098360 this_sequence A098362 A098363 A098364
%K A098361 nonn,tabl
%O A098361 0,4
%A A098361 Douglas Stones (dssto1(AT)student.monash.edu.au), Sep 04 2004

%I A050977
%S A050977 1,2,1,2,6,2,6,5,2,4,6,4,16,6,9,6,5,22,2,4,18,6,14,3,8,10,16,6,36,9,4,
%T A050977 20,6,42,5,22,46,4,42,16,4,52,18,6,18,14,29,30,3,6,16,10,22,16,22,5,6,
%U A050977 72,36,9,30,4,39,54,20,82,6,42,14,10,44,12,22,6,46,8,96,42,30,25,16
%N A050977 Haupt-exponents of 5 modulo integers relatively prime to 5.
%H A050977 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Haupt-Exponent.html">Link to a section of The World of Mathematics.</a>
%H A050977 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/MultiplicativeOrder.html">Multiplicative Order</a>
%Y A050977 Cf. A002326, A002329.
%Y A050977 Sequence in context: A008305 A133644 A098361 this_sequence A053448 A060550 A099206
%Y A050977 Adjacent sequences: A050974 A050975 A050976 this_sequence A050978 A050979 A050980
%K A050977 nonn
%O A050977 1,2
%A A050977 Eric Weisstein (eric(AT)weisstein.com)

%I A053448
%S A053448 1,1,2,1,2,6,2,6,5,2,4,6,4,16,6,9,6,5,22,2,4,18,6,14,3,8,10,16,6,36,9,
%T A053448 4,20,6,42,5,22,46,4,42,16,4,52,18,6,18,14,29,30,3,6,16,10,22,16,22,5,
%U A053448 6,72,36,9,30,4,39,54,20,82,6,42,14,10,44,12,22,6,46,8,96,42,30,25,16
%N A053448 Multiplicative order of 5 mod n, where GCD(n, 5) = 1.
%Y A053448 Sequence in context: A133644 A098361 A050977 this_sequence A060550 A099206 A121341
%Y A053448 Adjacent sequences: A053445 A053446 A053447 this_sequence A053449 A053450 A053451
%K A053448 nonn
%O A053448 1,3
%A A053448 David W. Wilson (davidwwilson(AT)comcast.net)

%I A060550
%S A060550 0,0,0,1,0,1,2,1,2,6,2,6,12,6,12,28,12,28,56,28,56,120,56,120,240,120,
%T A060550 240,496,240,496,992,496,992,2016,992,2016,4032,2016,4032,8128,4032,
%U A060550 8128,16256,8128,16256,32640,16256,32640,65280,32640
%N A060550 a(n) is the number of distinct patterns (modulo geometric D_3-operations) with no other than strict 120 degree rotational symmetry which can be formed by an equilateral triangular arrangement of closely packed black and white cells satisfying the local matching rule of Pascal's triangle modulo 2, where n is the number of cells in each edge of the arrangement.
%C A060550 The matching rule is such that any elementary top-down triangle of three neighboring cells in the arrangement contains either one or three white cells.
%D A060550 A. Barb\'{e}, Symmetric patterns in the cellular automaton that generates Pascal's triangle modulo 2, Discr.Appl.Math. 105(2000),1-38.
%H A060550 <a href="http://www.research.att.com/~njas/sequences/Sindx_Ce.html#cell">Index entries for sequences related to cellular automata</a>
%F A060550 a(n)= 2^[floor(n/3)+(n mod 3)mod 2-1]-2^{floor[(n+3)/6]+d(n)-1}, with d(n)=1 if n mod 6=1 else d(n)=0
%Y A060550 a(n) = [A060547(n)-A060548(n)]/2, a(n) = 2^[A008611(n-1)-1]+2^[A008615(n+1)-1], for n >= 1.
%Y A060550 Sequence in context: A098361 A050977 A053448 this_sequence A099206 A121341 A126093
%Y A060550 Adjacent sequences: A060547 A060548 A060549 this_sequence A060551 A060552 A060553
%K A060550 easy,nonn
%O A060550 1,7
%A A060550 Andr\'{e} Barb\'{e} (Andre.Barbe(AT)esat.kuleuven.ac.be), Apr 03 2001

%I A099206
%S A099206 0,1,1,1,1,1,1,1,2,1,2,6,2,6,15,6,15,38,15,38,97,38,97,247,97,247,629,
%T A099206 247,629,1602,629,1602,4080,1602,4080,10391,4080,10391,26464,10391,
%U A099206 26464,67399,26464,67399,171653,67399,171653,437169,171653,437169
%N A099206 Absolute value of a vector matrix Markov sequence with same polynomial as Kenyon's tile: x^3-2*x-x-1==0.
%H A099206 Stewart R. Hinsley, <a href="http://www.meden.demon.co.uk/Fractals/cubic8.html">A Tile Associated with the 8th Unit Cubic Pisot Number</a>
%H A099206 Richard Kenyon, <a href="http://arXiv.org/abs/math.MG/9505210">The Construction of Self-Similar Tilings</a>
%H A099206 Richard Kenyon, <a href="http://www.math.ubc.ca/~kenyon/papers/">Papers</a>
%F A099206 M = {{0, 1, 0}, {0, 0, 1}, {-1, 1, -2}}; v[0]={0, 1, 1}; a(n) = Abs[vector components of M^n*v[0]].
%t A099206 M = {{0, 1, 0}, {0, 0, 1}, {-1, 1, -2}}; v[0] = {0, 1, 1}; v[1] = {1, 1, -1}; v[n_] := v[n] = M.v[n - 1]; a = Flatten[Table[v[n], {n, 0, 17}]]; Abs[a]
%Y A099206 Sequence in context: A050977 A053448 A060550 this_sequence A121341 A126093 A065279
%Y A099206 Adjacent sequences: A099203 A099204 A099205 this_sequence A099207 A099208 A099209
%K A099206 nonn
%O A099206 1,9
%A A099206 Roger Bagula (rlbagulatftn(AT)yahoo.com), Mar 19 2005

%I A121341
%S A121341 0,1,1,2,1,2,6,3,1,1,2,3,6,7,2,4,16,2,18,2,6,3,22,4,2,7,3,8,28,2,15,5,2,
%T A121341 17,7,3,3,19,6,3,5,7,21,4,2,23,46,5,42,2,16,8,13,4,3,9,18,29,58,3,60,16,
%U A121341 6,6,7,3,33,18,22,7,35,4,8,4,3,20,6,7,13,4,9,6,41,8,17,22,28,5,44,2,6
%N A121341 Number of decimal places before 1/n either recurs or terminates.
%C A121341 In this sequence, the repeating decimals (e.g. 1/7) are treated differently from nonrepeating decimals (e.g. 1/5). If they are treated the same, then a(2)=2, a(4)=3, a(5)=2, a(8)=4, a(10)=2, ... and we obtain A054710. The two sequence differ only for n=2^j 5^k.
%H A121341 T. D. Noe, <a href="http://www.research.att.com/~njas/sequences/b121341.txt">Table of n, a(n) for n = 1..1000</a>
%H A121341 <a href="http://www.research.att.com/~njas/sequences/Sindx_1.html#1overn">Index entries for sequences related to decimal expansion of 1/n</a>
%e A121341 1/592=0.0016891891891... starts with 4 decimals (0016, zeros counted) and has period 3 (decimals 891) to yield a(592)=4+3=7.
%t A121341 EndingZeros[rep_List] := Module[{cnt=0, i=Length[rep]}, While[rep[[i]]==0, i--; cnt++ ]; cnt]; a[n_Integer] := Module[{lst,e,rep,len,initDigits}, {lst,e}=RealDigits[1/n]; If[VectorQ[lst], len=Length[lst]-e, rep=lst[[ -1]]; initDigits=Length[lst]-1; len=initDigits+Length[rep]-e-EndingZeros[rep]]; len]; Table[a[n],{n,100}]
%Y A121341 A007732 is the length of the periods and serves as a lower bound. Cf. A061075.
%Y A121341 Sequence in context: A053448 A060550 A099206 this_sequence A126093 A065279 A069627
%Y A121341 Adjacent sequences: A121338 A121339 A121340 this_sequence A121342 A121343 A121344
%K A121341 base,easy,nice,nonn
%O A121341 1,4
%A A121341 Anthony C Robin (anthony_robin(AT)hotmail.com), Aug 29 2006
%E A121341 More terms from T. D. Noe (noe(AT)sspectra.com), Aug 30 2006. Additional comments from R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Aug 30 2006

%I A126093
%S A126093 1,0,1,1,2,1,2,6,4,1,6,18,15,6,1,18,57,54,28,8,1,57,186,193,118,45,10,1,
%T A126093 186,622,690,474,218,66,12,1,622,2120,2476,1856,976,362,91,14,1,2120,
%U A126093 7338,8928,7164,4170,1791,558,120,16,1
%N A126093 Inverse binomial matrix applied to A110877.
%C A126093 Triangle T(n,k), 0<=k<=n, read by rows defined by : T(0,0)=1, T(n,k)=0 if k<0 or if k>n, T(n,0)=T(n-1,1), T(n,k)=T(n-1,k-1)+2*T(n-1,k)+T(n-1,k+1) for k>=1.
%C A126093 Diagonal sums are A065601 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Mar 05 2007
%C A126093 This triangle belongs to the family of triangles defined by: T(0,0)=1, T(n,k)=0 if k<0 or if k>n, T(n,0)=x*T(n-1,0)+T(n-1,1), T(n,k)=T(n-1,k-1)+y*T(n-1,k)+T(n-1,k+1) for k>=1 . Other triangles arise by choosing different values for (x,y): (0,0) -> A053121; (0,1) -> A089942; (0,2) -> A126093; (0,3) -> A126970; (1,0)-> A061554; (1,1) -> A064189; (1,2) -> A039599; (1,3) -> A110877; ((1,4) -> A124576; (2,0) -> A126075; (2,1) -> A038622; (2,2) -> A039598; (2,3) -> A124733; (2,4) -> A124575; (3,0) -> A126953; (3,1) -> A126954; (3,2) -> A111418; (3,3) -> A091965; (3,4) -> A124574; (4,3) -> A126791; (4,4) -> A052179; (4,5) -> A126331; (5,5) -> A125906 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Sep 25 2007
%F A126093 Sum_{k, 0<=k<=n}T(m,k)*T(n,k)=T(m+n,0)=A000957(m+n+1).
%F A126093 Sum[k, 0<=k<=n-1}A126093(n,k)=A026641(n), for n>=1 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Mar 05 2007
%F A126093 Sum_{k, 0<=k<=n}T(n,k)*(3k+1)=4^n . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Mar 22 2007
%e A126093 Triangle begins:
%e A126093 1;
%e A126093 0, 1;
%e A126093 1, 2, 1;
%e A126093 2, 6, 4, 1;
%e A126093 6, 18, 15, 6, 1;
%e A126093 18, 57, 54, 28, 8, 1;
%e A126093 57, 186, 193, 118, 45, 10, 1;
%e A126093 186, 622, 690, 474, 218, 66, 12, 1;
%e A126093 622, 2120, 2476, 1856, 976, 362, 91, 14, 1;
%e A126093 2120, 7338, 8928, 7164, 4170, 1791, 558, 120, 16, 1;
%Y A126093 Sequence in context: A060550 A099206 A121341 this_sequence A065279 A069627 A072137
%Y A126093 Adjacent sequences: A126090 A126091 A126092 this_sequence A126094 A126095 A126096
%K A126093 nonn,tabl
%O A126093 0,5
%A A126093 Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Mar 03 2007

%I A065279
%S A065279 1,1,1,2,1,2,6,4,2,4,2,4,12,4,14,8,4,8,4,8,4,8,4,8,24,8,26,8,28,8,30,
%T A065279 16,8,16,8,16,8,16,8,16,8,16,8,16,8,16,8,16,48,16,50,16,52,16,54,16,56,
%U A065279 16,58,16,60,16,62,32,16,32,16,32,16,32,16,32,16,32,16,32,16,32,16,32
%N A065279 The siteswap sequence (the deltas p[i]-i, i in ]-inf,+inf[, folded from Z to N, mapping 0->1, 1->2, -1->3, 2->4, -2->5, etc.) for A065278.
%Y A065279 The bisection of even terms (the positive half of Z) is given by A053644, and the bisection of odd terms (the nonpositive half of Z) is given by A065280.
%Y A065279 Sequence in context: A099206 A121341 A126093 this_sequence A069627 A072137 A061569
%Y A065279 Adjacent sequences: A065276 A065277 A065278 this_sequence A065280 A065281 A065282
%K A065279 nonn
%O A065279 1,4
%A A065279 Antti Karttunen Oct 28 2001

%I A069627
%S A069627 1,1,1,1,1,1,1,1,1,2,1,2,6,4,5,3,3,5,4,6,2,1,2,6,4,5,3,3,5,4,6,2,1,2,6
%N A069627 Duplicate of A072137.
%Y A069627 Sequence in context: A121341 A126093 A065279 this_sequence A072137 A061569 A094965
%Y A069627 Adjacent sequences: A069624 A069625 A069626 this_sequence A069628 A069629 A069630
%K A069627 dead
%O A069627 1,10

%I A072137
%S A072137 0,1,1,1,1,1,1,1,1,1,2,1,2,6,4,5,3,3,5,4,6,2,1,2,6,4,5,3,3,5,4,6,2,1,2,
%T A072137 6,4,5,3,3,5,4,6,2,1,2,6,4,5,3,3,5,4,6,2,1,2,6,4,5,3,3,5,4,6,2,1,2,6,4,
%U A072137 5,3,3,5,4,6,2,1,2,6,4,5,3,3,5,4,6,2,1,2,6,4,5,3,3,5,4,6,2,1,2,1,2,6,4
%N A072137 Length of the preperiodic part of the 'Reverse and Subtract' trajectory of n.
%C A072137 'Reverse and Subtract' (cf. A070837, A070838) is defined by x -> |x - reverse(x)|, where reverse(x) is the digit reversal of x.
%C A072137 For every n the trajectory eventually becomes periodic, since 'Reverse and Subtract' does not increase the number of digits and so the set of available terms is finite. For small n the period length is 1, the periodic part consists of 0's, the last term of the preperiodic part is a palindrome.
%C A072137 The first n with period length 2 and a nontrivial periodic part is 1012 (cf. A072140).
%C A072137 This sequence is a weak analogue of A033665, which uses 'Reverse and Add'.
%e A072137 a(15) = 4 since 15 -> |15- 51| = 36 -> |36 - 63| = 27 -> |27 - 72| = 45 -> |45 - 54| = 9.
%Y A072137 Cf. A033665, A070837, A070838, A072138, A072139, A072140, A072141, A072146, A072147.
%Y A072137 Sequence in context: A126093 A065279 A069627 this_sequence A061569 A094965 A025277
%Y A072137 Adjacent sequences: A072134 A072135 A072136 this_sequence A072138 A072139 A072140
%K A072137 base,easy,nonn,nice
%O A072137 0,11
%A A072137 Klaus Brockhaus (klaus-brockhaus(AT)t-online.de), Jun 24 2002

%I A061569
%S A061569 0,0,0,2,1,2,6,4,21,33,38,50,74,81
%N A061569 Number of irreducible representations of the symmetric group S_n such that their degree is divisible by 3.
%C A061569 The total number of irreducible representations of S_n is the partition function partition(n) (sequence A000041) and the number of irreducible representations of the symmetric group S_n with their degree not divisible by 3 is given in A060840 so a(n) = A000041(n) - A060840(n).
%e A061569 a(3) = 0 because the degrees of the irreducible representations of S_3 are 1,1,2.
%Y A061569 A000041, A060840.
%Y A061569 Sequence in context: A065279 A069627 A072137 this_sequence A094965 A025277 A074727
%Y A061569 Adjacent sequences: A061566 A061567 A061568 this_sequence A061570 A061571 A061572
%K A061569 nonn
%O A061569 1,4
%A A061569 Ola Veshta (olaveshta(AT)my-deja.com), May 18 2001

%I A094965
%S A094965 2,1,2,6,6,7,1,4,1,1,0,7,5,2,9,7,4,2,8,2,4,4,4,3,0,8,0,6,3,7,2,1,0,0,0,
%T A094965 8,4,1,8,7,4,2,8,9,0,6,8,3,9,7,8,2,5,2,8,4,6,2,5,2,2,4,5,6,4,3,6,3,9,5,
%U A094965 2,8,2,3,9,3,0,3,6,9,0,3,6,8,5,4,9,8,8,0,3,4,3,9,5,3,3,0,1,1,6,9,8,8,1
%N A094965 A continued fraction transformation of E.
%C A094965 The number, C, has the continued fraction which is the decimal expansion is that of E.
%e A094965 C = 2.126671411...
%p A094965 RealDigits[ FromContinuedFraction[ RealDigits[E, 10, 125][[1]]], 10, 111][[1]]
%Y A094965 Cf. A001113.
%Y A094965 Sequence in context: A069627 A072137 A061569 this_sequence A025277 A074727 A059587
%Y A094965 Adjacent sequences: A094962 A094963 A094964 this_sequence A094966 A094967 A094968
%K A094965 cons,easy,nonn
%O A094965 1,1
%A A094965 Robert G. Wilson v (rgwv(AT)rgwv.com), May 26 2004

%I A025277
%S A025277 0,0,1,1,0,1,2,1,2,6,6,7,20,30,34,75,140,182,322,644,972,1554,3024,5091,
%T A025277 8052,14784,26378,43032,75504,136994,232232,399399,720356,1257256,2161874,
%U A025277 3852576,6831552,11858418,20949304,37350768
%N A025277 a(n) = a(1)*a(n-1) + a(2)*a(n-2) + ...+ a(n-1)*a(1) for n >= 5.
%Y A025277 Sequence in context: A072137 A061569 A094965 this_sequence A074727 A059587 A070236
%Y A025277 Adjacent sequences: A025274 A025275 A025276 this_sequence A025278 A025279 A025280
%K A025277 nonn
%O A025277 1,7
%A A025277 Clark Kimberling (ck6(AT)evansville.edu)

%I A074727
%S A074727 1,1,1,2,1,2,6,7,4,1,2,4,16,1,6,6,2,3,1,3,3,6,3,5,1,2,1,2,2,2,15,1,15,
%T A074727 1,7,3,2,21,5,15,4,16,1,8,1,7,1,2,7,7,2,1,20,2,15,1,6,1,1,8,22,2,1,20,
%U A074727 64,3,1,31,14,22,19,66,7,1,14,1,15,10,7,2,6,19,1,4,8,2,1,7,18,3,2,1,2
%N A074727 Number of steps needed to reach a prime when the following map is repeatedly applied to n: if n is even then 2n + SOD(n) + 1, otherwise 2n - SOD(n) - 1, where SOD(n) is the sum of the digits of n; or -1 if no prime is ever reached.
%e A074727 a(10) = 4 because 10 -> 22 -> 49 -> 84 -> 181.
%Y A074727 Sequence in context: A061569 A094965 A025277 this_sequence A059587 A070236 A020825
%Y A074727 Adjacent sequences: A074724 A074725 A074726 this_sequence A074728 A074729 A074730
%K A074727 nonn
%O A074727 2,4
%A A074727 Jason Earls (zevi_35711(AT)yahoo.com), Sep 04 2002

