The Database of Integer Sequences, Part 10
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
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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 A096871
%S A096871 2,1,3,5,86,75,1393,2494,162402,148157,7489051,13853391,2009866406,
%T A096871 1878178855,185066460993,174321510430,89622746262146,28317869903523,
%U A096871 13807296146243251,26310320926601689,10551742962933162102
%N A096871 Denominator of the fraction whose continued fraction is 0, barover{1, 2, ..., n}.
%e A096871 For n=4, the continued fraction of (1/5)*(-9 + 2*Sqrt(39)) = 0, {1,2,3,4} where what is between the parenthesis is infinity repeated.
%t A096871 Table[ Denominator[ FromContinuedFraction[{0, Range[n]}]], {n, 21}]
%Y A096871 Sequence in context: A092944 A049902 A096631 this_sequence A077890 A077877 A081520
%Y A096871 Adjacent sequences: A096868 A096869 A096870 this_sequence A096872 A096873 A096874
%K A096871 nonn
%O A096871 1,1
%A A096871 Robert G. Wilson v (rgwv(AT)rgwv.com), Jul 12 2004
%I A077890
%S A077890 1,1,0,2,1,3,6,0,11,11,12,34,11,57,78,36,193,121,264,506,23,1035,990,1080,
%T A077890 3059,899,5220,7018,3421,17457,10614,24300,45529,3071,94128,87986,100271,
%U A077890 276243,75702,476784,628187,325381,1581756,930994,2232517,4094505,370530
%V A077890 1,1,0,-2,-1,3,6,0,-11,-11,12,34,11,-57,-78,36,193,121,-264,-506,23,1035,990,-1080,
%W A077890 -3059,-899,5220,7018,-3421,-17457,-10614,24300,45529,-3071,-94128,-87986,100271,
%X A077890 276243,75702,-476784,-628187,325381,1581756,930994,-2232517,-4094505,370530
%N A077890 Expansion of (1-x)^(-1)/(1+x^2+2*x^3).
%Y A077890 Sequence in context: A049902 A096631 A096871 this_sequence A077877 A081520 A129116
%Y A077890 Adjacent sequences: A077887 A077888 A077889 this_sequence A077891 A077892 A077893
%K A077890 sign
%O A077890 0,4
%A A077890 njas, Nov 17 2002
%I A077877
%S A077877 1,2,1,3,6,0,16,23,8,69,75,72,292,224,431,1170,531,2241,4474,524,10664,
%T A077877 16185,4620,47655,54601,45328,202184,166128,283569,818010,417001,1502587,
%U A077877 3154598,566424,7245360,11532807,2391488,32702461,39452291,28344120,139951164
%V A077877 1,2,1,-3,-6,0,16,23,-8,-69,-75,72,292,224,-431,-1170,-531,2241,4474,524,-10664,
%W A077877 -16185,4620,47655,54601,-45328,-202184,-166128,283569,818010,417001,-1502587,
%X A077877 -3154598,-566424,7245360,11532807,-2391488,-32702461,-39452291,28344120,139951164
%N A077877 Expansion of (1-x)^(-1)/(1-x+2*x^2+x^3).
%Y A077877 Sequence in context: A096631 A096871 A077890 this_sequence A081520 A129116 A010251
%Y A077877 Adjacent sequences: A077874 A077875 A077876 this_sequence A077878 A077879 A077880
%K A077877 sign
%O A077877 0,2
%A A077877 njas, Nov 17 2002
%I A081520
%S A081520 1,1,2,1,3,6,1,2,4,6,1,5,10,15,20,1,2,3,4,6,8,1,7,14,21,28,35,42,1,2,4,
%T A081520 6,8,10,12,14,1,3,6,9,12,15,18,21,24,1,2,4,5,6,8,10,12,14,15,1,11,22,33,
%U A081520 44,55,66,77,88,99,110,1,2,3,4,6,8,9,10,12,14,15,16,1,13,26,39,52,65,78
%N A081520 Triangle read by rows in which row n gives n smallest numbers having common factors with n.