%I A059587
%S A059587 1,1,1,2,1,2,6,7,4,1,6,24,48,68,73,56,28,8,1,24,120,360,940,2251,4704,
%T A059587 8176,11488,12876,11440,8008,4368,1820,560,120,16,1,120,720,3000,12720,
%U A059587 56660,247016,987252,3480536,10647035,28163200,64592320,129068160
%N A059587 T(n,m)=(1/m!)*Sum_{i=0..m} stirling1(m,i)*(2^i)*(2^i+1)*...*(2^i+n-1).
%F A059587 T(n, m)=Sum_{i=0..n} |stirling1(n, i)|*binomial(2^i, m).
%e A059587 [1, 1], [1, 2, 1], [2, 6, 7, 4, 1], [6, 24, 48, 68, 73, 56, 28, 8, 1], ...
%p A059587 with(combinat): for n from 0 to 10 do for m from 0 to 2^n do printf(`%d,`,sum(abs(stirling1(n,i))*binomial(2^i, m), i=0..n)) od: od:
%Y A059587 Cf. A059084, (row sums) A059588.
%Y A059587 Sequence in context: A094965 A025277 A074727 this_sequence A070236 A020825 A110422
%Y A059587 Adjacent sequences: A059584 A059585 A059586 this_sequence A059588 A059589 A059590
%K A059587 easy,nonn
%O A059587 0,4
%A A059587 Vladeta Jovovic (vladeta(AT)Eunet.yu), Jan 23 2001
%E A059587 More terms from James A. Sellers (sellersj(AT)math.psu.edu), Jan 24 2001

%I A070236
%S A070236 0,1,2,1,2,6,7,5,0,6,7,6,7,15,22,15,16,12,13,10,19,31,32,30,11,25,10,5,
%T A070236 6,28,29,15,28,46,57,46,47,67,82,76,77,107,108,99,80,104,105,92,51,33,
%U A070236 52,41,42,30,45,35,56,86,87,86,87,119,90,59,76,122,123,108,133,179,180
%N A070236 Sum(k=1,n,core(k)-phi(k)) where core(k) is the square-free part of k.
%C A070236 Always =>0.
%F A070236 a(n) = A069891(n)- A002088(n) asymptotically : a(n)=(Pi^2/30-3/Pi^2)*n^2+0(nln(n))
%o A070236 (PARI) for(n=1,100,print1(sum(i=1,n,core(i)-eulerphi(i)),","))
%Y A070236 Sequence in context: A025277 A074727 A059587 this_sequence A020825 A110422 A131804
%Y A070236 Adjacent sequences: A070233 A070234 A070235 this_sequence A070237 A070238 A070239
%K A070236 easy,nonn
%O A070236 1,3
%A A070236 Benoit Cloitre (benoit7848c(AT)orange.fr), May 08 2002

%I A020825
%S A020825 1,2,1,2,6,7,8,1,2,5,1,8,1,6,6,4,8,6,7,5,9,4,5,3,2,3,1,0,5,8,0,6,1,
%T A020825 0,8,8,9,7,4,9,1,7,6,2,4,2,7,5,6,9,4,7,1,8,6,5,8,7,8,3,3,4,0,3,8,5,
%U A020825 1,4,5,7,1,6,0,6,8,7,5,3,5,7,6,2,8,1,9,9,5,9,8,9,7,2,9,7,2,6,0,0,8
%N A020825 Decimal expansion of 1/sqrt(68).
%Y A020825 Sequence in context: A074727 A059587 A070236 this_sequence A110422 A131804 A032085
%Y A020825 Adjacent sequences: A020822 A020823 A020824 this_sequence A020826 A020827 A020828
%K A020825 nonn,cons
%O A020825 0,2
%A A020825 njas

%I A110422
%S A110422 1,2,1,2,6,8,6,8,15,18,15,18,28,32,28,32,45,50,45,50,66,72,66,72,91,98,91,
%T A110422 98,120,128,120,128,153,162,153,162,190,200,190,200,231,242,231,242,276,288,
%U A110422 276,288,325,338,325,338,378,392,378,392,435,450,435,450,496,512,496,512,561
%V A110422 1,2,-1,-2,6,8,-6,-8,15,18,-15,-18,28,32,-28,-32,45,50,-45,-50,66,72,-66,-72,91,98,-91,
%W A110422 -98,120,128,-120,-128,153,162,-153,-162,190,200,-190,-200,231,242,-231,-242,276,288,
%X A110422 -276,-288,325,338,-325,-338,378,392,-378,-392,435,450,-435,-450,496,512,-496,-512,561
%N A110422 a(n)=sum((-1)^(r+1)*(n-r)*r,r=1..floor(n/2)).
%C A110422 a(4n)=-a(4n-2); a(4n+1)=-a(4n-1). If sum in definition is not alternating one obtains A023855. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Aug 08 2005
%F A110422 a(2n)=(1/2)n-(-1)^n*(1/2)n^2; a(2n-1)=(1/2)n-(1/4)+(-1)^n*(1/4)(2n^2-2n+1). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Aug 08 2005
%e A110422 a(8)=-6 because 7*1-6*2+5*3-4*4=-6.
%p A110422 a:=n->sum((-1)^(r+1)*(n-r)*r,r=1..floor(n/2)): seq(a(n),n=2..70); (Deutsch)
%Y A110422 Cf. A023855.
%Y A110422 Sequence in context: A059587 A070236 A020825 this_sequence A131804 A032085 A032163
%Y A110422 Adjacent sequences: A110419 A110420 A110421 this_sequence A110423 A110424 A110425
%K A110422 easy,sign
%O A110422 2,2
%A A110422 Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Aug 01 2005
%E A110422 Corrected and extended by Emeric Deutsch (deutsch(AT)duke.poly.edu), Aug 08 2005

%I A131804
%S A131804 0,0,1,1,1,2,1,2,6,8,7,9,15,18,17,20,28,32,31,35,45,50,49,54,66,72,71,
%T A131804 77,91,98,97,104,120,128,127,135,153,162,161,170,190,200,199,209,231,
%U A131804 242,241,252,276,288,287,299,325,338,337,350,378,392,391,405,435,450
%V A131804 0,0,-1,-1,1,2,1,2,6,8,7,9,15,18,17,20,28,32,31,35,45,50,49,54,66,72,71,77,91,98,97,
%W A131804 104,120,128,127,135,153,162,161,170,190,200,199,209,231,242,241,252,276,288,287,299,
%X A131804 325,338,337,350,378,392,391,405,435,450
%N A131804 Antidiagonal sums of triangular array T: T(j,k) = -(k+1)/2 for odd k, T(j,k) = 0 for k = 0, T(j,k) = j+1-k/2 for even k > 0; 0 <= k <= j.
%C A131804 T is obtained by replacing the values of the second, fourth, sixth, ... column of the triangular array defined in A129819 by the corresponding negative values.
%C A131804 Interleaving of A000384, A001105, A056220 and A014107 (starting at the second term).
%C A131804 Main diagonal of T is in A001057, row sums are in A131805.
%F A131804 a(0) = 0, a(1) = 0, a(2) = -1, a(3) = -1, a(4) = 1, a(5) = 2, a(6) = 1; for n > 6, a(n) = 3*a(n-1) - 5*a(n-2) + 7*a(n-3) - 7*a(n-4) + 5*a(n-5) - 3*a(n-6) + a(n-7);
%F A131804 G.f.: x^2*(-1+2*x-x^2+x^3)/((1-x)^3*(1+x^2)^2).
%e A131804 First seven rows of T are
%e A131804 [ 0 ],
%e A131804 [ 0, -1 ],
%e A131804 [ 0, -1, 2 ],
%e A131804 [ 0, -1, 3, -2 ],
%e A131804 [ 0, -1, 4, -2, 3 ],
%e A131804 [ 0, -1, 5, -2, 4, -3 ],
%e A131804 [ 0, -1, 6, -2, 5, -3, 4 ]
%o A131804 (MAGMA) m:=62; M:=ZeroMatrix(IntegerRing(), m, m); for j:=1 to m do for k:=2 to j do if k mod 2 eq 0 then M[j, k]:=-k div 2; else M[j, k]:=j-(k div 2); end if; end for; end for; [ &+[ M[j-k+1, k]: k in [1..(j+1) div 2] ]: j in [1..m] ];
%o A131804 (PARI) {for(n=0, 61, r=n%4; k=(n-r)/4; a=if(r==0, k*(2*k-1), if(r==1, 2*k^2, if(r==2, 2*k^2-1, k*(2*k+1)-1))); print1(a, ","))}
%Y A131804 Cf. A129819, A000384 (n*(2*n-1)), A001105 (2*n^2), A056220 (2*n^2-1), A014107 (n*(2*n-3)), A001057, A131805.
%Y A131804 Sequence in context: A070236 A020825 A110422 this_sequence A032085 A032163 A038078
%Y A131804 Adjacent sequences: A131801 A131802 A131803 this_sequence A131805 A131806 A131807
%K A131804 sign
%O A131804 0,6
%A A131804 Klaus Brockhaus (klaus-brockhaus(AT)t-online.de), Jul 18 2007

%I A032085
%S A032085 2,1,2,6,12,28,56,120,240,496,992,2016,4032,8128,16256,32640,
%T A032085 65280,130816,261632,523776,1047552,2096128,4192256,8386560,
%U A032085 16773120,33550336,67100672,134209536,268419072,536854528
%N A032085 Number of reversible strings with n beads of 2 colors. If more than 1 bead, not palindromic.
%D A032085 S. J. Cyvin et al., Theory of polypentagons, J. Chem. Inf. Comput. Sci., 33 (1993), 466-474.
%H A032085 C. G. Bower, <a href="http://www.research.att.com/~njas/sequences/transforms2.html">Transforms (2)</a>
%H A032085 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=1022">Encyclopedia of Combinatorial Structures 1022</a>
%F A032085 "BHK" (reversible, identity, unlabeled) transform of 2, 0, 0, 0...
%F A032085 a(n) = 2^(n-1)-2^floor((n-1)/2), n > 1. - Vladeta Jovovic (vladeta(AT)Eunet.yu), Nov 11 2001
%F A032085 For n>1 g.f. = 1/((1-2*x)*(1-2*x^2)) - Mohammed Bouayoun (bouyao(AT)wanadoo.fr), Mar 25 2004
%Y A032085 Cf. A005418. Essentially the same as A122746.
%Y A032085 Row sums of triangle A034877.
%Y A032085 Sequence in context: A020825 A110422 A131804 this_sequence A032163 A038078 A000139
%Y A032085 Adjacent sequences: A032082 A032083 A032084 this_sequence A032086 A032087 A032088
%K A032085 nonn
%O A032085 1,1
%A A032085 Christian G. Bower (bowerc(AT)usa.net)

%I A032163
%S A032163 2,1,2,6,18,63,232,868,3338,13105,52270,210822,859528,3535424,
%T A032163 14654628,61150659,256674992,1083008336,4590945532,19542842196,
%U A032163 83504907276,358032884540,1539885406796,6641914100221
%N A032163 Number of series-reduced planted compound windmills with n leaves of 2 colors where any 2 submills extending from the same node are different.
%H A032163 <a href="http://www.research.att.com/~njas/sequences/Sindx_Mo.html#mobiles">Index entries for sequences related to mobiles</a>
%F A032163 Doubles (index 2+) under "CGK" (necklace, element, unlabeled) transform.
%Y A032163 Sequence in context: A110422 A131804 A032085 this_sequence A038078 A000139 A052621
%Y A032163 Adjacent sequences: A032160 A032161 A032162 this_sequence A032164 A032165 A032166
%K A032163 nonn
%O A032163 1,1
%A A032163 Christian G. Bower (bowerc(AT)usa.net)

%I A038078
%S A038078 1,2,1,2,6,20,69,270,1026,4120,16794,70230,298306,1288912,5642559,
%T A038078 25007756,111998920,506348902,2308338456,10602357346,49026021552,
%U A038078 228085486580,1067020210339,5016982766202,23698640081356
%N A038078 Number of identity trees with 2-colored nodes.
%H A038078 <a href="http://www.research.att.com/~njas/sequences/Sindx_Tra.html#trees">Index entries for sequences related to trees</a>
%F A038078 G.f.: B(x)-B^2(x)/2-B(x^2)/2, where B(x) is g.f. for A038077.
%Y A038078 Cf. A000220. A038077-A038080.
%Y A038078 Sequence in context: A131804 A032085 A032163 this_sequence A000139 A052621 A131057
%Y A038078 Adjacent sequences: A038075 A038076 A038077 this_sequence A038079 A038080 A038081
%K A038078 nonn
%O A038078 0,2
%A A038078 Christian G. Bower (bowerc(AT)usa.net), Jan 04 1999.

%I A000139 M1660 N0651
%S A000139 2,1,2,6,22,91,408,1938,9614,49335,260130,1402440,7702632,42975796,
%T A000139 243035536,1390594458,8038677054,46892282815,275750636070,1633292229030,
%U A000139 9737153323590,58392041019795,352044769046880,2132866978427640
%N A000139 Number of 2-stack sortable permutations on n letters.
%C A000139 The number of rooted non-separable planar maps with n edges. - Valery A. Liskovets (liskov(AT)im.bas-net.by), Mar 17 2005
%C A000139 The shifted sequence starting with a(1): Number of quadrangular dissections of a square, counted by the number of vertices. Rooted, non-separable planar maps with no multiple edges, in which each non-root face has degree 4.
%C A000139 Number of left ternary trees having n nodes (n>=1). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Jul 23 2006
%D A000139 W. G. Brown, Enumeration of non-separable planar maps, Canad. J. Math., 15:3 (1963), 526-545.
%D A000139 A. Del Lungo, F. Del Ristoro, and J.-G. Penaud, Left ternary trees and non-separable rooted planar maps, Theor. Comp. Sci., 233, 2000, 201-215.
%D A000139 J. L. Gross and J. Yellen, eds., Handbook of Graph Theory, CRC Press, 2004; p. 714.
%D A000139 O. Guibert, Stack words, ..., Discr. Math., 210 (2000), 71-85.
%D A000139 W. F. Lunnon, Counting polyominoes, pp. 347-372 of A. O. L. Atkin and B. J. Birch, editors, Computers in Number Theory. Academic Press, NY, 1971.
%D A000139 R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 6.41.
%D A000139 W. T. Tutte, A census of planar maps, Canad. J. Math., 15 (1963), 249-271.
%D A000139 J. West, Sorting twice through a stack. Conference on Formal Power Series and Algebraic Combinatorics (Bordeaux, 1991). Theoret. Comput. Sci. 117 (1993), no. 1-2, 303-313.
%D A000139 D. Zeilberger, A proof of Julian West's conjecture ..., Discrete Math., 102 (1992), 85-93.
%H A000139 T. D. Noe, <a href="http://www.research.att.com/~njas/sequences/b000139.txt">Table of n, a(n) for n=0..200</a>
%H A000139 M. Bousqet-Melou and A. Jehanne, <a href="http://arXiv.org/abs/math.CO/0504018">Polynomial equations with one catalytic variable, algebraic series, and map enumeration</a>
%H A000139 I. Gessel and G. Xin, <a href="http://arXiv.org/abs/math.CO/0505217">The generating function of ternary trees and continued fractions</a>
%H A000139 P. Zinn-Justin and J.-B. Zuber, <a href="http://arXiv.org/abs/math-ph/0303049">Matrix integrals and the generation and counting of virtual tangles and links</a>, p. 11.
%F A000139 2*C(3n, 2n+1)/(n(n+1)), or 2*(3*n)!/((2*n+1)!*((n+1)!)).
%F A000139 Using Stirling's formula in A000142 it easy to get the asymptotic expression a(n) ~ (27/4)^n / (sqrt(Pi*n / 3) * (2n + 1) * (n + 1)). - Dan Fux (dan.fux(AT)OpenGaia.com or danfux(AT)OpenGaia.com), Apr 13 2001
%F A000139 G.f. A(z) = 2 + zB(z), where B(z) = 1 - 8z + 2z(5-6z)B - 2z^2(1+3z)B^2 - z^4B^3.
%p A000139 A000139 := n->2*(3*n)!/((2*n+1)!*((n+1)!)); [seq(f(i),i=0..30)];
%Y A000139 Cf. A000142.
%Y A000139 Cf. A000309, A006335, A004677.
%Y A000139 Sequence in context: A032085 A032163 A038078 this_sequence A052621 A131057 A051852
%Y A000139 Adjacent sequences: A000136 A000137 A000138 this_sequence A000140 A000141 A000142
%K A000139 nonn,easy,nice
%O A000139 0,1
%A A000139 njas

%I A052621
%S A052621 2,1,2,6,48,120,720,5040,80640,362880,3628800,39916800,958003200,
%T A052621 6227020800,87178291200,1307674368000,41845579776000,355687428096000,
%U A052621 6402373705728000,121645100408832000,4865804016353280000
%N A052621 A simple regular expression in a labeled universe.
%H A052621 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=567">Encyclopedia of Combinatorial Structures 567</a>
%F A052621 E.g.f.: -(x^3+x^2+x+2)/(-1+x^4)
%F A052621 Recurrence: {a(1)=1, a(3)=6, a(2)=2, a(0)=2, (-n^4-35*n^2-50*n-24-10*n^3)*a(n)+a(n+4)}
%F A052621 Sum(1/4*(_alpha^3+_alpha^2+2*_alpha+1)*_alpha^(-1-n), _alpha=RootOf(-1+_Z^4))*n!
%F A052621 2n! if n is 0 mod 4, n! otherwise.
%p A052621 spec := [S,{S=Union(Sequence(Z), Sequence(Prod(Z,Z,Z,Z)))},labeled]: seq(combstruct[count](spec,size=n), n=0..20);
%Y A052621 Sequence in context: A032163 A038078 A000139 this_sequence A131057 A051852 A054495
%Y A052621 Adjacent sequences: A052618 A052619 A052620 this_sequence A052622 A052623 A052624
%K A052621 easy,nonn
%O A052621 0,1
%A A052621 encyclopedia(AT)pommard.inria.fr, Jan 25 2000

%I A131057
%S A131057 0,2,1,2,7,1,2,1,5,3,1,19,11,1,19,19,11,1,19,23,1,1,47,1,1,29,3,29,2,59,
%T A131057 73,1,43,1,13,17,41,1,2,5,3,53,79,7,1,53,23,1
%N A131057 Least nonnegative k such that n!-k is semiprime.
%C A131057 Semiprime analogue of A033933. After n = 3, a(n) is never again 0.
%F A131057 a(n) = MIN{k such that n!-k is in A001358 and k>=0}.
%e A131057 a(3) = 0 because 3! - 0 = 6 - 0 = 6 = 2 * 3.
%e A131057 a(4) = 2 because 4! - 2 = 24 - 2 = 22 = 2 * 11.
%e A131057 a(5) = 1 because 5! - 1 = 119 = 7 * 17.
%Y A131057 Cf. A001358, A033933.
%Y A131057 Sequence in context: A038078 A000139 A052621 this_sequence A051852 A054495 A007966
%Y A131057 Adjacent sequences: A131054 A131055 A131056 this_sequence A131058 A131059 A131060
%K A131057 easy,nonn
%O A131057 3,2
%A A131057 Jonathan Vos Post (jvospost2(AT)yahoo.com), Sep 24 2007