%e A081520 1; 1,2; 1,3,6; 1,2,4,6; 1,5,10,15,20; 1,2,3,4,6,8; 1,7,14,21,28,35,42; ...
%Y A081520 Cf. A081518, A081519.
%Y A081520 Sequence in context: A096871 A077890 A077877 this_sequence A129116 A010251 A051537
%Y A081520 Adjacent sequences: A081517 A081518 A081519 this_sequence A081521 A081522 A081523
%K A081520 nonn,tabl
%O A081520 1,3
%A A081520 Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Mar 27 2003
%E A081520 More terms from Ryan Propper (rpropper(AT)stanford.edu), Nov 05 2005
%I A129116
%S A129116 1,1,2,1,3,6,1,2,15,24,1,2,3,105,120,1,2,3,4,945,720,1,2,3,4,10,10395,
%T A129116 5040
%N A129116 Multifactorial array A[k,n] = k-tuple factorial of n, for positive n, by antidiagonals.
%C A129116 The term "Quintuple factorial numbers" is also used for the sequences A008546, A008548, A052562, A047055, A047056 which have a different definition. The definition given here is the one commonly used. This problem exists for the other rows as well. "n!2" = n!!, "n!3" = n!!!, "n!4" = n!!!!, etcetera. Main diagonal is A[n,n] = n!n = n.
%H A129116 Eric Weisstein's World of Mathematics, Multifactorial.
%F A129116 A[k,n] = n!k, by antidiagonals.
%e A129116 Table begins:
%e A129116 k / A[k,n]
%e A129116 1.|.1.2.6.24.120.720.5040.40320.362880.3628800...=A000142.
%e A129116 2.|.1.2.3..8..15..48..105...384....945....3840...=A006882.
%e A129116 3.|.1.2.3..4..10..18...28....80....162.....280...=A007661.
%e A129116 4.|.1.2.3..4...5..12...21....32.....45.....120...=A007662.
%e A129116 5.|.1.2.3..4...5...6...14....24.....36......50...=A085157.
%e A129116 6.|.1.2.3..4...5...6....7....16.....27......40...=A085158.
%Y A129116 Cf. A000142, A006882, A007661, A007662, A085157, A085158.
%Y A129116 Sequence in context: A077890 A077877 A081520 this_sequence A010251 A051537 A036038
%Y A129116 Adjacent sequences: A129113 A129114 A129115 this_sequence A129117 A129118 A129119
%K A129116 easy,nonn,tabl
%O A129116 1,3
%A A129116 Jonathan Vos Post (jvospost2(AT)yahoo.com), May 24 2007
%I A010251
%S A010251 2,1,3,6,1,3,17,1,7,3,3,11,2,92,1,3,1,3,1,2,2,26,2,1,20,
%T A010251 1,4,2,10,43,2,1,4,2,3,1,11,2,1,1,4,1,1,1,10,1,1,1,2,1,
%U A010251 12,35,1,1,2,1,1,5,1,3,204,4,1,1,5,2,13,3,2,5,1,6,253,3
%N A010251 Continued fraction for cube root of 21.
%H A010251 G. Xiao, Contfrac
%Y A010251 Sequence in context: A077877 A081520 A129116 this_sequence A051537 A036038 A078760
%Y A010251 Adjacent sequences: A010248 A010249 A010250 this_sequence A010252 A010253 A010254
%K A010251 nonn,cofr
%O A010251 0,1
%A A010251 njas
%I A051537
%S A051537 1,2,1,3,6,1,4,2,12,1,5,10,15,20,1,6,3,2,6,30,1,7,14,21,28,35,42,1,8,4,
%T A051537 24,2,40,12,56,1,9,18,3,36,45,6,63,72,1,10,5,30,10,2,15,70,20,90,1,11,
%U A051537 22,33,44,55,66,77,88,99,110,1,12,6,4,3,60,2,84,6,12,30,132,1,13,26,39
%N A051537 Triangle T(i,j) read by rows: the j-th term of the i-th row is LCM(i,j)/GCD(i,j).
%C A051537 The first term of the k-th row is k. The first leading diagonal contains all 1's. The second leading diagonal contains twice triangular numbers = n*(n-1). a(p) = (p^3 - p^2 + 2)/2, where p is a prime. Proof: The p-th row is p, 2p, 3p, ..., (p-2)*p, (p-1)*p, 1 The sum = p*( 1+2+3+...+ (p-2) + (p-1)) + 1 = p*(p-1)*(p)/2 + 1 etc. - Robert G. Wilson v (rgwv(AT)rgwv.com), May 10 2002
%C A051537 In the array T(i,j)=T(j,i), the natural extension of the triangle, each set of rows and columns with common indices [d1,d2,...,ds] define a group multiplication table on their grid, if the d1,d2,..., ds are the set of divisors of a square-free number [A. Jorza]. - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), May 03 2007
%D A051537 A. Jorza, Groups of Divisors, Solution to problem 10893, Amer. Math. Monthly, 2003, 441-443.
%e A051537 1; 2, 1; 3, 6, 1; 4, 2, 12, 1; 5, 10, 15, 20, 1; 6, 3, 2, 6, 30, 1; 7, 14, 21, 28, 35, 42, 1; 8, 4, 24, 2, 40, 12, 56, 1; .....
%t A051537 Flatten[ Table[ LCM[i, j] / GCD[i, j], {i, 1, 13}, {j, 1, i}]]
%Y A051537 Diagonals give A002378, A070260, A070261, A070262. Row sums give A056789.
%Y A051537 Sequence in context: A081520 A129116 A010251 this_sequence A036038 A078760 A103280
%Y A051537 Adjacent sequences: A051534 A051535 A051536 this_sequence A051538 A051539 A051540
%K A051537 nonn,tabl
%O A051537 1,2
%A A051537 njas and Amarnath Murthy (amarnath_murthy(AT)yahoo.com), May 10, 2002
%E A051537 More terms from Robert G. Wilson v (rgwv(AT)rgwv.com), May 10 2002
%I A036038
%S A036038 1,1,2,1,3,6,1,4,6,12,24,1,5,10,20,30,60,120,1,6,15,20,30,60,90,120,180,
%T A036038 360,720,1,7,21,35,42,105,140,210,210,420,630,840,1260,2520,5040,1,8,28,
%U A036038 56,70,56,168,280,420,560,336,840,1120,1680,2520,1680,3360,5040,6720
%N A036038 Triangle of multinomial coefficients.
%C A036038 The number of terms in the n-th row is the number of partition of n. - Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Sep 21 2002
%C A036038 For each n, the partitions are ordered by length and then lexicographically, which is different from the usual practice of ordering all partitions lexicographically. (T. D. Noe)
%D A036038 Abramowitz and Stegun, Handbook, p. 831, column labeled "M_1".
%H A036038 T. D. Noe, Rows n=1..25 of triangle, flattened
%H A036038 M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, December 1972 [alternative scanned copy].
%e A036038 1; 1,2; 1,3,6; 1,4,6,12,24; 1,5,10,20,30,60,120; ...
%t A036038 Flatten[Table[Apply[Multinomial, Reverse[Sort[Partitions[i], Length[ #1]>Length[ #2]&]], {1}], {i,9}]] - T. D. Noe (noe(AT)sspectra.com), Nov 03 2006
%Y A036038 Cf. A036036-A036040. Different from A078760. Row sums give A005651.
%Y A036038 Sequence in context: A129116 A010251 A051537 this_sequence A078760 A103280 A046899
%Y A036038 Adjacent sequences: A036035 A036036 A036037 this_sequence A036039 A036040 A036041
%K A036038 nonn,easy,nice,tabf
%O A036038 1,3
%A A036038 njas
%E A036038 More terms from David W. Wilson (davidwwilson(AT)comcast.net) and wouter.meeussen(AT)pandora.be.