%I A051852
%S A051852 1,1,2,1,2,7,1,2,3,8,39,1,2,3,9,10,52,311,1,2,3,4,10,11,19,67,68,467,
%T A051852 3268,1,2,3,4,11,12,13,21,84,85,94,669,670,5350,42799,1,2,3,4,5,12,13,
%U A051852 14,23,24,103,104,105,114,205,923,924,934,8304,8305,74734,672605,1,2,3
%N A051852 A051851(n)/row_index_of(n).
%e A051852 1; 1, 2; 1, 2, 7; 1, 2, 3, 8, 39; 1, 2, 3, 9, 10, 52, 311;
%p A051852 with(combinat); rows_upto_u := proc(u) local a,n; a := []; for n from 1 to u do a := [op(a),op(map(divby, sort(map(list_in_base_b,partition(n),(n+1))), n))]; od; RETURN(a); end;
%Y A051852 A subset of A051850 (but note the relative indexing...)
%Y A051852 Sequence in context: A000139 A052621 A131057 this_sequence A054495 A007966 A011241
%Y A051852 Adjacent sequences: A051849 A051850 A051851 this_sequence A051853 A051854 A051855
%K A051852 easy,nonn
%O A051852 1,3
%A A051852 Antti Karttunen Dec 13 1999

%I A054495
%S A054495 1,1,1,2,1,2,7,1,3,2,11,4,1,7,3,2,17,6,19,4,1,11,23,3,5,2,9,14,29,6,31,
%T A054495 4,11,17,7,12,37,19,3,5,41,2,43,22,9,23,47,6,49,10,17,4,53,18,11,7,19,
%U A054495 29,59,12,61,31,3,8,5,22,67,34,23,14,71,9,73,37,15,38,77,6,79,10,27,41
%N A054495 Smallest k such that n/k is a Fibonacci number.
%F A054495 a(n)=n/A054495(n)
%e A054495 a(10)=2 because 10/1=10 is not a Fibonacci number but 10/2=5 is.
%Y A054495 Cf. A000045, A005086, A037943.
%Y A054495 Sequence in context: A052621 A131057 A051852 this_sequence A007966 A011241 A021051
%Y A054495 Adjacent sequences: A054492 A054493 A054494 this_sequence A054496 A054497 A054498
%K A054495 easy,nonn
%O A054495 1,4
%A A054495 Henry Bottomley (se16(AT)btinternet.com), Apr 04 2000

%I A007966
%S A007966 0,1,1,1,2,1,2,7,2,3,1,1,3,1,7,3,4,1,2,1,4,3,2,23,4,5,1,1,7,1,5,31,
%T A007966 16,11,17,5,6,1,2,3,2,1,6,1,11,5,23,47,6,7,1,1,4,1,2,11,7,3,1,1,15,
%U A007966 1,31,7,8,1,2,1,4,23,5,71,8,1,1,25,19,7,26,79,8,9,1,1,3,1,2,3,4,1,9
%N A007966 First factor in happy factorization of n.
%H A007966 J. H. Conway, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">On Happy Factorizations</a>, J. Integer Sequences, Vol. 1, 1998, #1.
%Y A007966 Sequence in context: A131057 A051852 A054495 this_sequence A011241 A021051 A113246
%Y A007966 Adjacent sequences: A007963 A007964 A007965 this_sequence A007967 A007968 A007969
%K A007966 nonn
%O A007966 0,5
%A A007966 J. H. Conway (conway(AT)math.princeton.edu)

%I A011241
%S A011241 1,1,2,1,2,7,3,7,3,5,4,2,5,9,4,5,5,3,9,7,9,0,5,1,8,1,8,0,4,6,2,8,9,
%T A011241 2,2,3,2,8,1,8,3,9,3,5,0,5,4,7,1,6,8,9,1,0,1,9,9,8,2,8,3,0,9,7,7,0,
%U A011241 3,8,2,4,6,1,8,6,5,0,6,9,0,9,3,7,4,3,4,7,7,2,8,8,1,0,4,2,1,8,8,1,5
%N A011241 Decimal expansion of 17th root of 7.
%Y A011241 Sequence in context: A051852 A054495 A007966 this_sequence A021051 A113246 A056887
%Y A011241 Adjacent sequences: A011238 A011239 A011240 this_sequence A011242 A011243 A011244
%K A011241 nonn,cons
%O A011241 1,3
%A A011241 njas

%I A021051
%S A021051 0,2,1,2,7,6,5,9,5,7,4,4,6,8,0,8,5,1,0,6,3,8,2,9,7,8,7,2,3,4,0,4,2,
%T A021051 5,5,3,1,9,1,4,8,9,3,6,1,7,0,2,1,2,7,6,5,9,5,7,4,4,6,8,0,8,5,1,0,6,
%U A021051 3,8,2,9,7,8,7,2,3,4,0,4,2,5,5,3,1,9,1,4,8,9,3,6,1,7,0,2,1,2,7,6,5
%N A021051 Decimal expansion of 1/47.
%Y A021051 Sequence in context: A054495 A007966 A011241 this_sequence A113246 A056887 A095062
%Y A021051 Adjacent sequences: A021048 A021049 A021050 this_sequence A021052 A021053 A021054
%K A021051 nonn,cons
%O A021051 0,2
%A A021051 njas

%I A113246
%S A113246 1,1,2,1,2,7,8,1,2,7,8,19,20,25,26,1,2,7,8,19,20,25,26,55,56,61,62,73,
%T A113246 74,79,80,1,2,7,8,19,20,25,26,55,56,61,62,73,74,79,80,163,164,169,170,
%U A113246 181,182,187,188,217,218,223,224,235,236,241,242
%N A113246 a(2^n+a) = a(2^(n-1)+a) [if 0 <= a < 2^(n-1)], 3^n-a(2^n-a) [if 2^(n-1) <= a < 2^n].
%C A113246 Cantor tree of endpoints. Row #R (with 2^R values), divided by 3^R, gives some endpoints of the Cantor set.
%H A113246 <a href="http://planetmath.org/encyclopedia/CantorSet.html">PlanetMath Cantor Set</a>
%e A113246 1
%e A113246 1,2
%e A113246 1,2,7,8
%e A113246 1,2,7,8,19,20,25,26
%Y A113246 Sequence in context: A007966 A011241 A021051 this_sequence A056887 A095062 A114346
%Y A113246 Adjacent sequences: A113243 A113244 A113245 this_sequence A113247 A113248 A113249
%K A113246 nonn
%O A113246 1,3
%A A113246 Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Jan 07 2006
%E A113246 Edited by njas, Jan 26 2006

%I A056887
%S A056887 0,1,2,1,2,7,8,7,6,27,54,57,56,233,232,7,6,187,588,1485,1242,6177,3184,
%T A056887 639,640,1441,3472,13033,13032,20745,24120,1593,1594,20891,60476,9069,
%U A056887 26964
%V A056887 0,-1,-2,1,2,7,-8,-7,-6,27,-54,57,-56,-233,-232,-7,-6,187,-588,1485,-1242,-6177,-3184,
%W A056887 -639,640,-1441,-3472,13033,13032,-20745,-24120,1593,1594,20891,-60476,9069,-26964
%N A056887 Determinant of n X n Hankel matrix whose entries are t(i+j), 0 <= i, j < n, where t is the Thue-Morse sequence.
%Y A056887 Sequence in context: A011241 A021051 A113246 this_sequence A095062 A114346 A032068
%Y A056887 Adjacent sequences: A056884 A056885 A056886 this_sequence A056888 A056889 A056890
%K A056887 sign
%O A056887 1,3
%A A056887 Jeffrey Shallit (shallit(AT)graceland.uwaterloo.ca), Sep 04 2000

%I A095062
%S A095062 1,1,2,1,2,7,12,14,27,50,91,178,335,611,1156,2147,4042,7831,14724,28227,
%T A095062 53736,102482,196303,376121,723408,1393572,2683465,5180304,10009707,
%U A095062 19366479,37509260,72706948,141074303
%N A095062 Number of fib00 primes (A095082) in range ]2^n,2^(n+1)].
%H A095062 A. Karttunen and J. Moyer, <a href="http://www.research.att.com/~njas/sequences/a095062.c.txt">C-program for computing the initial terms of this sequence</a>
%H A095062 <a href="http://www.research.att.com/~njas/sequences/Sindx_Pri.html#primesubsetpop2">Index entries for sequences related to occurrences of various subsets of primes in range ]2^n,2^(n+1)]</a>
%Y A095062 a(n) = A095060(n)-A095067(n) = A095065(n)+A095068(n).
%Y A095062 Sequence in context: A021051 A113246 A056887 this_sequence A114346 A032068 A103410
%Y A095062 Adjacent sequences: A095059 A095060 A095061 this_sequence A095063 A095064 A095065
%K A095062 nonn
%O A095062 1,3
%A A095062 Antti Karttunen (his-firstname.his-surname(AT)iki.fi), Jun 01 2004

%I A114346
%S A114346 2,1,2,7,14,21,26,29,29,27,23,19,15,11,8,5,3,2,1,0,0
%N A114346 The integer difference between n+1 dimensional surface area and n dimensional volume.
%C A114346 This sequence is important in the n dimensional ( topological dimension) theory of particles and has a maximum at n=8. I had noticed that at a given set of scale of radius there were near integer relationships between these two. q=(v[5]*e0)^(1/5) in esu ( electric charge) s[5]*q^4-v[4]*q^4 --> 3*G for G the gravitational constant.
%D A114346 D.M.Y Sommerville, An Introduction to the Geometry of n dimensions,Dover Publications,1858, pages136-137
%F A114346 v[n_] = Pi^(n/2)/Gamma[n/2 + 1] s[n_] = 2*Pi^(n/2)/Gamma[n/2] a(n) = Floor[Abs[s[n] - v[n + 1]]]
%t A114346 v[n_]=Pi^(n/2)/Gamma[n/2+1] s[n_]=2*Pi^(n/2)/Gamma[n/2] a=Table[Floor[Abs[s[n]-v[n+1]]],{n,0,20}]
%Y A114346 Sequence in context: A113246 A056887 A095062 this_sequence A032068 A103410 A030651
%Y A114346 Adjacent sequences: A114343 A114344 A114345 this_sequence A114347 A114348 A114349
%K A114346 nonn,uned
%O A114346 0,1
%A A114346 Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Feb 08 2006

%I A032068
%S A032068 2,1,2,7,22,85,344,1408,5914,25394,110818,488688,2178594,9797950,
%T A032068 44406134,202591861,929716110,4288808485,19876422076,92501558277,
%U A032068 432109230664,2025444499805,9523528005822,44906724639089
%N A032068 Number of series-reduced dyslexic planted planar trees with n leaves of 2 colors where any 2 subtrees extending from the same node are different.
%H A032068 <a href="http://www.research.att.com/~njas/sequences/Sindx_Ro.html#rooted">Index entries for sequences related to rooted trees</a>
%F A032068 Doubles (index 2+) under "BGK" (reversible, element, unlabeled) transform.
%Y A032068 Sequence in context: A056887 A095062 A114346 this_sequence A103410 A030651 A137305
%Y A032068 Adjacent sequences: A032065 A032066 A032067 this_sequence A032069 A032070 A032071
%K A032068 nonn
%O A032068 1,1
%A A032068 Christian G. Bower (bowerc(AT)usa.net)

%I A103410
%S A103410 2,1,2,7,56,2212,2595782,3374959180831,5695183504489239067484387,
%T A103410 16217557574922386301420531277071365103168734284282,
%U A103410 131504586847961235687181874578063117114329409897598970946516793776220805297959867258692249572750581
%N A103410 Number of products of distinct elements in generation n, starting with two elements.
%C A103410 The binary operation must be commutative, idempotent, and non-associative. - David Wasserman (dwasserm(AT)earthlink.net), Apr 15 2008
%F A103410 a(n)=a(n-1)(a(0)+a(1)+...+a(n-2))+C(a(n-1), 2).
%e A103410 The word "product" means a binary operation * . For example, using * = average, given by a*b=(a+b)/2, generation G(0) consisting of 0 and 1 yields successive generations:
%e A103410 G(1): 0*1=1/2, whence a(1)=1
%e A103410 G(2): 1/4=0*(1/2), 3/4=1*(1/2), whence a(2)=2
%e A103410 G(3): 1/8=0*(1/4), 5/8=1*(1/4), 3/8=(1/2)*(1/4), 3/8=0*(3/4),
%e A103410 7/8=1*(3/4), 5/8=(1/2)*(3/4), 1/2=(1/4)*(3/4), whence a(3)=7.
%e A103410 To summarize, for n>=3, G(n) consists of a(n-1)*(a(0)+a(1)+...+a(n-2)) products a*b where a runs through G(0), G(1),...,G(n-2) and b runs through G(n-1), together with C(a(n-1),2) products a*b where a and b run through G(n-1).
%o A103410 (PARI) print1("2,");a=2;s=0;for(n=1,12,aa=a*s+binomial(a,2);print1(aa",");s+=a;a=aa) - Herman Jamke (hermanjamke(AT)fastmail.fm), May 01 2008
%Y A103410 Essentially the same as A002658.
%Y A103410 Sequence in context: A095062 A114346 A032068 this_sequence A030651 A137305 A119419
%Y A103410 Adjacent sequences: A103407 A103408 A103409 this_sequence A103411 A103412 A103413
%K A103410 nonn
%O A103410 0,1
%A A103410 Clark Kimberling (ck6(AT)evansville.edu), Feb 04 2005
%E A103410 One more term from David Wasserman (dwasserm(AT)earthlink.net), Apr 15 2008
%E A103410 One more term from Herman Jamke (hermanjamke(AT)fastmail.fm), May 01 2008

%I A030651
%S A030651 2,1,2,8,1,2,1,1,2,1,1,3,1,1,9,1,4,1,3,1,7,1,1,3,6,2,2,1,1,32,3,3,3,
%T A030651 1,24,2,2,25,1,2,1,6,2,1,1,3,1,9,3,2,1,6,7,2,8,2,5,1,5,1,2,2,2,2,4,
%U A030651 3,1,5,1,15,1,1,2,4,3,3,1,5,1,4,1,8,1,3,1,1,8,2,1,2,1,514,1,2,1,1,1
%N A030651 Continued fraction for GAMMA(1/3).
%H A030651 G. Xiao, <a href="http://wims.unice.fr/~wims/en_tool~number~contfrac.en.html">Contfrac</a>
%H A030651 <a href="http://www.research.att.com/~njas/sequences/Sindx_Con.html#confC">Index entries for continued fractions for constants</a>
%F A030651 Note that 3*GAMMA(1/3)*GAMMA(2/3)=2*Pi*sqrt(3).
%Y A030651 Cf. A030652.
%Y A030651 Sequence in context: A114346 A032068 A103410 this_sequence A137305 A119419 A109529
%Y A030651 Adjacent sequences: A030648 A030649 A030650 this_sequence A030652 A030653 A030654
%K A030651 nonn,cofr
%O A030651 1,1
%A A030651 Paolo Dominici (pl.dm(AT)libero.it)

%I A137305
%S A137305 0,2,1,2,8,5,1,7,4,2,20,11,8,26,17,5,23,14,1,19,10,7,22,16,4,22,13,2,56,
%T A137305 29,20,74,47,11,65,38,8,62,35,26,80,53,17,71,44,5,59,32,23,77,50,14,68,
%U A137305 41,1,55,28,19,73,46,10
%N A137305 Write n in base 3, complement, reverse.
%C A137305 This is to A036044 as A007089 is to A007088. Fixed points begin: 5, 7, 11, 19, 22, 29, 44, 50, 55.
%F A137305 a(3^n) = 2. a(2*3^n) = 1.
%e A137305 53 -> 1222 -> 2111 -> 1112 -> 41.
%Y A137305 Cf. A007089, A036044.
%Y A137305 Sequence in context: A032068 A103410 A030651 this_sequence A119419 A109529 A022694
%Y A137305 Adjacent sequences: A137302 A137303 A137304 this_sequence A137306 A137307 A137308
%K A137305 base,easy,nonn
%O A137305 0,2
%A A137305 Jonathan Vos Post (jvospost3(AT)gmail.com), Apr 20 2008

%I A119419
%S A119419 2,1,2,9,2,7,1,2,1,19,1,8,1,1,5,126,3,2,1,15,1,5,1,1,2,26,1,2,1,7,5,5,2,
%T A119419 9,5,1,1,1,1,9,3,2,1,7,2,1,16,2,2,1,5,1,1,1,2,4,1,1,4,1,3,2,17,1,3,18,1,
%U A119419 1,5,1,2,10,6,43
%V A119419 -2,1,2,9,2,7,1,2,1,19,1,8,1,1,5,126,3,2,1,15,1,5,1,1,2,26,1,2,1,7,5,5,2,9,5,1,1,1,1,9,
%W A119419 3,2,1,7,2,1,16,2,2,1,5,1,1,1,2,4,1,1,4,1,3,2,17,1,3,18,1,1,5,1,2,10,6,43
%N A119419 Continued fraction expansion of the imaginary part of (-Exp[ -1])^(-Exp[ -1]).
%C A119419 (-Exp[ -1])^(-Exp[ -1]) is the value of z^z where Abs[z^z] achieves its unique local maximum. A119418 gives the continued fraction expansion of the corresponding real part. A119420 gives the decimal expansion of the corresponding real part. A119421 gives the decimal expansion.
%Y A119419 Cf. A119418, A119420, A119421.
%Y A119419 Sequence in context: A103410 A030651 A137305 this_sequence A109529 A022694 A002079
%Y A119419 Adjacent sequences: A119416 A119417 A119418 this_sequence A119420 A119421 A119422
%K A119419 cofr,sign
%O A119419 0,1
%A A119419 Joseph Biberstine (jrbibers(AT)indiana.edu), May 17 2006; corrected May 24 2006

%I A109529
%S A109529 1,2,1,2,9,2,9,34,9,34,131,34,131,504,131,504,1939,504,1939,7460,1939,
%T A109529 7460,28701,7460,28701,110422,28701,110422,424829,110422,424829,1634454,
%U A109529 424829,1634454,6288271,1634454,6288271,24193004,6288271,24193004
%N A109529 A switched vector Markov between three matrices with the same characteristic polynomial: x^3-x^2-x-1 The second stream of three that results.
%C A109529 CharacteristicPolynomial[M1, x] CharacteristicPolynomial[M2, x] CharacteristicPolynomial[M3, x]
%F A109529 M[n] =M1 if Mod[n, 3]=1 M[n] =M2 if Mod[n, 3]=2 M[n] =M3 if Mod[n, 3]=0 v[n]=M[n].v[n-1] a(n) = v[n][[2]]
%t A109529 M1 = {{0, 1, 0}, {0, 0, 1}, {1, 1, 1}}; M2 = {{1, 1, 1}, {1, 0, 0}, {0, 1, 0}}; M3 = {{0, 1, 0}, {1, 1, 1}, {1, 0, 0}}; M[n_] = If[Mod[n, 3] == 1, M3, If[Mod[n, 3] == 2, M2, M1]]; v[0] = {0, 1, 1}; v[n_] := v[n] = M[n].v[n - 1] a = Table[v[n][[2]], {n, 0, 100}]
%Y A109529 Cf. A000213.
%Y A109529 Sequence in context: A030651 A137305 A119419 this_sequence A022694 A002079 A078357
%Y A109529 Adjacent sequences: A109526 A109527 A109528 this_sequence A109530 A109531 A109532
%K A109529 nonn,uned
%O A109529 0,2
%A A109529 Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Jun 18 2005