%I A078760
%S A078760 1,1,1,2,1,3,6,1,4,6,12,24,1,5,10,20,30,60,120,1,6,15,30,20,60,120,90,
%T A078760 180,360,720,1,7,21,42,35,105,210,140,210,420,840,630,1260,2520,5040,1,
%U A078760 8,28,56,56,168,336,70,280,420,840,1680,560,1120,1680,3360,6720,2520
%N A078760 Combinations of a partition: number of ways to label a partition (of size n) with numbers 1 to n.
%C A078760 This is a function of the individual partitions of an integer. The number of values in each line is given by A000041; thus lines 0 to 5 of the sequence are (1), (1), (1,2), (1,3,6), (2,4,6,12,24). The partitions in each line are ordered with the largest part sizes first, so the line 4 indices are [4], [3,1], [2,2], [2,1,1], and [1,1,1,1]. Note that exponents are often used to represent repeated values in a partition, so the last index could instead be written [1^4]. The combination function (sequence A007318) C(n,m) = C([m,n-m]).
%H A078760 T. D. Noe, Rows n=0..25 of triangle, flattened
%H A078760 Index entries for triangles and arrays related to Pascal's triangle.
%F A078760 C([]) = (\Sigma a_i)! / \Pi a_i !
%e A078760 C([2,1]) = 3 for the labelings ({1,2},{3}), ({1,3},{2}) and ({2,3},{2}).
%t A078760 Flatten[Table[Apply[Multinomial, Partitions[i], {1}], {i,0,25}] (from T. D. Noe, Oct 14 2007)
%Y A078760 Different from A036038.
%Y A078760 Sequence in context: A010251 A051537 A036038 this_sequence A103280 A046899 A035206
%Y A078760 Adjacent sequences: A078757 A078758 A078759 this_sequence A078761 A078762 A078763
%K A078760 nice,easy,nonn,tabl
%O A078760 0,4
%A A078760 Frank Adams-Watters (FrankTAW(AT)Netscape.net), Jan 08 2003
%I A103280
%S A103280 1,1,2,1,3,6,1,4,9,16,1,5,14,27,44,1,6,21,48,81,120,1,7,30,85,164,243,
%T A103280 328,1,8,41,144,341,560,729,896,1,9,54,231,684,1365,1912,2187,2448,1,10,
%U A103280 69,352,1289,3240,5461,6528,6561,6688,1,11,86,513,2276,7175,15336,21845
%N A103280 Array read by antidiagonals, generated by the matrix M = [1,1,1;1,N,1;1,1,1];.
%C A103280 Consider the matrix M = [1,1,1;1,N,1;1,1,1];
%C A103280 Characteristic polynomial of M is x^3 + (-N - 2)*x^2 + (2*N - 2)*x.
%C A103280 Now (M^n)[1,2] is equivalent to the recursion a(1) = 1, a(2) = N+2, a(n) = (N+2)a(n-1)+(2N-2)a(n-2). (This also holds for negative N and fractional N.)
%C A103280 a(n+1)/a(n) converges to the upper root of the characteristic polynomial ((N + 2) + sqrt((N - 2)^2 + 8))/2 for n to infinity.
%C A103280 Columns of array follow the polynomials:
%C A103280 0
%C A103280 1
%C A103280 N + 2
%C A103280 N^2 + 2*N + 6
%C A103280 N^3 + 2*N^2 + 8*N + 16
%C A103280 N^4 + 2*N^3 + 10*N^2 + 24*N + 44
%C A103280 N^5 + 2*N^4 + 12*N^3 + 32*N^2 + 76*N + 120
%C A103280 N^6 + 2*N^5 + 14*N^4 + 40*N^3 + 112*N^2 + 232*N + 328
%C A103280 N^7 + 2*N^6 + 16*N^5 + 48*N^4 + 152*N^3 + 368*N^2 + 704*N + 896
%C A103280 N^8 + 2*N^7 + 18*N^6 + 56*N^5 + 196*N^4 + 528*N^3 + 1200*N^2 + 2112*N + 2448
%C A103280 etc.