%I A022694
%S A022694 1,2,1,2,9,2,10,16,38,98,53,116,340,434,463,990,2378,2792,
%T A022694 3660,7058,11454,18900,24104,36206,81623,119400,128194,
%U A022694 248062,447066,576154,880401,1415926,2297516,3724290,4854450
%V A022694 1,-2,-1,-2,9,-2,10,-16,38,-98,53,-116,340,-434,463,-990,2378,-2792,
%W A022694 3660,-7058,11454,-18900,24104,-36206,81623,-119400,128194,
%X A022694 -248062,447066,-576154,880401,-1415926,2297516,-3724290,4854450
%N A022694 Expansion of Product (1+m*q^m)^(-2); m=1..inf.
%Y A022694 Sequence in context: A137305 A119419 A109529 this_sequence A002079 A078357 A086382
%Y A022694 Adjacent sequences: A022691 A022692 A022693 this_sequence A022695 A022696 A022697
%K A022694 sign
%O A022694 0,2
%A A022694 njas

%I A002079 M0122 N0049
%S A002079 2,1,2,9,96,2690,226360,64646855,68339572672
%N A002079 N-equivalence classes of threshold gates of exactly n variables.
%D A002079 S. Muroga, Threshold Logic and Its Applications. Wiley, NY, 1971, p. 38, Table 2.3.2. - Row 8.
%D A002079 S. Muroga, T. Tsuboi and C. R. Baugh, Enumeration of threshold functions of eight variables, IEEE Trans. Computers, 19 (1970), 818-825.
%Y A002079 Cf. A002078.
%Y A002079 Sequence in context: A119419 A109529 A022694 this_sequence A078357 A086382 A062345
%Y A002079 Adjacent sequences: A002076 A002077 A002078 this_sequence A002080 A002081 A002082
%K A002079 nonn
%O A002079 0,1
%A A002079 njas

%I A078357
%S A078357 1,1,2,1,2,10,1,5,2,250,1,106,1138,2,25,146,1
%N A078357 Minimal solutions of certain Pell equations.
%C A078357 a(n) gives minimal (positive) solution of Pell equation b(n)^2 - D(n)*a(n)^2 = -4 with D(n)= A077426(n). The companion sequence is a(n)=A078356(n).
%C A078357 For the general solution of Pell equation b^2 - D(n)*a^2 = -4 see a comment in A078356 (with a and b interchanged).
%C A078357 For the conversion of the (x,y) values of Perron's table to the (b(n),a(n)) values see a A078356 comment.
%D A078357 O. Perron, "Die Lehre von den Kettenbruechen, Bd.I", Teubner, 1954, 1957 (Sec. 30, Satz 3.35, p. 109 and table p. 108).
%Y A078357 Sequence in context: A109529 A022694 A002079 this_sequence A086382 A062345 A077098
%Y A078357 Adjacent sequences: A078354 A078355 A078356 this_sequence A078358 A078359 A078360
%K A078357 nonn,more
%O A078357 1,3
%A A078357 Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Nov 29 2002

%I A086382
%S A086382 2,1,2,10,1,10,2,1,2,12,1,10,2,1,2,10,1,12,2,1,2,10,1,10,2,1,2,16,1,12,
%T A086382 2,1,2,10,1,10,2,1,2,16,1,10,2,1,2,10,1,12,2,1,2,10,1,10,2,1,2,12,1,12,
%U A086382 2,1,2,10,1,10,2,1,2,36,1,10,2,1,2,10,1,12,2,1,2,10,1,10,2,1,2,12,1,12
%N A086382 k divides F(k*n^2+1)-F(k+1) for 1<=k<=a(n) where F(k) is the k-th Fibonacci number.
%F A086382 a(3n)=1; a( A047235(n))=2
%o A086382 (PARI) a(n)=if(n<0,0,m=1; while((fibonacci(m*n^2+1)-fibonacci(m+1))%m==0,m++); m-1)
%Y A086382 Sequence in context: A022694 A002079 A078357 this_sequence A062345 A077098 A069238
%Y A086382 Adjacent sequences: A086379 A086380 A086381 this_sequence A086383 A086384 A086385
%K A086382 nonn
%O A086382 2,1
%A A086382 Benoit Cloitre (benoit7848c(AT)orange.fr), Sep 06 2003

%I A062345
%S A062345 1,2,1,2,10,2,19,2,25,2,156,2,149,2,580,2,716,2,6461,2,2485,2,123256,2,64,
%T A062345 2,8638,2,722190,2,3804214,2,1783536,2,3550696,2
%N A062345 Length of period of continued fraction expansion of square root of 3^n-1.
%e A062345 The period of sqrt(242) contains 10 terms: [1,1,3,1,14,1,3,1,1,30]
%p A062345 with(numtheory): [seq(nops(cfrac(sqrt(3^k-1),'periodic','quotients')[2]),k=1..16)];
%t A062345 Table[Length[Last[ContinuedFraction[Sqrt[ -1+3^u]]]], {u, 1, 36}]
%Y A062345 Cf. A059866, A059926, A059927, A062328.
%Y A062345 Sequence in context: A002079 A078357 A086382 this_sequence A077098 A069238 A052579
%Y A062345 Adjacent sequences: A062342 A062343 A062344 this_sequence A062346 A062347 A062348
%K A062345 nonn
%O A062345 1,2
%A A062345 Labos E. (labos(AT)ana.sote.hu), Jul 13 2001

%I A077098
%S A077098 2,1,2,10,2,26,2,2,2,84531,2,531160,2,4738,2
%N A077098 Quotient cycle length in continued fraction expansion of sqrt(-1+n^n).
%t A077098 Table[Length[Part[ContinuedFraction[Sqrt[ -1+u^u]], 2]], {u, 2, 15}]
%Y A077098 Sequence in context: A078357 A086382 A062345 this_sequence A069238 A052579 A048296
%Y A077098 Adjacent sequences: A077095 A077096 A077097 this_sequence A077099 A077100 A077101
%K A077098 cofr,nonn
%O A077098 2,1
%A A077098 Labos E. (labos(AT)ana.sote.hu), Nov 05 2002

%I A069238
%S A069238 2,1,2,10,700,700,9800,3185000,85358000,1484210000,4904900000,213514756000,
%T A069238 10932576200000,651421552600000,491216647558000000,59347135259594000000,
%U A069238 308654469531044000000,582291574342534420000000,3395537788696824680000000
%N A069238 Numerator of coefficient G_n defined by Sum_{ (m,m') != (0,0)} 1/(m+m'*sqrt(-2))^(2*n) = (4*w)^(2*n)*G_n/(2*n)!, where 2w is one of the periods of the associated Weierstrass P-function.
%D A069238 E. Dintzl, Ueber die Zahlen im Koerper k(sqrt(-2)), welche den Bernoulli'schen Zahlen analog sind, Sitz. K. Akad. Wiss. Wien, Math.-Naturw. Klasse, 108 (1909), 1-29.
%F A069238 For n >= 2, G_n = A069182(n-1)*(2*n)/(2^(2*n-1)*(-1+(-2)^n)).
%e A069238 G_1, G_2, ... = 2/3, 1/3, 2/3, 10/3, 700/33, 700/3, 9800/3, 3185000/51, ...
%Y A069238 Cf. A069239, A069182, A069240.
%Y A069238 Sequence in context: A086382 A062345 A077098 this_sequence A052579 A048296 A016542
%Y A069238 Adjacent sequences: A069235 A069236 A069237 this_sequence A069239 A069240 A069241
%K A069238 nonn,frac
%O A069238 1,1
%A A069238 njas, Apr 13 2002

%I A052579
%S A052579 2,1,2,12,24,120,1440,5040,40320,725760,3628800,39916800,958003200,
%T A052579 6227020800,87178291200,2615348736000,20922789888000,355687428096000,
%U A052579 12804747411456000,121645100408832000,2432902008176640000
%N A052579 A simple regular expression in a labeled universe.
%H A052579 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=523">Encyclopedia of Combinatorial Structures 523</a>
%F A052579 E.g.f.: -(x^2+x+2)/(-1+x)/(1+x+x^2)
%F A052579 Recurrence: {a(1)=1, a(2)=2, a(0)=2, (-11*n-6-n^3-6*n^2)*a(n)+a(n+3)}
%F A052579 (4/3+Sum(1/3*_alpha^(-n), _alpha=RootOf(_Z^2+_Z+1)))*n!
%p A052579 spec := [S,{S=Union(Sequence(Prod(Z,Z,Z)), Sequence(Z))},labeled]: seq(combstruct[count](spec,size=n), n=0..20);
%Y A052579 Sequence in context: A062345 A077098 A069238 this_sequence A048296 A016542 A007402
%Y A052579 Adjacent sequences: A052576 A052577 A052578 this_sequence A052580 A052581 A052582
%K A052579 easy,nonn
%O A052579 0,1
%A A052579 encyclopedia(AT)pommard.inria.fr, Jan 25 2000

%I A048296
%S A048296 0,2,1,2,14,1,1,2,3,5,1,3,1,5,1,1,2,3,5,46,2,2,4,4,2,1,6,1,1,4,2,2,
%T A048296 1,109,1,1,4,9,3,45,8,4,1,2,1,13,13,1,1,2,1,1,2,1,4,2,3,1,17,1,1,1,
%U A048296 6,42,1,3,1,1,4,1,1,1,1,1,2,4,5,4,1,26,1,1,74,1,1,2,1,2,2,1,1,10,1
%N A048296 Continued fraction for Artin's constant.
%D A048296 See A005596 for further references.
%H A048296 <a href="http://www.research.att.com/~njas/sequences/Sindx_Ar.html#Artin">Index entries for sequences related to Artin's conjecture</a>
%Y A048296 Cf. A005596.
%Y A048296 Sequence in context: A077098 A069238 A052579 this_sequence A016542 A007402 A058260
%Y A048296 Adjacent sequences: A048293 A048294 A048295 this_sequence A048297 A048298 A048299
%K A048296 cofr,nonn,nice
%O A048296 0,2
%A A048296 Fred Lunnon (fred(AT)csa5.cs.may.ie), Simon Plouffe (plouffe(AT)math.uqam.ca)

%I A016542
%S A016542 2,1,2,14,1,1,14,13,1,86,1,19,1,5,1,1,1,5,1,2,1,1,9,1,16,
%T A016542 1,35,7,1,2,1,1,8,1,5,4,3,2,1,1,1,7,12,12,3,1,4,2,1,1,3,
%U A016542 2,1,4,1,3,7,1,1,1,1,4,3,1,22,1,238,2,13,1,5,1,7,6,6,2
%N A016542 Continued fraction for ln(29/2).
%H A016542 G. Xiao, <a href="http://wims.unice.fr/~wims/en_tool~number~contfrac.en.html">Contfrac</a>
%Y A016542 Sequence in context: A069238 A052579 A048296 this_sequence A007402 A058260 A115507
%Y A016542 Adjacent sequences: A016539 A016540 A016541 this_sequence A016543 A016544 A016545
%K A016542 nonn,cofr
%O A016542 1,1
%A A016542 njas

%I A007402 M0123
%S A007402 1,1,2,1,2,15,1,5,7,971,20,276
%N A007402 No-3-in-line problem for equilateral triangle array of side n.
%D A007402 R. H. Buchholz, personal communication.
%Y A007402 Sequence in context: A052579 A048296 A016542 this_sequence A058260 A115507 A051502
%Y A007402 Adjacent sequences: A007399 A007400 A007401 this_sequence A007403 A007404 A007405
%K A007402 nonn,nice
%O A007402 1,3
%A A007402 njas, Mira Bernstein

%I A058260
%S A058260 0,0,0,0,2,1,2,16,58
%N A058260 Number of minimal forbidden posets for split semiorders on n points.
%D A058260 P. C. Fishburn and J. A. Reeds, Counting split semiorders, Order, 18(2): 119-128, 2001.
%H A058260 <a href="http://www.research.att.com/~njas/sequences/Sindx_Pos.html#posets">Index entries for sequences related to posets</a>
%Y A058260 Cf. A000111, A058259.
%Y A058260 Sequence in context: A048296 A016542 A007402 this_sequence A115507 A051502 A121721
%Y A058260 Adjacent sequences: A058257 A058258 A058259 this_sequence A058261 A058262 A058263
%K A058260 nonn,hard,nice
%O A058260 1,5
%A A058260 njas, Dec 06 2000

%I A115507
%S A115507 1,1,2,1,2,18,72,42,126,1080,5400,42240,253440,3578400,28627200,
%T A115507 16914240,54396720,620978400,5417102880,40328285760
%N A115507 Number of circular permutations where every next element in binary notation has ones at the same or adjacent positions of previous element's binary ones.
%Y A115507 Cf. A115506, A115509.
%Y A115507 Sequence in context: A016542 A007402 A058260 this_sequence A051502 A121721 A136156
%Y A115507 Adjacent sequences: A115504 A115505 A115506 this_sequence A115508 A115509 A115510
%K A115507 nonn
%O A115507 1,3
%A A115507 Max Alekseyev (maxal(AT)cs.ucsd.edu), Jan 31 2006

%I A051502
%S A051502 2,1,2,23,3904,134156284,288230371925149328
%N A051502 Number of asymmetric types of Boolean functions of n variables under action of complementing group C(n,2).
%H A051502 <a href="http://www.research.att.com/~njas/sequences/Sindx_Bo.html#Boolean">Index entries for sequences related to Boolean functions</a>
%F A051502 1/(2^n)*Sum((-1)^j*2^C(j, 2)*[ n, j ]*2^(2^(n-j)), j=0..n), where [ n, j ] is Gaussian 2-binomial coefficient.
%Y A051502 Sequence in context: A007402 A058260 A115507 this_sequence A121721 A136156 A141238
%Y A051502 Adjacent sequences: A051499 A051500 A051501 this_sequence A051503 A051504 A051505
%K A051502 easy,nonn
%O A051502 0,1
%A A051502 Vladeta Jovovic (vladeta(AT)Eunet.yu)

%I A121721
%S A121721 1,0,2,1,2,25,9,1,25,117,36,7,24,117,356,100,34,16,116,356,850,225,98,11,108,
%T A121721 355,850,1737,441,223,75,81,347,849,1737,3185,784,439,200,17,320,841,1736,3185,
%U A121721 5392,1296,782,416,108,256,814,1728,3184,5392,8586,2025,1294,759,324,131,750
%V A121721 1,0,2,-1,2,25,-9,1,25,117,-36,-7,24,117,356,-100,-34,16,116,356,850,-225,-98,-11,108,
%W A121721 355,850,1737,-441,-223,-75,81,347,849,1737,3185,-784,-439,-200,17,320,841,1736,3185,
%X A121721 5392,-1296,-782,-416,-108,256,814,1728,3184,5392,8586,-2025,-1294,-759,-324,131,750
%N A121721 Triangle read by rows: T[n, m] = Sum[m^3 - 3*m^2*k + 3*m*k^2 - k^3, {k, 0, n - 1}] + m^4.
%e A121721 1
%e A121721 0, 2
%e A121721 -1, 2, 25
%e A121721 -9, 1, 25, 117
%e A121721 -36, -7, 24, 117, 356
%e A121721 -100, -34, 16, 116, 356, 850
%t A121721 g[n_, m_] = Sum[m^3 - 3*m^2*k + 3*m*k^2 - k^3, {k, 0, n - 1}] + m^4 t[n_, m_] = If[n == 0, 1, g[n, m]] a = Table[Table[t[n, m], {m, 0, n}], {n, 0, 10}] Flatten[a]
%Y A121721 Sequence in context: A058260 A115507 A051502 this_sequence A136156 A141238 A094690
%Y A121721 Adjacent sequences: A121718 A121719 A121720 this_sequence A121722 A121723 A121724
%K A121721 sign,tabl
%O A121721 1,3
%A A121721 Roger Bagula (rlbagulatftn(AT)yahoo.com), Sep 08 2006
%E A121721 Edited by njas, Oct 01 2006

%I A136156
%S A136156 2,1,2,30,24,1,720,15,560,12,3628800,10,479001600,360,8,315,
%T A136156 20922789888000,280,6402373705728000,6,240,1814400,
%U A136156 1124000727777607680000,5,1596672,239500800,1478400,180,304888344611713860501504000000,4,265252859812191058636308480000000
%N A136156 Quotients p!/n arising in A139171.
%t A136156 f[n_] := Block[{m = 1}, While[ !IntegerQ[ Prime@ m!/n], m++ ]; Prime@ m!/n]; Array[f, 31] - Robert G. Wilson v, (rgwv(AT)rgwv.com), Apr 17 2008
%Y A136156 Sequence in context: A115507 A051502 A121721 this_sequence A141238 A094690 A010249
%Y A136156 Adjacent sequences: A136153 A136154 A136155 this_sequence A136157 A136158 A136159
%K A136156 nonn
%O A136156 1,1
%A A136156 njas, Apr 13 2008
%E A136156 More terms from Robert G. Wilson v, (rgwv(AT)rgwv.com), Apr 17 2008

%I A141238
%S A141238 2,1,2,37,16,181,220,273,959
%N A141238 Sequence of k such that starting with P(0)=23 then k(n)*P(n-1)*(k(n)*P(n-1)-1)-1 is the least prime = P(n).
%e A141238 1*23*(1*23-1)-1=505 composite
%e A141238 2*23*(2*23-1)-1=2069 prime so k(1)=2 P(1)=2069
%Y A141238 Cf. A141233 A141234 A141235 A141236 A141237 A141239 A141240.
%Y A141238 Sequence in context: A051502 A121721 A136156 this_sequence A094690 A010249 A002431
%Y A141238 Adjacent sequences: A141235 A141236 A141237 this_sequence A141239 A141240 A141241
%K A141238 nonn
%O A141238 1,1
%A A141238 Pierre CAMI (pierrecami(AT)tele2.fr), Jun 16 2008

%I A094690
%S A094690 1,2,1,2,37,17,4,4,6,11,25,2,1,2,1,9,1,2,1,13,4,1,2,3,1,1,8,1,3,1,2,1,1,
%T A094690 2,2,1,1,2,1,1,1,3,1,7,2,1,3,1,1,9,2,1,1,5,6,5,1,2,10,1,1,1,5,1,1,2,8,1,
%U A094690 6,1,1,2,2,1,3,1,1,1,4,30,1,4,66,1,2,76,4,6,1,1,9,3,1,1,1,28,1,11,1,2
%N A094690 Continued fraction expansion of (1+(+1+ ... )^(1/e))^(1/e))^(1/e)).
%C A094690 Increasing partial quotients are: 1,2,37,66,76,84,145,686,983,2434,5120
%t A094690 f[n_] := N[(n + 1)^(1/E), 1024]; ContinuedFraction[ Nest[f, 1, 2500], 100]
%Y A094690 Cf. A094689.
%Y A094690 Sequence in context: A121721 A136156 A141238 this_sequence A010249 A002431 A062963
%Y A094690 Adjacent sequences: A094687 A094688 A094689 this_sequence A094691 A094692 A094693
%K A094690 cofr,nonn
%O A094690 1,2
%A A094690 Robert G. Wilson v (rgwv(AT)rgwv.com), May 18 2004