%F A103280 T(N, 1)=1, T(n, 2)=N+2, T(N, n)=(N+2)*T(N, n-1)-(2*N-2)*T(N, n-2)))
%e A103280 Array begins:
%e A103280 1,2,6,16,44,120,328,896,2448,6688,...
%e A103280 1,3,9,27,81,243,729,2187,6561,19683, ...
%e A103280 1,4,14,48,164,560,1912,6528,22288,76096,...
%e A103280 1,5,21,85,341,1365,5461,21845,87381,349525,...
%e A103280 1,6,30,144,684,3240,15336,72576,343440,1625184,...
%e A103280 1,7,41,231,1289,7175,39913,221991,1234633,6866503,...
%e A103280 ...
%o A103280 (PARI) T12(N, n) = if(n==1,1,if(n==2,N+2,(N+2)*T12(N,n-1)-(2*N-2)*T12(N,n-2))) for(k=0,10,print1(k,": ");for(i=1,10,print1(T12(k,i),","));print())
%Y A103280 Cf. A103279 (for (M^n)[1, 1]), A002605 and A080953 (for N=0), A000244 (for N=1), A007070 (for N=2), A002450 (for N=3), A002450 (for N=4), A030192 (for N=5) A006131 (for N=-1), A000400 (bisection for N=-2), A015443 (for N=-3), A083102 (for N=-4).
%Y A103280 Sequence in context: A051537 A036038 A078760 this_sequence A046899 A035206 A115196
%Y A103280 Adjacent sequences: A103277 A103278 A103279 this_sequence A103281 A103282 A103283
%K A103280 nonn,tabl
%O A103280 0,3
%A A103280 Lambert Klasen (lambert.klasen(AT)gmx.net), Jan 27 2005
%I A046899
%S A046899 1,1,2,1,3,6,1,4,10,20,1,5,15,35,70,1,6,21,56,126,252,1,7,28,84,210,
%T A046899 462,924,1,8,36,120,330,792,1716,3432,1,9,45,165,495,1287,3003,6435,
%U A046899 12870,1,10,55,220,715,2002,5005,11440,24310,48620,1,11,66,286,1001
%N A046899 Triangle in which n-th row is {C(n+k,k), k=0..n}, n >= 0.
%C A046899 Row sums = A134391: (1, 3, 10, 35, 126, 362, 1726,...). - Gary W. Adamson (qntmpkt(AT)yahoo.com), Oct 23 2007
%D A046899 H. W. Gould, A class of binomial sums and a series transformation, Utilitas Math., 45 (1994), 71-83.
%e A046899 1
%e A046899 1, 2
%e A046899 1, 3, 6
%e A046899 1, 4, 10, 20
%e A046899 1, 5, 15, 35, 70
%e A046899 1, 6, 21, 56, 126, 252
%e A046899 1, 7, 28, 84, 210, 462, 924
%e A046899 1, 8, 36, 120, 330, 792, 1716, 3432
%e A046899 1, 9, 45, 165, 495, 1287, 3003, 6435, 12870
%e A046899 1, 10, 55, 220, 715, 2002, 5005, 11440, 24310, 48620
%e A046899 1, 11, 66, 286, 1001, 3003, 8008, 19448, 43758, 92378, 184756
%p A046899 for n from 0 to 10 do seq( binomial(n+m,n), m = 0 .. n) od; - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Dec 09 2007
%Y A046899 Cf. A046900, A134391.
%Y A046899 Sequence in context: A036038 A078760 A103280 this_sequence A035206 A115196 A093346
%Y A046899 Adjacent sequences: A046896 A046897 A046898 this_sequence A046900 A046901 A046902
%K A046899 nonn,tabl,easy,nice
%O A046899 0,3
%A A046899 njas
%E A046899 More terms from James A. Sellers (sellersj(AT)math.psu.edu)
%I A035206
%S A035206 1,2,1,3,6,1,4,12,6,12,1,5,20,20,30,30,20,1,6,30,30,15,60,120,20,60,90,
%T A035206 30,1,7,42,42,42,105,210,105,105,140,420,140,105,210,42,1,8,56,56,56,28,
%U A035206 168,336,336,168,168,280,840,420,840,70,280,1120,560,168,420,56,1,9,72
%N A035206 Number of multisets associated with least integer of each prime signature.
%C A035206 Multiplying by 1; 1,2; 1,3,6; 1,4,6,12,24; ... (A036038) yields 1; 2,2; 3,18,6; 4,48,36,144,24; ... in which the groups sum to 1; 4; 27; 256; .... (A000312).
%C A035206 a(n,k) enumerates distributions of n identical objects (balls) into m of alltogether n distinguishable boxes. The k-th partition of n, taken in the Abramowitz-Stegun (A-St) order, specifies the occupation of the m =m(n,k)= A036043(n,k) boxes. m = m(n,k) is the number of parts of the k-th partition of n. For the A-St ordering see pp.831-2 of the reference given in A117506. W. Lang, Nov 13 2007.
%C A035206 The sequence of row lengths is p(n)= A000041(n) (partition numbers) [1, 2, 3, 5, 7, 11, 15, 22, 30, 42,...].
%C A035206 For the A-St order of partitions see the Abramowitz-Stegun reference given in A117506.
%H A035206 M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, December 1972 [alternative scanned copy].
%H A035206 W. Lang, First 10 rows and more.
%F A035206 a(n,k) = A048996(n,k)*binomial(n,m(n,k)),n>=1, k=1,...,p(n), and m(n,k):=A036043(n,k) gives the number of parts of the k-th partition of n.
%e A035206 1; 2,1; 3,6,1; 4,12,6,12,1; 5,20,20,30,30,20,1; ...
%e A035206 a(5,5) relates to the partition (1,2^2) of n=5. Here m=3, and 5 indistinguishable (identical)
%e A035206 balls are put into boxes b1,...,b5 with m=3 boxes occupied; one with one ball and two with two balls.
%e A035206 Therefore a(5,5) = binomial(5,3)*3!/(1!*2!) = 10*3 = 30. W. Lang, Nov 13 2007.
%Y A035206 Cf. A036038, A048996, A049009.
%Y A035206 Cf. A001700 (row sums).
%Y A035206 Cf. A103371(n-1, m-1) (triangle obtained after summing in every row the numbers with like part numbers m).
%Y A035206 Sequence in context: A078760 A103280 A046899 this_sequence A115196 A093346 A115597
%Y A035206 Adjacent sequences: A035203 A035204 A035205 this_sequence A035207 A035208 A035209
%K A035206 nonn,tabf,easy
%O A035206 1,2
%A A035206 Alford Arnold (Alford1940(AT)aol.com)
%E A035206 More terms from Joshua Zucker (joshua.zucker(AT)stanfordalumni.org), Jul 27 2006
%I A115196
%S A115196 1,1,2,1,3,6,1,4,15,13,1,5,30,82,37,1,6,51,301,578,106,1,7,80,842,
%T A115196 4985,6021,409,1,8,117,1995,27107,142276,101267,1896,1,9,164,4210,
%U A115196 112225,1724440,7269487,2882460,12171
%N A115196 Triangle read by rows formed from nonzero entries in table of number of graphs on n nodes with clique number k.
%H A115196 Keith M. Briggs, Combinatorial Graph Theory
%e A115196 Table: number of graphs on n nodes with clique number k
%e A115196 n = .1...2...3...4....5....6.....7......8........9.......10.
%e A115196 k ----------------------------------------------------------
%e A115196 2....0...1...2...6...13...37...106....409.....1896....12171 = A052450
%e A115196 3....0...0...1...3...15...82...578...6021...101267..2882460 = A052451
%e A115196 4....0...0...0...1...4....30...301...4985...142276..7269487 = A052452
%e A115196 5....0...0...0...0...1....5.....51....842....27107..1724440 = A077392
%e A115196 6....0...0...0...0...0....1......6.....80.....1995...112225 = A077393
%e A115196 7....0...0...0...0...0....0......1......7......117.....4210 = A077394
%e A115196 8....0...0...0...0...0....0......0......1........8......164.