%I A010249
%S A010249 2,1,2,63,1,2,2,2,1,95,2,1,1,2,7,4,2,3,1,2,3,127,1,4,1,
%T A010249 3,1,4,4,12,2,1,1,1,5,1,18,2,3,1,1,1,1,1,2,1,1,8,1,4,1,
%U A010249 4,8,1,1,2,1,1,11,1,12,6,1,1,3,2,3,3,1,2,8,1,3,2,1,6,2
%N A010249 Continued fraction for cube root of 19.
%H A010249 G. Xiao, <a href="http://wims.unice.fr/~wims/en_tool~number~contfrac.en.html">Contfrac</a>
%Y A010249 Sequence in context: A136156 A141238 A094690 this_sequence A002431 A062963 A127139
%Y A010249 Adjacent sequences: A010246 A010247 A010248 this_sequence A010250 A010251 A010252
%K A010249 nonn,cofr
%O A010249 0,1
%A A010249 njas

%I A002431 M0124 N0050
%S A002431 1,1,1,2,1,2,1382,4,3617,87734,349222,310732,472728182,2631724,
%T A002431 13571120588,13785346041608,7709321041217,303257395102,52630543106106954746,
%U A002431 616840823966644,522165436992898244102,6080390575672283210764,10121188937927645176372
%V A002431 1,-1,-1,-2,-1,-2,-1382,-4,-3617,-87734,-349222,-310732,-472728182,-2631724,
%W A002431 -13571120588,-13785346041608,-7709321041217,-303257395102,-52630543106106954746,
%X A002431 -616840823966644,-522165436992898244102,-6080390575672283210764,-10121188937927645176372
%N A002431 Numerators in Taylor series for cot x.
%C A002431 Can be written as numerators of multiples of Bernoulli numbers.
%D A002431 M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, December 1972, p. 75 (4.3.70).
%D A002431 G. W. Caunt, Infinitesimal Calculus, Oxford Univ. Press, 1914, p. 477.
%D A002431 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 88.
%D A002431 A. Fletcher, J. C. P. Miller, L. Rosenhead and L. J. Comrie, An Index of Mathematical Tables. Vols. 1 and 2, 2nd ed., Blackwell, Oxford and Addison-Wesley, Reading, MA, 1962, Vol. 1, p. 74.
%D A002431 H. Rademacher, Topics in Analytic Number Theory, Springer, 1973, Chap. 1, p. 19.
%D A002431 H. A. Rothe, in C. F. Hindenburg, editor, Sammlung Combinatorisch-Analytischer Abhandlungen, Vol. 2, Chap. XI. Fleischer, Leipzig, 1800, p. 331.
%H A002431 T. D. Noe, <a href="http://www.research.att.com/~njas/sequences/b002431.txt">Table of n, a(n) for n=-1..100</a>
%H A002431 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, December 1972 [alternative scanned copy].
%H A002431 M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards Applied Math. Series 55, Tenth Printing, December 1972, p. 75 (4.3.70).
%H A002431 <a href="http://www.research.att.com/~njas/sequences/Sindx_Be.html#Bernoulli">Index entries for sequences related to Bernoulli numbers.</a>
%H A002431 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Cotangent.html">Cotangent</a>
%F A002431 a(n)=-numerator(A000182[ n ]/(4^n-1)), n>0.
%F A002431 cot x = Sum_{k=0..inf} (-1)^k B_{2k} 4^k x^(2k-1) / (2k)!.
%e A002431 x^(-1)-1/3*x-1/45*x^3-2/945*x^5-1/4725*x^7-2/93555*x^9+O(x^11).
%Y A002431 Cf. A036278 (denominators), A000182.
%Y A002431 Sequence in context: A141238 A094690 A010249 this_sequence A062963 A127139 A071431
%Y A002431 Adjacent sequences: A002428 A002429 A002430 this_sequence A002432 A002433 A002434
%K A002431 sign,easy,nice
%O A002431 -1,4
%A A002431 njas

%I A062963
%S A062963 1,0,2,1,3,0,0,2,5,0,6,3,4,0,8,0,9,0,6,5,11,0,0,6,0,0,14,4,15,0,10,8,12,0,
%T A062963 18,9,12,0,20,6,21,0,0,11,23,0,0,0,16,0,26,0,20,0,18,14,29,0,30,15,0,0,24,10,
%U A062963 33,0,22,12,35,0,36,18,0,0,30,12,39,0,0,20,41,0,32,21,28,0,44,0,36,0,30,23,36
%V A062963 -1,0,-2,1,-3,0,0,2,-5,0,-6,3,4,0,-8,0,-9,0,6,5,-11,0,0,6,0,0,-14,-4,-15,0,10,8,12,0,
%W A062963 -18,9,12,0,-20,-6,-21,0,0,11,-23,0,0,0,16,0,-26,0,20,0,18,14,-29,0,-30,15,0,0,24,-10,
%X A062963 -33,0,22,-12,-35,0,-36,18,0,0,30,-12,-39,0,0,20,-41,0,32,21,28,0,-44,0,36,0,30,23,36
%N A062963 Mu(n) * H(n) where H(n) is A023022.
%o A062963 (PARI) H(n)=eulerphi(n)/2; j=[]; for(n=3,200,j=concat(j,moebius(n)*H(n))); j
%Y A062963 Sequence in context: A094690 A010249 A002431 this_sequence A127139 A071431 A140699
%Y A062963 Adjacent sequences: A062960 A062961 A062962 this_sequence A062964 A062965 A062966
%K A062963 easy,sign
%O A062963 3,3
%A A062963 Jason Earls (zevi_35711(AT)yahoo.com), Jul 22 2001

%I A127139
%S A127139 1,2,1,3,0,1,0,2,0,1,5,0,0,0,1,6,3,2,0,0,1,7,0,0,0,0,0,1,0,0,0,2,0,0,0,
%T A127139 1,0,0,3,0,0,0,0,0,1,10,5,0,0,2,0,0,0,0,1
%V A127139 1,-2,1,-3,0,1,0,-2,0,1,-5,0,0,0,1,6,-3,-2,0,0,1,-7,0,0,0,0,0,1,0,0,0,-2,0,0,0,1,0,0,
%W A127139 -3,0,0,0,0,0,1,10,-5,0,0,-2,0,0,0,0,1
%N A127139 Inverse triangle of A126988, row sums = A023900.
%C A127139 Row sums = A023900: (1, -1, -2, -1, -4, 2, -6, -1,...) Left column = A055615: (1, -2, -3, 0, -5, 6, -7,...) A127139 * [1, 2, 3,...] = [1, 0, 0, 0...] A127139 * [1, 0, 0, 0,...] = A055615 A127140 is the square of A127139
%F A127139 Inverse triangle of A126988
%e A127139 First few rows of the triangle are:
%e A127139 1;
%e A127139 -2, 1;
%e A127139 -3, 0, 1;
%e A127139 0, -2, 0, 1;
%e A127139 -5, 0, 0, 0, 1;
%e A127139 6, -3, -2, 0, 0, 1;
%e A127139 -7, 0, 0, 0, 0, 0, 1;
%e A127139 ...
%Y A127139 Cf. A126988, A055615, A023900, A127140.
%Y A127139 Sequence in context: A010249 A002431 A062963 this_sequence A071431 A140699 A140256
%Y A127139 Adjacent sequences: A127136 A127137 A127138 this_sequence A127140 A127141 A127142
%K A127139 tabl,sign
%O A127139 1,2
%A A127139 Gary W. Adamson (qntmpkt(AT)yahoo.com), Jan 06 2007

%I A071431
%S A071431 0,1,1,0,2,1,3,0,1,1,3,2,2,3,4,1,5,3,2,2,3,1,1,0,3,1,2,0,1,1,4,4,2,6,4,
%T A071431 1,1,0,2,1,3,0,1,1,3,2,2,3,4,4,5,7,2,2,3,1,1,0,3,1,2,0,1,1,4,4,3,6,4,1,
%U A071431 1,0,2,1,3,0,1,1,3,2,2,3,4,4,5,7,2,2,3,1,1,0,3,1,2,0,1,1,4,4,3
%N A071431 Sprague-Grundy values for octal game .17.
%C A071431 Sequence is eventually periodic with period 34. The only exceptions are n=0, 15, 17 and 32.
%C A071431 Winning Ways, p. 93 erroneously states that for odd n a(n)=A002187(n-1) xor 1. The first exception to this rule is n=49 and many others follow.
%D A071431 E. R. Berlekamp, J. H. Conway and R. K. Guy, Winning Ways, Academic Press, NY, 2 vols., 1982; see Chapter 4, p. 93.
%H A071431 Achim Flammenkamp, <a href="http://www.uni-bielefeld.de/~achim/octal_sparse.html">Octal games</a>
%Y A071431 Sequence in context: A002431 A062963 A127139 this_sequence A140699 A140256 A126206
%Y A071431 Adjacent sequences: A071428 A071429 A071430 this_sequence A071432 A071433 A071434
%K A071431 nonn
%O A071431 0,5
%A A071431 njas and Sue Pope (pope(AT)research.att.com), May 29 2002
%E A071431 More terms from Joseph S. Myers (jsm(AT)polyomino.org.uk), Jun 04, 2002
%E A071431 Edited by Christian G. Bower (bowerc(AT)usa.net), Oct 23 2002

%I A140699
%S A140699 1,2,1,3,0,1,2,2,0,0,5,0,0,0,1,1,3,2,0,0,1,7,0,0,0,0,0,1,2,2,0,0,0,0,0,
%T A140699 0,3,0,3,0,0,0,0,0,0,1,5,0,0,2,0,0,0,0,1,11,0,0,0,0,0,0,0,0,0,1,1,1,2,0,
%U A140699 0,2,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,0,0,0,1,1,7,0,0,0,0,2,0,0,0,0,0,0,1
%V A140699 1,2,-1,3,0,-1,2,-2,0,0,5,0,0,0,-1,1,-3,-2,0,0,1,7,0,0,0,0,0,-1,2,-2,0,0,0,0,0,0,3,0,
%W A140699 -3,0,0,0,0,0,0,1,-5,0,0,-2,0,0,0,0,1,11,0,0,0,0,0,0,0,0,0,-1,1,-1,-2,0,0,2,0,0,0,0,0,
%X A140699 0,13,0,0,0,0,0,0,0,0,0,0,0,-1,1,-7,0,0,0,0,-2,0,0,0,0,0,0,1,1,0,-5,0,-3,0,0,0,0,0,0,0
%N A140699 Triangle read by rows: A054524*A140256.
%C A140699 Row products are A140700. Similar to table A140256. Can perhaps be seen as taking the moebius function of A126988 times the mangoldt function of A126988.
%F A140699 a(n) = A054524*A140256
%e A140699 1;
%e A140699 2, -1;
%e A140699 3, 0, -1;
%e A140699 2, -2, 0, 0;
%e A140699 5, 0, 0, 0, -1;
%e A140699 1, -3, -2, 0, 0, 1;
%e A140699 7, 0, 0, 0, 0, 0, -1;
%e A140699 2, -2, 0, 0, 0, 0, 0, 0;
%e A140699 3, 0, -3, 0, 0, 0, 0, 0, 0;
%e A140699 1, -5, 0, 0, -2, 0, 0, 0, 0, 1;
%e A140699 11,0, 0, 0, 0, 0, 0, 0, 0, 0, -1;
%Y A140699 Cf. A054524, A140256, A126988, A140700.
%Y A140699 Sequence in context: A062963 A127139 A071431 this_sequence A140256 A126206 A119709
%Y A140699 Adjacent sequences: A140696 A140697 A140698 this_sequence A140700 A140701 A140702
%K A140699 sign,tabl
%O A140699 1,2
%A A140699 Mats O. Granvik and Gary W. Adamson (mgranvik(AT)abo.fi), May 24 2008

%I A140256
%S A140256 1,2,1,3,0,1,2,2,0,1,5,0,0,0,1,1,3,2,0,0,1,7,0,0,0,0,0,1,2,2,0,2,0,0,0,
%T A140256 1,3,0,3,0,0,0,0,0,1,1,5,0,0,2,0,0,0,0,1,11,0,0,0,0,0,0,0,0,0,1,1,1,2,3,
%U A140256 0,2,0,0,0,0,0,1,13,0,0,0,0,0,0,0,0,0,0,0,1,1,7,0,0,0,0,2,0,0,0,0,0,0,1
%N A140256 Triangle by columns: A014963 interleaved with (k-1) zeros.
%C A140256 T(n,k) = 0 if k does not divide n.
%C A140256 n-th row = "n" followed by (n-2) zeros and 1 if n is prime.
%C A140256 Row sums = A140255: (1, 3, 4, 5, 6, 7, 8, 7, 7, 9, 12,...).
%C A140256 Similar to A138618.
%H A140256 T. Tao, <a href="http://terrytao.wordpress.com/2007/04/05/simons-lecture-i-structure-and-randomness-in-fourier-analysis-and-number-theory/">Simons Lecture I: Structure and randomness in Fourier analysis and number theory</a>.
%H A140256 Wikipedia, <a href="http://en.wikipedia.org/wiki/Fundamental_theorem_of_arithmetic">Fundamental theorem of arithmetic</a>.
%F A140256 Triangle by columns, n terms in n-th row; A014963 interleaved with (k-1) zeros. A014963 = (1, 2, 3, 2, 5, 1, 7, 2, 3, 1, 11, 1, 13, 1,...).
%F A140256 a(n) = A014963(A126988).
%e A140256 First few rows of the triangle are:
%e A140256 1;
%e A140256 2, 1;
%e A140256 3, 0, 1;
%e A140256 2, 2, 0, 1;
%e A140256 5, 0, 0, 0, 1;
%e A140256 1, 3, 2, 0, 0, 1;
%e A140256 7, 0, 0, 0, 0, 0, 1;
%e A140256 2, 2, 0, 2, 0, 0, 0, 1;
%e A140256 3, 0, 3, 0, 0, 0, 0, 0, 1;
%e A140256 1, 5, 0, 0, 2, 0, 0, 0, 0, 1;
%e A140256 11, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
%e A140256 1, 1, 2, 3, 0, 2, 0, 0, 0, 0, 0, 1;
%e A140256 ...
%e A140256 Column 2 = (1, 0, 2, 0, 3, 0, 2, 0, 5, 0, 1, 0, 7,...).
%o A140256 (Excel cell formula) =if(row()>=column();if(mod(row();column())=0;lookup(roundup(row()/column();0);A000027;A014963);0);"")
%Y A140256 Cf. A140255, A014963.
%Y A140256 Sequence in context: A127139 A071431 A140699 this_sequence A126206 A119709 A120251
%Y A140256 Adjacent sequences: A140253 A140254 A140255 this_sequence A140257 A140258 A140259
%K A140256 nonn,tabl
%O A140256 1,2
%A A140256 Gary W. Adamson and Mats Granvik (qntmpkt(AT)yahoo.com), May 16 2008, Jun 11 2008

%I A126206
%S A126206 0,1,0,1,0,1,1,1,0,2,1,3,0,1,2,2,2,1,1,3,0,3,3,3,2,2,2,3,1,2,2,3,4,3,1,
%T A126206 3,3,4,2,4,2,4,2,2,3,4,3,3,3,3,2,2,3,5,2,4,2,4,4,3,3,3,4,4,6,5,5,4,2,9,
%U A126206 5,2,4,6,4,7,4,2,5,4,3,4,8,4,7,9,2,8,4,7,3,10,9,5,3,5,8,8,3,10,4
%N A126206 Number of 4's in decimal expansion of 4^n.
%e A126206 a(11)=3 because 4^11=4194304 with three 4's.
%p A126206 P:=proc(n) local i,k,x,y,w,cont; y:=4; k:=y; for i from 0 by 1 to n do x:=y^i; cont:=0; while x>0 do w:=x-trunc(x/10)*10; if w=k then cont:=cont+1; fi; x:=trunc(x/10); od; print(cont); od; end: P(100);
%Y A126206 Cf. A065710, A126205, A126207, A126208, A126209, A126210, A126211.
%Y A126206 Sequence in context: A071431 A140699 A140256 this_sequence A119709 A120251 A071490
%Y A126206 Adjacent sequences: A126203 A126204 A126205 this_sequence A126207 A126208 A126209
%K A126206 easy,nonn
%O A126206 0,10
%A A126206 Paolo P. Lava & Giorgio Balzarotti (ppl(AT)spl.at), Dec 20 2006

%I A119709
%S A119709 0,1,0,1,2,1,3,0,1,2,4,0,1,2,5,0,1,2,3,6,1,3,7,0,1,2,4,8,0,1,2,4,9,0,1,
%T A119709 2,5,10,0,1,2,3,5,11,0,1,2,3,4,6,12,0,1,2,3,5,6,13,0,1,2,3,6,7,14,1,3,7,
%U A119709 15,0,1,2,4,8,16,0,1,2,4,8,17
%N A119709 Table where n-th row (of A078822(n) terms) contains the distinct nonnegative integers which, when written in binary, are substrings of n written in binary.
%e A119709 12 in binary is 1100. Within this binary representation there is 0 (occurring twice), 1 (occurring twice), 10 (= 2 in decimal), 11 (= 3 in decimal), 100 (= 4 in decimal), 110 (= 6 in decimal), and 1100 (= 12 in decimal).
%e A119709 So row 12 = (0,1,2,3,4,6,12).
%Y A119709 Cf. A078822.
%Y A119709 Sequence in context: A140699 A140256 A126206 this_sequence A120251 A071490 A127094
%Y A119709 Adjacent sequences: A119706 A119707 A119708 this_sequence A119710 A119711 A119712
%K A119709 easy,nonn
%O A119709 0,5
%A A119709 Leroy Quet (qq-quet(AT)mindspring.com), Jun 10 2006

%I A120251
%S A120251 0,0,1,0,2,1,3,0,1,2,5,1,8,3,2,0,13,1,21,2,3,5,34,1,3,8,1,3,55,2,89,0,5,
%T A120251 13,5,1,144,21,8,2,233,3,377,5,2,34,610,1,4,3,13,8,987,1,8,3,21,55,1597,
%U A120251 2,2584,89,3,0,13,5,4181,13,34,5,6765,1,10946,144,3,21,7,8,17711,2
%N A120251 A120249[n] modulo A120250[n].
%C A120251 a[n] = 0 precisely when n is a power of 2.
%F A120251 a[n] = Mod[A120249[n], A120250[n]]
%e A120251 a[n] = A120249[2646] modulo A120250[2646] = 42 modulo 19 = 4
%t A120251 Table[If[n == 1, 0, (fl = FactorInteger[n]; pq = Table[1, {i, 1, PrimePi[Last[fl][[1]]]}]; While[Length[fl] > 0, pp = First[fl]; fl = Drop[fl, 1]; pq[[PrimePi[pp[[1]]]]] = pp[[2]] + 1;]; Mod[Numerator[FromContinuedFraction[pq]], Denominator[FromContinuedFraction[pq]]])], {n, 1, 80}]
%Y A120251 Cf. Corresponding denominators in A120250.
%Y A120251 Sequence in context: A140256 A126206 A119709 this_sequence A071490 A127094 A126988
%Y A120251 Adjacent sequences: A120248 A120249 A120250 this_sequence A120252 A120253 A120254
%K A120251 frac,hard,nonn
%O A120251 1,5
%A A120251 Joseph Biberstine (jrbibers(AT)indiana.edu), Jun 12 2006