%e A115196 9....0...0...0...0...0....0......0......0........1........9
%e A115196 10...0...0...0...0...0....0......0......0........0........1.
%Y A115196 Sequence in context: A103280 A046899 A035206 this_sequence A093346 A115597 A103371
%Y A115196 Adjacent sequences: A115193 A115194 A115195 this_sequence A115197 A115198 A115199
%K A115196 nonn,tabl
%O A115196 2,3
%A A115196 njas, based on email from Keith M. Briggs, Apr 03 2006
%I A093346
%S A093346 1,1,2,1,3,6,1,4,15,22,1,5,28,95,91,1,6,45,252,690,408,1,7,66,525,2618,
%T A093346 5481,1938,1,8,91,946,7095,29848,46376,9614,1,9,120,1547,15741,105417,
%U A093346 363216,411255,49335,1,10,153,2360,30576,288288,1673535
%N A093346 Array read by antidiagonals: T(r,n) = number of two-stack sortable r-permutations.
%H A093346 D. Xu, Generalizations of two-stack-sortable permutations, PhD thesis.
%F A093346 T(r, n) = 2(r+1) * ((2r+1)n)!/[n!*(2rn+2)! ]
%o A093346 (PARI) T(r,n)=2*(r+1)*((2*r+1)*n)!/(n!*(2*r*n+2)!)
%Y A093346 Sequence in context: A046899 A035206 A115196 this_sequence A115597 A103371 A120257
%Y A093346 Adjacent sequences: A093343 A093344 A093345 this_sequence A093347 A093348 A093349
%K A093346 nonn,tabl
%O A093346 1,3
%A A093346 Ralf Stephan, Apr 26 2004
%I A115597
%S A115597 1,1,2,1,3,6,1,4,16,12,1,5,31,84,34,1,6,52,318,579,87,1,7,81,867,
%T A115597 5366,5721,302,1,8,118,2028,28722,155291,87381,1118,1,9,165,4251,
%U A115597 115391,1919895,7855628,2104349,5478
%N A115597 Triangle read by rows: let a(n,k) = number of graphs on n nodes with chromatic number k; T(n,k) = a(n,n-k), n >= 2, k=0..n-2.
%H A115597 Keith M. Briggs, Combinatorial Graph Theory
%e A115597 Table of values of a(n,k): number of graphs on n nodes with chromatic number k
%e A115597 n. = .1...2...3...4....5....6.....7......8........9.......10
%e A115597 k.----------------------------------------------------------
%e A115597 2....0...1...2...6...12...34....87....302.....1118.....5478... = A076278
%e A115597 3....0...0...1...3...16...84...579...5721....87381..2104349... = A076279
%e A115597 4....0...0...0...1....4...31...318...5366...155291..7855628... = A076280
%e A115597 5....0...0...0...0....1....5....52....867....28722..1919895... = A076281
%e A115597 6....0...0...0...0....0....1.....6.....81.....2028...115391... = A076282
%e A115597 7....0...0...0...0....0....0.....1......7......118.....4251
%e A115597 8....0...0...0...0....0....0.....0......1........8......165
%e A115597 9....0...0...0...0....0....0.....0......0........1........9
%e A115597 10...0...0...0...0....0....0.....0......0........0........1
%e A115597 Triangle begins:
%e A115597 1
%e A115597 1 2
%e A115597 1 3 6
%e A115597 1 4 16 12
%e A115597 1 5 31 84 34
%e A115597 1 6 52 318 579 87
%e A115597 1 7 81 867 5366 5721 302
%e A115597 1 8 118 2028 28722 155291 87381 1118
%e A115597 1 9 165 4251 115391 1919895 7855628 2104349 5478
%Y A115597 Cf. A076278-A076282.