%I A071490
%S A071490 1,0,2,1,3,0,1,3,4,0,2,3,4,2,1,3,2,0,3,4,1,3,4,6,2,0,1,2,5,3,1,6,7,
%T A071490 8,1,3,4,1,6,0
%N A071490 Sprague-Grundy values for octal game .324.
%D A071490 E. R. Berlekamp, J. H. Conway and R. K. Guy, Winning Ways, Academic Press, NY, 2 vols., 1982; see Chapter 4.
%Y A071490 Sequence in context: A126206 A119709 A120251 this_sequence A127094 A126988 A130026
%Y A071490 Adjacent sequences: A071487 A071488 A071489 this_sequence A071491 A071492 A071493
%K A071490 nonn
%O A071490 1,3
%A A071490 njas and Sue Pope (pope(AT)research.att.com), May 29 2002

%I A127094
%S A127094 1,2,1,3,0,1,4,0,2,1,5,0,0,0,1,6,0,0,3,2,1,7,0,0,0,0,0,1,8,0,0,0,4,0,2,
%T A127094 1,9,0,0,0,0,0,3,1,10,0,0,0,0,5,0,0,2,1,11,0,0,0,0,0,0,0,0,0,1
%N A127094 Triangle, row sums = sigma(n), reversal of A127093.
%C A127094 Row sums = sigma(n), A000203: (1, 3, 4, 7, 6, 12...).
%F A127094 Reversed rows of A127093
%F A127094 Also mod(n;k-n-1)-mod(n+1;k-n-1)+1. - Mats Granvik (mgranvik(AT)abo.fi), Sep 2 2007
%e A127094 First few rows of the triangle are:
%e A127094 1;
%e A127094 2, 1;
%e A127094 3, 0, 1;
%e A127094 4, 0, 2, 1;
%e A127094 5, 0, 0, 0, 1;
%e A127094 6, 0, 0, 3, 2, 1;
%e A127094 ...
%Y A127094 Cf. A127093, A000203, A126988.
%Y A127094 Sequence in context: A119709 A120251 A071490 this_sequence A126988 A130026 A113287
%Y A127094 Adjacent sequences: A127091 A127092 A127093 this_sequence A127095 A127096 A127097
%K A127094 nonn
%O A127094 1,2
%A A127094 Gary W. Adamson (qntmpkt(AT)yahoo.com), Jan 05 2007

%I A126988
%S A126988 1,2,1,3,0,1,4,2,0,1,5,0,0,0,1,6,3,2,0,0,1,7,0,0,0,0,0,1,8,4,0,2,0,0,0,
%T A126988 1,9,0,3,0,0,0,0,0,1,10,5,0,0,2,0,0,0,0,1,11,0,0,0,0,0,0,0,0,0,1,12,6,4,
%U A126988 3,0,2,0,0,0,0,0,1
%N A126988 Triangle read by rows: k-th column (k=0,1,2...) is (1,2,3,...) interspersed with n consecutive zeros starting after the "1".
%C A126988 Row sums = A000203, (sigma(n)): 1, 3, 4, 7, 6, 12, 8, 15,... sigma(n) is the sum of the divisors of the integer n. The sequence of parsed terms in sigma(n) is the reversal of non-zero row terms in the triangle A126988.
%C A126988 T(n,k)=n/k if k is a divisor of n; T(n,k)=0 if k is not a divisor of n (1<=k<=n). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Jan 17 2007
%C A126988 The nonzero entries of row n are the divisors of n in decreasing order. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Jan 17 2007
%D A126988 David Wells, "Prime Numbers, the Most Mysterious Figures in Math", John Wiley & Sons, Inc, 2005, Appendix B.
%F A126988 G.f. of column k = z^k/(1-z^k)^2 (k=1,2,...). G.f.=G(t,z)=Sum(t^k*z^k/(1-z^k)^2,k=1..infinity). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Jan 17 2007
%e A126988 First few rows of the triangle are:
%e A126988 1;
%e A126988 2, 1;
%e A126988 3, 0, 1;
%e A126988 4, 2, 0, 1;
%e A126988 5, 0, 0, 0, 1;
%e A126988 6, 3, 2, 0, 0, 1;
%e A126988 7, 0, 0, 0, 0, 0, 1;
%e A126988 8, 4, 0, 2, 0, 0, 0, 1;
%e A126988 9, 0, 3, 0, 0, 0, 0, 0, 1;
%e A126988 10, 5, 0, 0, 2, 0, 0, 0, 0, 1;
%e A126988 ...
%e A126988 sigma(12) = 28 = (from tables): (1 + 2 + 3 + 4 + 6 + 12).
%e A126988 sigma(12) = 28, from 12-th row of A126988 = (12 + 6 + 4 + 3 + 2 + 1), deleting the zeros, from left to right.
%p A126988 T:=proc(n,k) if type(n/k,integer)=true then n/k else 0 fi end: for n from 1 to 12 do seq(T(n,k),k=1..n) od; # yields sequence in triangular form - Emeric Deutsch (deutsch(AT)duke.poly.edu), Jan 17 2007
%Y A126988 Cf. A000203.
%Y A126988 Sequence in context: A120251 A071490 A127094 this_sequence A130026 A113287 A096798
%Y A126988 Adjacent sequences: A126985 A126986 A126987 this_sequence A126989 A126990 A126991
%K A126988 nonn,tabl
%O A126988 1,2
%A A126988 Gary W. Adamson (qntmpkt(AT)yahoo.com), Dec 31 2006
%E A126988 Edited by njas, Jan 24 2007

%I A130026
%S A130026 1,2,1,3,0,1,4,3,0,1,5,0,0,0,1,6,5,4,0,0,1,7,0,0,0,0,0,1,8,7,0,5,0,0,0,
%T A130026 1,9,0,7,0,0,0,0,0,1,10,9,0,0,6,0,0,0,0,1
%N A130026 Triangle (n,k) by columns, arithmetic sequences interspersed with k zeros.
%C A130026 Row sums = A081307: (1, 3, 4, 8, 6, 16, 8, 21, 17,...).
%F A130026 Arithmetic sequences by columns, (1,2,3,...); (1,3,5,...); (1,4,7,...); interspersed with k zeros, k=0,1,2,...
%e A130026 First few rows of the triangle are:
%e A130026 1;
%e A130026 2, 1;
%e A130026 3, 0, 1;
%e A130026 4, 3, 0, 1;
%e A130026 5, 0, 0, 0, 1;
%e A130026 6, 5, 4, 0, 0, 1;
%e A130026 7, 0, 0, 0, 0, 0, 1;
%e A130026 ...
%Y A130026 Cf. A081307, A130027.
%Y A130026 Sequence in context: A071490 A127094 A126988 this_sequence A113287 A096798 A137587
%Y A130026 Adjacent sequences: A130023 A130024 A130025 this_sequence A130027 A130028 A130029
%K A130026 nonn,tabl
%O A130026 0,2
%A A130026 Gary W. Adamson (qntmpkt(AT)yahoo.com), May 02 2007

%I A113287
%S A113287 1,2,1,3,0,1,4,4,4,1,5,10,10,0,1,6,18,24,12,6,1,7,28,49,42,21,0,1,8,40,
%T A113287 88,104,72,24,8,1,9,54,144,216,198,108,36,0,1,10,70,220,400,460,340,160,
%U A113287 40,10,1,11,88,319,682,946,880,550,220,55,0,1
%V A113287 1,2,1,-3,0,1,4,4,4,1,-5,-10,-10,0,1,6,18,24,12,6,1,-7,-28,-49,-42,-21,0,1,8,40,88,104,
%W A113287 72,24,8,1,-9,-54,-144,-216,-198,-108,-36,0,1,10,70,220,400,460,340,160,40,10,1,-11,
%X A113287 -88,-319,-682,-946,-880,-550,-220,-55,0,1
%N A113287 Triangle T, read by rows, where row n of T equals row n of matrix (n+1)-th power of triangle A112555.
%C A113287 Remarkably, the matrix logarithm (A113290) is an integer triangle. Matrix m-th power of A112555 = I + m*(A112555 - I) where I = identity matrix.
%F A113287 G.f.: A(x, y) = 1/(1-x*y) + x*(x+2)/((1-x*y)^2*(1+x+x*y)^2).
%e A113287 Triangle begins:
%e A113287 1;
%e A113287 2,1;
%e A113287 -3,0,1;
%e A113287 4,4,4,1;
%e A113287 -5,-10,-10,0,1;
%e A113287 6,18,24,12,6,1;
%e A113287 -7,-28,-49,-42,-21,0,1;
%e A113287 8,40,88,104,72,24,8,1;
%e A113287 -9,-54,-144,-216,-198,-108,-36,0,1;
%e A113287 10,70,220,400,460,340,160,40,10,1; ...
%o A113287 (PARI) {T(n,k)=local(x=X+X*O(X^n),y=Y+Y*O(Y^k)); polcoeff(polcoeff(1/(1-x*y)+x*(x+2)/((1-x*y)^2*(1+x+x*y)^2),n,X),k,Y)}
%Y A113287 Cf. A112555, A113288 (inverse), A113290 (log), A113291, A072374.
%Y A113287 Sequence in context: A127094 A126988 A130026 this_sequence A096798 A137587 A137639
%Y A113287 Adjacent sequences: A113284 A113285 A113286 this_sequence A113288 A113289 A113290
%K A113287 sign,tabl
%O A113287 0,2
%A A113287 Paul D. Hanna (pauldhanna(AT)juno.com), Oct 23 2005

%I A096798
%S A096798 1,2,1,3,0,1,5,0,0,1,7,6,0,0,1,11,13,2,0,0,1,15,34,9,0,0,0,1,22,65,33,0,0,0,0,1,30,
%T A096798 128,102,12,0,0,0,0,1,42,225,255,62,2,0,0,0,0,1,56,394,591,232,15,0,0,0,0,0,1,77,
%U A096798 649,1265,721,100,0,0,0,0,0,0,1,101,1064,2559,1972,455,18,0,0,0,0,0,0,1
%V A096798 1,2,1,3,0,1,5,0,0,1,7,-6,0,0,1,11,-13,2,0,0,1,15,-34,9,0,0,0,1,22,-65,33,0,0,0,0,1,30,
%W A096798 -128,102,-12,0,0,0,0,1,42,-225,255,-62,2,0,0,0,0,1,56,-394,591,-232,15,0,0,0,0,0,1,77,
%X A096798 -649,1265,-721,100,0,0,0,0,0,0,1,101,-1064,2559,-1972,455,-18,0,0,0,0,0,0,1
%N A096798 Triangle, read by rows, where T(n,k) = (k/n)*Sum_{d|n} A096797(d,k).
%C A096798 The first column forms the partition numbers (A000041).
%e A096798 Rows begin:
%e A096798 [1],
%e A096798 [2,1],
%e A096798 [3,0,1],
%e A096798 [5,0,0,1],
%e A096798 [7,-6,0,0,1],
%e A096798 [11,-13,2,0,0,1],
%e A096798 [15,-34,9,0,0,0,1],
%e A096798 [22,-65,33,0,0,0,0,1],
%e A096798 [30,-128,102,-12,0,0,0,0,1],
%e A096798 [42,-225,255,-62,2,0,0,0,0,1],
%e A096798 [56,-394,591,-232,15,0,0,0,0,0,1],
%e A096798 [77,-649,1265,-721,100,0,0,0,0,0,0,1],
%e A096798 [101,-1064,2559,-1972,455,-18,0,0,0,0,0,0,1],
%e A096798 [135,-1681,4941,-4926,1680,-147,2,0,0,0,0,0,0,1],...
%Y A096798 Cf. A000041, A008284, A096797.
%Y A096798 Sequence in context: A126988 A130026 A113287 this_sequence A137587 A137639 A113288
%Y A096798 Adjacent sequences: A096795 A096796 A096797 this_sequence A096799 A096800 A096801
%K A096798 sign,tabl
%O A096798 1,2
%A A096798 Paul D. Hanna (pauldhanna(AT)juno.com), Jul 13 2004

%I A137587
%S A137587 1,2,1,3,0,1,5,2,0,1,6,1,0,0,1,11,3,2,0,0,1,12,2,1,0,0,0,1,20,6,1,2,0,0,
%T A137587 0,1,25,4,3,1,0,0,0,1,37,9,2,1,2,0,0,0,0,1,43,8,3,1,1,0,0,0,0,0,1,70,16,
%U A137587 6,3,1,2,0,0,0,0,0,1
%N A137587 Triangle read by rows: A051731 * A026794.
%C A137587 Left column = A083710 starting (1, 2, 3, 5, 6, 11, 12,...). Row sums = A047968.
%F A137587 Inverse mobius transform of the partition triangle, A026794
%e A137587 First few rows of the triangle are:
%e A137587 1;
%e A137587 2, 1;
%e A137587 3, 0, 1;
%e A137587 5, 2, 0, 1;
%e A137587 6, 1, 0, 0, 1;
%e A137587 11, 3, 2, 0, 0, 1;
%e A137587 12, 2, 1, 0, 0, 0, 1;
%e A137587 20, 6, 1, 2, 0, 0, 0, 1;
%e A137587 25, 4, 3, 1, 0, 0, 0, 0, 1;
%e A137587 ...
%Y A137587 Cf. A026794, A051731, A083710, A047968.
%Y A137587 Sequence in context: A130026 A113287 A096798 this_sequence A137639 A113288 A035215
%Y A137587 Adjacent sequences: A137584 A137585 A137586 this_sequence A137588 A137589 A137590
%K A137587 nonn,tabl
%O A137587 1,2
%A A137587 Gary W. Adamson (qntmpkt(AT)yahoo.com), Jan 27 2008

%I A137639
%S A137639 1,2,1,3,0,1,5,2,0,1,7,1,0,0,1,11,3,2,0,0,1,15,2,1,0,0,0,1,22,6,1,2,0,0,
%T A137639 0,1,30,5,3,1,0,0,0,0,1,42,9,2,1,2,0,0,0,0,1,56,9,3,1,1,0,0,0,0,0,1,77,
%U A137639 16,6,3,1,2,0,0,0,0,0,1
%N A137639 Triangle read by rows, A026794 * A051731.
%C A137639 Left border = A000041 starting with offset 1: (1, 2, 3, 5, 7, 11, 15,...). Row sums = A137640: (1, 3, 4, 8, 9, 17, 19,...).
%F A137639 A026794 * A051731 as infinite lower triangular matrices.
%e A137639 First few rows of the triangle are:
%e A137639 1;
%e A137639 2, 1;
%e A137639 3, 0, 1;
%e A137639 5, 2, 0, 1;
%e A137639 7, 1, 0, 0, 1;
%e A137639 11, 3, 2, 0, 0, 1;
%e A137639 15, 2, 1, 0, 0, 0, 1;
%e A137639 22, 6, 1, 2, 0, 0, 0, 1;
%e A137639 30, 5, 3, 1, 0, 0, 0, 0, 1;
%e A137639 ...
%Y A137639 Cf. A000041, A137640, A026794, A051731.
%Y A137639 Sequence in context: A113287 A096798 A137587 this_sequence A113288 A035215 A071467
%Y A137639 Adjacent sequences: A137636 A137637 A137638 this_sequence A137640 A137641 A137642
%K A137639 nonn,tabl
%O A137639 1,2
%A A137639 Gary W. Adamson (qntmpkt(AT)yahoo.com), Jan 31 2008

%I A113288
%S A113288 1,2,1,3,0,1,8,4,4,1,15,10,10,0,1,36,30,36,12,6,1,77,70,91,42,21,0,1,192,
%T A113288 184,256,152,96,24,8,1,459,450,648,432,306,108,36,0,1,1220,1210,1780,1280,
%U A113288 1000,460,200,40,10,1,3201,3190,4741,3542,2926,1540,770,220,55,0,1
%V A113288 1,-2,1,3,0,1,-8,-4,-4,1,15,10,10,0,1,-36,-30,-36,-12,-6,1,77,70,91,42,21,0,1,-192,
%W A113288 -184,-256,-152,-96,-24,-8,1,459,450,648,432,306,108,36,0,1,-1220,-1210,-1780,-1280,
%X A113288 -1000,-460,-200,-40,-10,1,3201,3190,4741,3542,2926,1540,770,220,55,0,1
%N A113288 Matrix inverse of triangle A113287.
%F A113288 T(n, 0) = (-1)^n*(n+1)*A072374(n-1) for n>=2, with T(1, 0)=-2, T(n, n)=1. T(n, 1) = (-1)^n*(n+1)*(A072374(n-1) - 1) for n>=2.
%e A113288 Triangle begins:
%e A113288 1;
%e A113288 -2,1;
%e A113288 3,0,1;
%e A113288 -8,-4,-4,1;
%e A113288 15,10,10,0,1;
%e A113288 -36,-30,-36,-12,-6,1;
%e A113288 77,70,91,42,21,0,1;
%e A113288 -192,-184,-256,-152,-96,-24,-8,1;
%e A113288 459,450,648,432,306,108,36,0,1;
%e A113288 -1220,-1210,-1780,-1280,-1000,-460,-200,-40,-10,1;
%e A113288 3201,3190,4741,3542,2926,1540,770,220,55,0,1; ...
%o A113288 (PARI) {T(n,k)=local(x=X+O(X^(n+2)),y=Y+O(Y^(n+2)),M=matrix(n+1,n+1,r,c, polcoeff(polcoeff(1/(1-x*y)+r*x/((1-x*y)*(1+x+x*y)),r-1,X),c-1,Y))); if(n<k,0,(M^-1)[n+1,k+1])}
%Y A113288 Cf. A113287, A113289 (row sums), A113290 (-log), A072374.
%Y A113288 Sequence in context: A096798 A137587 A137639 this_sequence A035215 A071467 A125073
%Y A113288 Adjacent sequences: A113285 A113286 A113287 this_sequence A113289 A113290 A113291
%K A113288 sign,tabl
%O A113288 0,2
%A A113288 Paul D. Hanna (pauldhanna(AT)juno.com), Oct 23 2005

%I A035215
%S A035215 1,2,1,3,0,2,0,4,1,0,1,3,0,0,0,5,2,2,0,0,0,2,0,4,1,0,1,
%T A035215 0,2,0,2,6,1,4,0,3,2,0,0,0,2,0,0,3,0,0,0,5,1,2,2,0,0,2,
%U A035215 0,0,0,4,0,0,0,4,0,7,0,2,2,6,0,0,0,4,0,4,1,0,0,0,0,0,1
%N A035215 Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)*p^(-s)+Kronecker(m,p)*p^(-2s))^(-1) for m = 33.
%o A035215 (PARI) direuler(p=2,101,1/(1-(kronecker(m,p)*(X-X^2))-X))
%Y A035215 Sequence in context: A137587 A137639 A113288 this_sequence A071467 A125073 A071461
%Y A035215 Adjacent sequences: A035212 A035213 A035214 this_sequence A035216 A035217 A035218
%K A035215 nonn
%O A035215 1,2
%A A035215 njas

%I A071467
%S A071467 1,1,0,0,2,1,3,0,2,1,1,0,2,2,3,3,4,2,5,1,4,2,2,3,3,4,2,0,1,1,2,0,3,
%T A071467 1,2,0,5,1,4,4
%N A071467 Sprague-Grundy values for octal game .136.
%D A071467 E. R. Berlekamp, J. H. Conway and R. K. Guy, Winning Ways, Academic Press, NY, 2 vols., 1982; see Chapter 4.
%Y A071467 Sequence in context: A137639 A113288 A035215 this_sequence A125073 A071461 A091829
%Y A071467 Adjacent sequences: A071464 A071465 A071466 this_sequence A071468 A071469 A071470
%K A071467 nonn
%O A071467 1,5
%A A071467 njas and Sue Pope (pope(AT)research.att.com), May 29 2002

%I A125073
%S A125073 0,1,1,0,1,2,1,3,0,2,1,1,1,2,2,0,1,1,1,1,2,2,1,4,0,2,3,1,1,3,1,0,2,2,2,
%T A125073 0,1,2,2,4,1,3,1,1,1,2,1,1,0,1,2,1,1,4,2,4,2,2,1,2,1,2,1,6,2,3,1,1,2,3,
%U A125073 1,3,1,2,1,1,2,3,1,1,0,2,1,2,2,2,2,4,1,2,2,1,2,2,2,1,1,1,1,0,1,3,1,4,3
%N A125073 a(n) = sum of the exponents in the prime-factorization of n which are triangular numbers.
%e A125073 The prime-factorzation of 360 is 2^3 *3^2 *5^1. There are two exponents in this factorization which are triangular numbers, 1 and 3. So a(360) = 1 + 3 = 4.
%t A125073 f[n_] := Plus @@ Select[Last /@ FactorInteger[n], IntegerQ[Sqrt[8# + 1]] &];Table[f[n], {n, 110}] (*Chandler*)
%Y A125073 Cf. A125072.
%Y A125073 Sequence in context: A113288 A035215 A071467 this_sequence A071461 A091829 A104512
%Y A125073 Adjacent sequences: A125070 A125071 A125072 this_sequence A125074 A125075 A125076
%K A125073 nonn
%O A125073 1,6
%A A125073 Leroy Quet (qq-quet(AT)mindspring.com), Nov 18 2006
%E A125073 Extended by Ray Chandler (rayjchandler(AT)sbcglobal.net), Nov 19 2006

%I A071461
%S A071461 1,0,0,1,1,0,2,1,3,0,2,1,3,0,1,1,3,0,2,3,3,2,2,3,4,2,5,0,4,2,5,3,2,
%T A071461 2,3,3,2,0,3,1
%N A071461 Sprague-Grundy values for octal game .124y.
%D A071461 E. R. Berlekamp, J. H. Conway and R. K. Guy, Winning Ways, Academic Press, NY, 2 vols., 1982; see Chapter 4.
%Y A071461 Sequence in context: A035215 A071467 A125073 this_sequence A091829 A104512 A131848
%Y A071461 Adjacent sequences: A071458 A071459 A071460 this_sequence A071462 A071463 A071464
%K A071461 nonn
%O A071461 1,7
%A A071461 njas and Sue Pope (pope(AT)research.att.com), May 29 2002

%I A091829
%S A091829 1,2,1,3,0,2,1,4,1,1,0,3,0,2,1,5,0,2,1,2,1,1,0,4,1,1,0,3,0,2,1,6,1,1,0,
%T A091829 3,0,2,1,3,0,2,1,2,1,1,0,5,0,2,1,2,1,1,0,4,1,1,0,3,0,2,1,7,0,2,1,2,1,1,
%U A091829 0,4,1,1,0,3,0,2,1,4,1,1,0,3,0,2,1,3,0,2,1,2,1,1,0,6,1,1,0,3,0,2,1,3,0
%N A091829 a(1)=1; a(2n)=a(n)+1, a(2n+1)=a(n) mod 2.
%F A091829 n>0 a(2*A059009(n)+1)=0; n>0 a(n) mod 2 = 1-A059448(n); a(2^n)=n+1
%o A091829 (PARI) a(n)=if(n<2,1,if(n%2,(a(n-1)+1)%2,a(n/2)+1))
%Y A091829 Sequence in context: A071467 A125073 A071461 this_sequence A104512 A131848 A035157
%Y A091829 Adjacent sequences: A091826 A091827 A091828 this_sequence A091830 A091831 A091832
%K A091829 nonn
%O A091829 1,2
%A A091829 Benoit Cloitre (benoit7848c(AT)orange.fr), Mar 09 2004

%I A104512
%S A104512 0,0,1,0,2,1,3,0,2,1,5,3,6,2,1,0,8,3,9,2,1,4,11,7,3,5,2,1,14,4,15,0,3,7,
%T A104512 2,1,18,8,4,6,20,3,21,2,1,10,23,15,4,8,6,3,26,2,1,5,7,13,29,4,30,14,3,0,
%U A104512 2,1,33,5,9,7,35,4,36,17,3,6,2,1,39,14,5,19,41,7,4,20,12,3,44,2,1,8,13
%N A104512 Minimum number a(n) which is the first of k>1 consecutive integers, the sum of which equals n, or 0 if impossible.
%C A104512 a(n)=0 iff n=2^k and a(n)=1 iff n is a triangular number (A000217).
%D A104512 Alfred S. Posamentier, Math Charmers, Tantalizing Tidbits for the Mind, Prometheus Books, NY, 2003, page 67.
%e A104512 a(18) = 3 because 3+4+5+6 = 5+6+7 = 18 but 3 < 5.
%t A104512 f[n_] := Block[{r = Ceiling[n/2]}, If[ IntegerQ[ Log[2, n]], 0, m = Range[r]; lst = Flatten[ Table[ m[[k]], {i, r}, {j, i + 1, r}, {k, i, j}], 1]; lst[[ Position[ Plus @@@ lst, n, 1, 1][[1, 1]], 1]]]]; Table[ f[n], {n, 93}] (from Robert G. Wilson v Feb 25 2005)
%Y A104512 Cf. A104513, A104514, A104515, A104516.
%Y A104512 Sequence in context: A125073 A071461 A091829 this_sequence A131848 A035157 A087469
%Y A104512 Adjacent sequences: A104509 A104510 A104511 this_sequence A104513 A104514 A104515
%K A104512 nonn
%O A104512 1,5
%A A104512 Alfred S. Posamentier (asp2(AT)juno.com) and Robert G. Wilson v (rgwv(AT)rgwv.com), Feb 23 2005

%I A131848
%S A131848 0,0,0,2,1,3,0,2,2,4,0,2,4,4,6,2,2,3,2,3,5,4,4,3,1,1,26,27,1,3,2,3,5,7,
%T A131848 34,33,33,34,37,39,1,3,0,43,1,43,46,1,2,4,49,50,0,1,54,55,51,54,53,55,
%U A131848 57,54,51,54,57,59,62,63,63,66,69,71,1,3,4,2,73,75,69,71
%N A131848 Least nonnegative number which when added to the n-th semiprime gives a multiple of n.
%C A131848 This is to semiprimes A001358 as A068901 is to primes A000040.
%F A131848 a(n) = MIN{k=>0 such that n|(k+A001358(n))}.
%e A131848 a(1) = 0 because 1 | (0+semiprime(1)=4).
%e A131848 a(6) = 3 because 6 | (3+semiprime(3)=3+15=18).
%e A131848 a(25) = 1 because 25 | (1+semiprime(25)=1+74=75).
%Y A131848 Cf. A000040, A001358, A068901.
%Y A131848 Sequence in context: A071461 A091829 A104512 this_sequence A035157 A087469 A022328
%Y A131848 Adjacent sequences: A131845 A131846 A131847 this_sequence A131849 A131850 A131851
%K A131848 easy,nonn
%O A131848 1,4
%A A131848 Jonathan Vos Post (jvospost2(AT)yahoo.com), Oct 04 2007
%E A131848 More terms from R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jan 15 2008

%I A035157
%S A035157 1,2,1,3,0,2,2,4,1,0,1,3,0,4,0,5,2,2,2,0,2,2,2,4,1,0,1,
%T A035157 6,2,0,0,6,1,4,0,3,2,4,0,0,2,4,2,3,0,4,2,5,3,2,2,0,0,2,
%U A035157 0,8,2,4,2,0,0,0,2,7,0,2,0,6,2,0,2,4,0,4,1,6,2,0,2,0,1
%N A035157 Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)*p^(-s)+Kronecker(m,p)*p^(-2s))^(-1) for m = -33.
%o A035157 (PARI) direuler(p=2,101,1/(1-(kronecker(m,p)*(X-X^2))-X))
%Y A035157 Sequence in context: A091829 A104512 A131848 this_sequence A087469 A022328 A025641
%Y A035157 Adjacent sequences: A035154 A035155 A035156 this_sequence A035158 A035159 A035160
%K A035157 nonn
%O A035157 1,2
%A A035157 njas

%I A087469
%S A087469 0,1,0,2,1,3,0,2,4,1,3,0,5,2,4,1,6,3,0,5,2,7,4,1,6,3,0,8,5,2,7,4,1,9,6,
%T A087469 3,0,8,5,2,10,7,4,1,9,6,3,0,11,8,5,2,10,7,4,1,12,9,6,3,0,11,8,5,2,13,10,
%U A087469 7,4,1,12,9,6,3,0,14,11,8,5,2,13,10,7,4,1,15,12,9,6,3,0,14,11,8,5,2,16
%N A087469 a(n) = number of the row (counting from initial row 0) of the array R in A087468 that contains n.
%C A087469 A fractal sequence.
%H A087469 Clark Kimberling and John E. Brown, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">Partial Complements and Transposable Dispersions</a>, J. Integer Seqs., Vol. 7, 2004.
%e A087469 Northwest corner of R:
%e A087469 1 3 7 12 19
%e A087469 2 5 10 16 24
%e A087469 4 8 14 21 30
%e A087469 6 11 18 26 36
%e A087469 9 15 23 32 43
%e A087469 a(11)=3 because 11 is in row 3.
%Y A087469 Cf. A087468, A087470.
%Y A087469 Sequence in context: A104512 A131848 A035157 this_sequence A022328 A025641 A025649
%Y A087469 Adjacent sequences: A087466 A087467 A087468 this_sequence A087470 A087471 A087472
%K A087469 nonn
%O A087469 1,4
%A A087469 Clark Kimberling (ck6(AT)evansville.edu), Sep 09 2003

%I A022328
%S A022328 0,1,0,2,1,3,0,2,4,1,3,0,5,2,4,1,6,3,0,5,2,7,4,1,6,3,0,8,5,2,7,4,1,9,6,3,
%T A022328 0,8,5,2,10,7,4,1,9,6,3,11,0,8,5,2,10,7,4,12,1,9,6,3,11,0,8,5,13,2,10,7,
%U A022328 4,12,1,9,6,14,3,11,0,8,5,13,2,10,7,15,4,12,1,9,6,14,3,11,0,8,16,5,13,2
%N A022328 Exponent of 2 (value of i) in n-th number of form 2^i*3^j.
%Y A022328 Sequence in context: A131848 A035157 A087469 this_sequence A025641 A025649 A025642
%Y A022328 Adjacent sequences: A022325 A022326 A022327 this_sequence A022329 A022330 A022331
%K A022328 nonn
%O A022328 1,4
%A A022328 Clark Kimberling (ck6(AT)evansville.edu)

%I A025641
%S A025641 0,1,0,2,1,3,0,2,4,1,3,0,5,2,4,1,6,3,0,5,2,7,4,1,6,3,8,0,5,2,7,4,9,1,6,3,
%T A025641 8,0,5,10,2,7,4,9,1,6,11,3,8,0,5,10,2,7,12,4,9,1,6,11,3,8,13,0,5,10,2,7,
%U A025641 12,4,9,14,1,6,11,3,8,13,0,5,10,15,2,7,12,4,9,14,1,6,11,16,3,8,13,0,5,10
%N A025641 Exponent of 3 (value of i) in n-th number of form 3^i*6^j.
%Y A025641 Differs from A025649 at a(114).
%Y A025641 Sequence in context: A035157 A087469 A022328 this_sequence A025649 A025642 A025643
%Y A025641 Adjacent sequences: A025638 A025639 A025640 this_sequence A025642 A025643 A025644
%K A025641 nonn
%O A025641 1,4
%A A025641 David W. Wilson (davidwwilson(AT)comcast.net)

%I A025649
%S A025649 0,1,0,2,1,3,0,2,4,1,3,0,5,2,4,1,6,3,0,5,2,7,4,1,6,3,8,0,5,2,7,4,9,1,6,3,
%T A025649 8,0,5,10,2,7,4,9,1,6,11,3,8,0,5,10,2,7,12,4,9,1,6,11,3,8,13,0,5,10,2,7,
%U A025649 12,4,9,14,1,6,11,3,8,13,0,5,10,15,2,7,12,4,9,14,1,6,11,16,3,8,13,0,5,10
%N A025649 Exponent of 4 (value of i) in n-th number of form 4^i*10^j.
%Y A025649 Differs from A025641 at a(114).
%Y A025649 Sequence in context: A087469 A022328 A025641 this_sequence A025642 A025643 A127478
%Y A025649 Adjacent sequences: A025646 A025647 A025648 this_sequence A025650 A025651 A025652
%K A025649 nonn
%O A025649 1,4
%A A025649 David W. Wilson (davidwwilson(AT)comcast.net)

%I A025642
%S A025642 0,1,0,2,1,3,0,2,4,1,3,5,0,2,4,6,1,3,5,7,0,2,4,6,8,1,3,5,7,0,9,2,4,6,8,1,
%T A025642 10,3,5,7,0,9,2,11,4,6,8,1,10,3,12,5,7,0,9,2,11,4,13,6,8,1,10,3,12,5,14,
%U A025642 7,0,9,2,11,4,13,6,15,8,1,10,3,12,5,14,7,0,16,9,2,11,4,13,6,15,8,1,17,10
%N A025642 Exponent of 3 (value of i) in n-th number of form 3^i*7^j.
%Y A025642 Sequence in context: A022328 A025641 A025649 this_sequence A025643 A127478 A127472
%Y A025642 Adjacent sequences: A025639 A025640 A025641 this_sequence A025643 A025644 A025645
%K A025642 nonn
%O A025642 1,4
%A A025642 David W. Wilson (davidwwilson(AT)comcast.net)

%I A025643
%S A025643 0,1,0,2,1,3,0,2,4,1,3,5,0,2,4,6,1,3,5,7,0,2,4,6,8,1,3,5,7,9,0,2,4,6,8,
%T A025643 10,1,3,5,7,9,11,0,2,4,6,8,10,12,1,3,5,7,9,11,13,0,2,4,6,8,10,12,14,1,3,
%U A025643 5,7,9,11,13,15,0,2,4,6,8,10,12,14,16,1,3,5,7,9,11,13,15,17,0,2,4,6,8,10
%N A025643 Exponent of 3 (value of i) in n-th number of form 3^i*8^j.
%Y A025643 Sequence in context: A025641 A025649 A025642 this_sequence A127478 A127472 A004563
%Y A025643 Adjacent sequences: A025640 A025641 A025642 this_sequence A025644 A025645 A025646
%K A025643 nonn
%O A025643 1,4
%A A025643 David W. Wilson (davidwwilson(AT)comcast.net)

%I A127478
%S A127478 1,2,1,3,0,2,4,2,0,2,5,0,0,0,4,6,3,4,0,0,2,7,0,0,0,0,0,6,8,4,0,4,0,0,0,
%T A127478 4,9,0,6,0,0,0,0,0,6,10,5,0,0,8,0,0,0,0,4
%N A127478 Triangle, row sums = A018804; right border = phi(n).
%C A127478 Row sums = A018804: (1, 3, 5, 8, 9, 15, 13,...); right border = phi(n), A000010. A127477 = A054522 * A054523
%F A127478 A054523 * A054522 as infinite lower triangular matrices.
%e A127478 First few rows of the triangle are:
%e A127478 1;
%e A127478 2, 1;
%e A127478 3, 0, 2;
%e A127478 4, 2, 0, 2;
%e A127478 5, 0, 0, 0, 4;
%e A127478 6, 3, 4, 0, 0, 2;
%e A127478 7, 0, 0, 0, 0, 0, 6;
%e A127478 8, 4, 0, 4, 0, 0, 0, 4;
%e A127478 ...
%Y A127478 Cf. A054522, A054523, A018804, A000010.
%Y A127478 Sequence in context: A025649 A025642 A025643 this_sequence A127472 A004563 A098035
%Y A127478 Adjacent sequences: A127475 A127476 A127477 this_sequence A127479 A127480 A127481
%K A127478 nonn,tabl,uned
%O A127478 1,2
%A A127478 Gary W. Adamson (qntmpkt(AT)yahoo.com), Jan 15 2007

%I A127472
%S A127472 1,2,1,3,0,2,4,3,0,2,5,0,0,0,4,6,3,4,0,0,2,7,0,0,0,0,0,6,8,7,0,6,0,0,0,
%T A127472 4,9,0,8,0,0,0,0,0,6,10,5,0,0,8,0,0,0,0,4
%N A127472 Triangle, row sums = A062949, right border = phi(n).
%C A127472 Right border = phi(n), A000010; row sums = A062949: (1, 3, 5, 9, 9, 15, 13, 25,...) A127471 = A051731 * A054522
%F A127472 A054522 * A051731 as infinite lower triangular matrices.
%e A127472 First few rows of the triangle are;
%e A127472 1;
%e A127472 2, 1;
%e A127472 3, 0, 2;
%e A127472 4, 3, 0, 2;
%e A127472 5, 0, 0, 0, 4;
%e A127472 6, 3, 4, 0, 0, 2;
%e A127472 7, 0, 0, 0, 0, 0, 6;
%e A127472 8, 7, 0, 6, 0, 0, 0, 4;
%e A127472 ...
%Y A127472 Cf. A054522, A051731, A062948, A000010, A127471.
%Y A127472 Sequence in context: A025642 A025643 A127478 this_sequence A004563 A098035 A079055
%Y A127472 Adjacent sequences: A127469 A127470 A127471 this_sequence A127473 A127474 A127475
%K A127472 nonn,tabl,uned
%O A127472 1,2
%A A127472 Gary W. Adamson (qntmpkt(AT)yahoo.com), Jan 15 2007

%I A004563
%S A004563 2,1,3,0,3,0,1,0,1,3,0,0,2,2,0,0,1,0,2,1,0,0,2,1,1,3,0,3,3,2,2,0,0,
%T A004563 1,2,1,0,0,3,0,2,0,2,2,0,1,2,1,0,0,3,1,0,2,2,2,2,3,0,2,0,3,2,1,3,2,
%U A004563 3,1,2,0,0,1,0,1,1,0,0,3,2,2,1,3,2,3,3,1,2,2,0,2,3,1,3,3,0,0,1,3,2
%N A004563 Expansion of sqrt(6) in base 4.
%Y A004563 Sequence in context: A025643 A127478 A127472 this_sequence A098035 A079055 A122170
%Y A004563 Adjacent sequences: A004560 A004561 A004562 this_sequence A004564 A004565 A004566
%K A004563 nonn,base,cons
%O A004563 1,1
%A A004563 njas

%I A098035
%S A098035 1,2,1,3,0,3,1,3,0,2,1,5,0,1,0,4,1,3,1,2,0,3,1,5,1,1,0,3,2,4,1,3,0,1,0,
%T A098035 6,0,1,0,3,2,2,1,4,2,3,1,6,1,0,1,3,1,2,0,2,0,4,1,6,0,2,1,2,0,4,1,1,2,2,
%U A098035 1,7,0,1,1,1,2,3,1,4,1,4,1,4,1,1,2,5,1,2,0,3,0,1,0,6,1,1,1,1,2,3,1,4,2
%V A098035 -1,-2,-1,-3,0,-3,-1,-3,0,-2,-1,-5,0,-1,0,-4,-1,-3,-1,-2,0,-3,-1,-5,1,-1,0,-3,-2,-4,-1,
%W A098035 -3,0,-1,0,-6,0,-1,0,-3,-2,-2,-1,-4,2,-3,-1,-6,-1,0,-1,-3,-1,-2,0,-2,0,-4,-1,-6,0,-2,1,
%X A098035 -2,0,-4,-1,-1,-2,-2,-1,-7,0,-1,1,-1,-2,-3,-1,-4,1,-4,-1,-4,1,-1,-2,-5,-1,-2,0,-3,0,-1
%N A098035 Sum{k|n} mu(k+1), where mu() is Moebius function.
%e A098035 12's divisors are 1, 2, 3, 4, 6, and 12. So a(12) = mu(2)+mu(3)+mu(4)+mu(5)+mu(7)+mu(13) = -1-1+0-1-1-1 = -5.
%t A098035 f[n_] := Plus @@ MoebiusMu[Divisors[n] + 1]; Table[ f[n], {n, 105}] (from Robert G. Wilson v Nov 01 2004)
%Y A098035 Cf. A008683, A098018.
%Y A098035 Sequence in context: A127478 A127472 A004563 this_sequence A079055 A122170 A066029
%Y A098035 Adjacent sequences: A098032 A098033 A098034 this_sequence A098036 A098037 A098038
%K A098035 sign
%O A098035 1,2
%A A098035 Leroy Quet (qq-quet(AT)mindspring.com), Oct 24 2004
%E A098035 More terms from Robert G. Wilson v (rgwv(AT)rgwv.com), Nov 01 2004

%I A079055
%S A079055 0,0,0,1,1,1,1,2,1,3,0,3,1,3,2,4,0,4,1,6,2,3,0,7,2,4,1,6,0,8,1,6,1,4,2,
%T A079055 10,0,3,2,10,0,9,1,7,4,4,0,14,2,8,0,8,0,9,2,10,1,4,0,18,1,4,4,11,2,11,0,
%U A079055 7,1,11,0,20,1,5,4,9,1,13,0,16,2,5,0,21,2,6,0,12,0,21,3,9,1,5,2,23,0,7
%N A079055 Numbers of prime pairs (p,q), p<=q, such that (p+q) divides n.
%o A079055 (PARI) a(n)=sum(i=1,n,sum(j=1,i,if(n%(prime(i)+prime(j)),0,1)))
%Y A079055 Sequence in context: A127472 A004563 A098035 this_sequence A122170 A066029 A141198
%Y A079055 Adjacent sequences: A079052 A079053 A079054 this_sequence A079056 A079057 A079058
%K A079055 nonn
%O A079055 1,8
%A A079055 Benoit Cloitre (benoit7848c(AT)orange.fr), Feb 02 2003

%I A122170
%S A122170 1,1,1,2,1,3,0,3,1,4,1,5,0,4,1,4,1,6,0,4,1,4,0,10,0,6,1,5,1,12,0,5,0,6,
%T A122170 1,13,0,7,1,9,1,13,0,7,1,6,0,13,0,9,1,7,0,14,0,12,1,7,1,19,0,7,0,10,1,
%U A122170 20,0,11,1,13,1,15,0,8,0,10,1,18,0,12,1,8,0,23,0,10,1,10,0,26,0,13,0,13
%N A122170 Number of primes p <= 2n such that p+n is also a prime.
%C A122170 a(n)=0 for n in A007921.
%e A122170 a(12)=5 because only the 5 primes p=5,7,11,17,19 below 24 form other primes p+12 = 17,19,23,29,31.
%t A122170 Table[Length[Select[Select[Range[2*n], PrimeQ], PrimeQ[ #+n]&]], {n, 100}] - Ryan Propper (rpropper(AT)stanford.edu), Nov 12 2006
%Y A122170 Sequence in context: A004563 A098035 A079055 this_sequence A066029 A141198 A092093
%Y A122170 Adjacent sequences: A122167 A122168 A122169 this_sequence A122171 A122172 A122173
%K A122170 nonn
%O A122170 1,4
%A A122170 Lekraj Beedassy (blekraj(AT)yahoo.com), Aug 23 2006
%E A122170 More terms from Ryan Propper (rpropper(AT)stanford.edu), Nov 12 2006

%I A066029
%S A066029 0,0,1,2,1,3,0,3,2,3,0,4,0,3,3,2,0,4,0,4,3,3,0
%N A066029 Number of applications of d to n needed to reach a fixed point of d (i.e. 1 or 2), where d(n) = number of divisors of n.
%F A066029 a(1) = 0; a(2) = 0; a(n) = 0 if n > 2 is prime; a(n) = 1 + a(d(n)) if n > 2 is composite.
%e A066029 a(1) = a(2) = 0 since 1 and 2 are fixed points of d (d(1) = 1, d(2) = 2). a(3) = 1 since d(3) = 2. a(4) = 2 since d(d(4)) = d(3) = 2. a(6) = 3 since d(d(d(6))) = d(d(4)) = d(3) = 2.
%Y A066029 Sequence in context: A098035 A079055 A122170 this_sequence A141198 A092093 A096269
%Y A066029 Adjacent sequences: A066026 A066027 A066028 this_sequence A066030 A066031 A066032
%K A066029 nonn
%O A066029 1,4
%A A066029 Joseph L. Pe (joseph_l_pe(AT)hotmail.com), Dec 11 2001

%I A141198
%S A141198 0,1,1,2,1,3,0,3,2,3,0,5,0,2,2,3,1,5,0,5,1,1,0,7,1,2,2,4,0,6,0,4,2,2,1,
%T A141198 7,0,2,1,6,0,5,0,3,3,1,0,8,0,4,2,3,0,6,1,5,1,1,0,10,0,2,2,4,2,4,0,4,1,
%U A141198 4,0,10,0,2,2,3,0,4,0,7,2,2,0,9,2,1,1,4,0,9,0,2,1,1,1,9,0,3,3,6,0,5,0
%N A141198 a(n) = the number of divisors of n that are each one more than a power of a prime.
%C A141198 1 is considered here to be a power of a prime. 0 is not considered here to be a power of a prime.
%H A141198 Diana Mecum, <a href="http://www.research.att.com/~njas/sequences/b141198.txt">Table of n, a(n) for n = 1..1049</a>
%e A141198 The divisors of 9 are 1,3,9. 1 is one more than 0, not a power of a prime. 3 is one more than 2, a power of a prime. And 9 is one more than 8, a power of a prime. There are therefore 2 such divisors that are each one more than a power of a prime. So a(9)=2.
%Y A141198 Cf. A141197.
%Y A141198 Sequence in context: A079055 A122170 A066029 this_sequence A092093 A096269 A073312
%Y A141198 Adjacent sequences: A141195 A141196 A141197 this_sequence A141199 A141200 A141201
%K A141198 nonn
%O A141198 1,4
%A A141198 Leroy Quet (qq-quet(AT)mindspring.com), Jun 12 2008
%E A141198 Corrected and extended by Diana Mecum (diana.mecum(AT)gmail.com), Jul 05 2007

%I A092093
%S A092093 1,2,1,3,0,3,6,2,6,0,5,10,3,9,0,7,14,4,12,0,9,18,5,15,0,11,22,6,18,0,13,
%T A092093 26,7,21,0,15,30,8,24,0,17,34,9,27,0,19,38,10,30,0,21,42,11,33,0,23,46,
%U A092093 12,36,0,25,50,13,39,0,27,54,14,42,0,29,58,15,45,0,31,62,16,48,0,33
%N A092093 Back and Forth Summant S(n, _5): a(n) = sum_{i = 0..floor(2n/5)} n-5i.
%D A092093 J. Dezert, editor, Smarandacheials, Mathematics Magazine, Aurora, Canada, No. 4/2004.
%D A092093 F. Smarandache, Back and Forth Factorials, Arizona State Univ., Special Collections, 1972.
%D A092093 F. Smarandache, Back and Forth Summants, Arizona State Univ., Special Collections, 1972.
%H A092093 J. Dezert, <a href="http://www.gallup.unm.edu/~smarandache/Smarandacheials.htm">Smaran dacheials</a>
%H A092093 J. Dezert, <a href="http://www.mathematicsmagazine.com/corresp/J_Dezert/JDezert.htm">S marandacheials, "Mathematics Magazine", Canada</a>
%H A092093 F. Smarandache, <a href="http://www.gallup.unm.edu/~smarandache/Summants.htm">Summants</a>
%F A092093 a(5n) = 0; a(5n+1) = 2n+1; a(5n+2) = 4n+2; a(5n+3) = n+1; a(5n+4) = 3n+3.
%o A092093 (PARI) S(n, k=5) = local(s, x); s = n; x = n - k; while (x >= -n, s = s + x; x = x - k); s;
%Y A092093 Cf. A092096, A092399.
%Y A092093 Other values of k: A000004 (k = 1, 2), A092092 (k = 3), A027656 (k = 4).
%Y A092093 Sequence in context: A122170 A066029 A141198 this_sequence A096269 A073312 A081171
%Y A092093 Adjacent sequences: A092090 A092091 A092092 this_sequence A092094 A092095 A092096
%K A092093 nonn,easy
%O A092093 1,2
%A A092093 Jahan Tuten (jahant(AT)indiainfo.com), Mar 29 2004
%E A092093 Edited and extended by David Wasserman (wasserma(AT)spawar.navy.mil), Dec 19 2005

%I A096269
%S A096269 2,1,3,0,4,0,3,0,4,0,4,0,3,0,3,0,4,0,4,0,4,0,4,0,3,0,3,0,3,0,3,0,4,0,4,
%T A096269 0,4,0,4,0,4,0,4,0,4,0,4,0,3,0,3,0,3,0,3,0,3,0,3,0,3,0,3,0,4,0,4,0,4,0,
%U A096269 4,0,4,0,4,0,4,0,4,0,4,0,4,0,4,0,4,0,4,0,4,0,4,0,4,0,3,0,3,0,3,0,3,0,3
%N A096269 a(n) = number of distinct palindromes of length n that occur in A096268.
%D A096269 D. Damanik, Local symmetries in the period-doubling sequence, Discrete Appl. Math., 100 (2000), 115-121.
%H A096269 J.-P. Allouche, M. Baake, J. Cassaigns and D. Damanik, <a href="http://www.lri.fr/~allouche/">Palindrome complexity</a>
%F A096269 For even n >= 4, a(n) = 0; for odd n >= 5, a(n) = a(2n-1) = a(2n+1).
%F A096269 For odd n >= 5, let x be the power of 2 closest to n; if n > x then a(n) = 4, and if n < x then a(n) = 3. - David Wasserman (dwasserm(AT)earthlink.net), Nov 01 2007
%Y A096269 Cf. A096268.
%Y A096269 Sequence in context: A066029 A141198 A092093 this_sequence A073312 A081171 A062778
%Y A096269 Adjacent sequences: A096266 A096267 A096268 this_sequence A096270 A096271 A096272
%K A096269 nonn,easy,base
%O A096269 1,1
%A A096269 njas, Jun 22 2004
%E A096269 More terms from David Wasserman (dwasserm(AT)earthlink.net), Nov 01 2007

%I A073312
%S A073312 0,0,0,0,1,0,1,0,2,1,3,0,4,1,2,1,5,0,6,1,4,1,7,0,7,2,5,3,11,0,11,3,7,3,
%T A073312 9,1,13,3,7,2,14,1,14,3,6,4,16,1,16,3,11,5,20,2,15,4,13,5,22,1,23,5,10,
%U A073312 6,18,2,25,6,15,2,26,2,27,6,11,7,24,2,29,4,17,8,31,1,23,8,17,8,33,1,28
%N A073312 Number of non square-free numbers in the reduced residue system of n.
%C A073312 a(n) + A073311(n) = A000010(n).
%e A073312 n=15, there are A000005(15)=8 residues: 1, 2, 4=2^2, 7, 8=2^3, 11, 13, and 14; two of them are not square-free: 4 and 8, therefore a(15)=2.
%Y A073312 Cf. A073311, A013929, A000010, A048864, A048865.
%Y A073312 Sequence in context: A141198 A092093 A096269 this_sequence A081171 A062778 A108202
%Y A073312 Adjacent sequences: A073309 A073310 A073311 this_sequence A073313 A073314 A073315
%K A073312 nonn
%O A073312 1,9
%A A073312 Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Jul 25 2002

%I A081171
%S A081171 1,0,2,1,3,0,4,2,1,5,2,1,6,3,7,4,2,1,8,4,2,1,9,6,3,10,5,11,8,4,2,1,12,6,
%T A081171 3,13,10,5,14,7,4,2,1,15,12,6,3,16,8,4,2,1,17,14,7,4,2,1,18,9,6,3,19,16,
%U A081171 8,4,2,1,20,10,5,21,18,9,6,3,22,11,8,4,2,1,23,20,10,5,24,12,6,3,25,22
%N A081171 Triangle in which n-th row gives trajectory of n (including n itself) under the map x -> x/2 if x is even, x -> 3x-3 if x is odd, stopping when reaching one of 1, 3, 5, 15, 51.
%o A081171 (PARI) xn1m3(n) = { for(x=1,n, x1=x; print1(x1" "); while(x1>1, if(x1%2==0,x1/=2,x1=3*x1-3); print1(x1" "); ) ) }
%Y A081171 Cf. A082399.
%Y A081171 Sequence in context: A092093 A096269 A073312 this_sequence A062778 A108202 A025480
%Y A081171 Adjacent sequences: A081168 A081169 A081170 this_sequence A081172 A081173 A081174
%K A081171 easy,nonn,tabf
%O A081171 1,3
%A A081171 Cino Hilliard (hillcino368(AT)gmail.com), Apr 16 2003

%I A062778
%S A062778 0,1,2,1,3,0,4,2,2,0,5,1,6,1,1,2,7,2,8,3,2,2,9,2,6,2,5,2,10,3,11,5,4,3,
%T A062778 4,2,12,3,4,2,13,3,14,5,6,4,15,4,11,5,6,5,16,4,8,5,6,5,17,2,18,6,8,7,9,
%U A062778 4,19,7,8,6,20,5,21,8,9,8,12,6,22,8,13,8,23,6,13,8,11,7,24,4,14,9,11,8
%N A062778 Values of Moebius-transform of PrimePi function.
%F A062778 a(n)=Sum {Pi[n/d]*mu[d]}, d divides n.
%e A062778 n=12, divisors=D(12)={1,2,3,4,6,12}, Pi[12/divisor)]={5,3,2,2,1,0}, mu[divisors]={1,-1,-1,0,1,0}, Sum=5*1-3*1-2*1+0+1*1+0=1, thus a(12)=1; for p=p[n] prime, Pi[p/divisor]={n,0}, mu[{1,p}]={1,-1}, Sum=1*n+0=n, so a(p[n])=n.
%t A062778 f[n_] := Block[{d = Divisors@n}, Plus @@ (MoebiusMu /@ (n/d)*PrimePi /@ d)]; Array[f, 94] (from Robert G. Wilson v (rgwv(at)rgwv.com), Dec 07 2005)
%Y A062778 Cf. A007444, A007445.
%Y A062778 Sequence in context: A096269 A073312 A081171 this_sequence A108202 A025480 A088002
%Y A062778 Adjacent sequences: A062775 A062776 A062777 this_sequence A062779 A062780 A062781
%K A062778 nonn
%O A062778 1,3
%A A062778 Labos E. (labos(AT)ana.sote.hu), Jul 18 2001

%I A108202
%S A108202 0,0,1,0,2,1,3,0,4,2,5,1,6,3,7,0,8,4,9,2,1,5,0,1,1,6,1,3,1,7,2,0,1,8,3,
%T A108202 4,1,9,4,2,1,1,5,5,1,0,6,1,1,1,7,6,1,1,8,3,1,1,9,7,2,2,0,0,2,1,1,8,2,3,
%U A108202 2,4,2,1,3,9,2,4,4,2,2,1,5,1,2,5,6,5,2,1,7,0,2,6,8,1,2,1,9,1,3,7,0,6,3
%N A108202 a(2n) = the natural counting digits (0, 1, 2, ... 8, 9, 1, 0, 1, 1, 1, 2, 1, 3, ...); a(2n+1) = a(n).
%C A108202 Fractal sequence (Kimberling-type) based upon the counting digits.
%Y A108202 Cf. A025480, A003602.
%Y A108202 Sequence in context: A073312 A081171 A062778 this_sequence A025480 A088002 A074894
%Y A108202 Adjacent sequences: A108199 A108200 A108201 this_sequence A108203 A108204 A108205
%K A108202 base,easy,nonn
%O A108202 0,5
%A A108202 Juliette Bruyndonckx and Alexandre Wajnberg (alexandre.wajnberg(AT)ulb.ac.be), Jun 15 2005
%E A108202 More terms from Joshua Zucker (joshua.zucker(AT)stanfordalumni.org), May 18 2006

%I A025480
%S A025480 0,0,1,0,2,1,3,0,4,2,5,1,6,3,7,0,8,4,9,2,10,5,11,1,12,6,13,3,14,7,15,0,
%T A025480 16,8,17,4,18,9,19,2,20,10,21,5,22,11,23,1,24,12,25,6,26,13,27,3,28,14,
%U A025480 29,7,30,15,31,0,32,16,33,8,34,17,35,4,36,18,37,9,38,19,39,2,40,20,41,10
%N A025480 a(2n) = n, a(2n+1) = a(n).
%C A025480 These are the nim-values for heaps of n beans in the game where you're allowed to take up to half of the beans in a heap. - R. K. Guy, Mar 30 2006
%C A025480 When n>0 is written as (2k+1)*2^j then k=A000265(n-1) and j=A007814(n), so: when n is written as (2k+1)*2^j-1 then k=A025480(n) and j=A007814(n+1), when n>1 is written as (2k+1)*2^j+1 then k=A025480(n-2) and j=A007814(n-1)
%C A025480 According to the comment from Deuard Worthen, this may be regarded as a triangle where row r=1,2,3... has length 2^(r-1) and values T[r,2k-1]=T[r-1,k], T[r,2k]=2^(r-1)+k-1, i.e. previous row gives 1st, 3rd, 5th... term and 2nd, 4th... terms are numbers 2^(r-1),...,2^r-1 (i.e. those following the last one from the previous row). - M. F. Hasler (www.univ-ag.fr/~mhasler), May 03 2008
%D A025480 L. Levine, Fractal sequences and restricted Nim, Ars Comb., Ars Combin. 80 (2006), 113-127.
%H A025480 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/sequences/b025480.txt">Table of n, a(n) for n = 0..10000</a>
%H A025480 L. Levine, <a href="http://arXiv.org/abs/math.CO/0409408">Fractal sequences and restricted Nim</a>
%H A025480 R. Stephan, <a href="http://www.research.att.com/~njas/sequences/somedcgf.html">Some divide-and-conquer sequences ...</a>
%H A025480 R. Stephan, <a href="http://www.research.att.com/~njas/sequences/a079944.ps">Tabl