%Y A115597 Sequence in context: A035206 A115196 A093346 this_sequence A103371 A120257 A059298
%Y A115597 Adjacent sequences: A115594 A115595 A115596 this_sequence A115598 A115599 A115600
%K A115597 nonn,tabl
%O A115597 2,3
%A A115597 njas, based on email from Keith Briggs, Mar 14 2006
%I A103371
%S A103371 1,2,1,3,6,1,4,18,12,1,5,40,60,20,1,6,75,200,150,30,1,7,126,525,700,315,
%T A103371 42,1,8,196,1176,2450,1960,588,56,1,9,288,2352,7056,8820,4704,1008,72,1,
%U A103371 10,405,4320,17640,31752,26460,10080,1620,90,1,11,550,7425,39600,97020
%N A103371 Number triangle T(n,k)=C(n,n-k)C(n+1,n-k).
%C A103371 Columns include A000027, A002411, A004302. Row sums are C(2n+1,n+1) or A001700.
%C A103371 T(n-1,k-1) is the number of possibilities to put n identical objects into k of alltogether n distinguishable boxes. See the partition array A035206 from which this triangle arises after summing over all entries related to partitions with fixed part number k.
%F A103371 Number triangle T(n, k)=C(n, n-k)C(n+1, n-k)=C(n, k)C(n+1, k+1); Column k of this triangle has g.f. sum{j=0..k, C(k, j)C(k+1, j)x^(k+j)}/(1-x)^(2k+2); coefficients of the numerators are the rows of the reverse triangle C(n, k)C(n+1, k).
%F A103371 T(n, k)=C(n, k)*sum{j=0..n, C(n-j, k), j, 0, n-k}; - Paul Barry (pbarry(AT)wit.ie), Jan 12 2006
%F A103371 T(n,k)= (n+1-k)*N(n+1,k+1), with N(n,k):=A001263(n,k), the Narayana triangle (with offset [1,1)]
%F A103371 O.g.f.: ((1-(1-y)*x)/sqrt((1-(1+y)*x)^2-4*x^2*y) -1)/2, (from o.g.f. of A001263, Narayana triangle). W. Lang, Nov 13 2007.
%F A103371 Matrix product of A007318 and A122899. O.g.f. for row n: (1-x)^n*P(n,1,0,(1+x)/(1-x)) = (1-x)^(n-1)*(Legendre_P(n,x) - Legendre_P(n+1,x)), where P(n,a,b,x) denotes the Jacobi polynomial. O.g.f. for column k: x^k/(1-x)^(k+2)*P(k,0,1,(1+x)/(1-x)). Compare with A008459. - Peter Bala (pbala(AT)toucansurf.com), Jan 24 2008
%e A103371 Rows start {1}, {2,1}, {3,6,1}, {4,18,12,1},...
%Y A103371 Cf. A008459, A122899.
%Y A103371 Sequence in context: A115196 A093346 A115597 this_sequence A120257 A059298 A059434
%Y A103371 Adjacent sequences: A103368 A103369 A103370 this_sequence A103372 A103373 A103374
%K A103371 easy,nonn,tabl
%O A103371 0,2
%A A103371 Paul Barry (pbarry(AT)wit.ie), Feb 03 2005
%I A120257
%S A120257 1,2,1,3,6,1,4,20,20,1,5,50,175,70,1,6,105,980,1764,252,1,7,196,4116,24696,
%T A120257 19404,924,1,8,336,14112,232848,731808,226512,3432,1,9,540,41580,1646568,
%U A120257 16818516,24293412,2760615,12870,1,10,825,108900,9343620,267227532,1447482465
%V A120257 1,2,-1,3,-6,-1,4,-20,-20,1,5,-50,-175,70,1,6,-105,-980,1764,252,-1,7,-196,-4116,24696,
%W A120257 19404,-924,-1,8,-336,-14112,232848,731808,-226512,-3432,1,9,-540,-41580,1646568,
%X A120257 16818516,-24293412,-2760615,12870,1,10,-825,-108900,9343620,267227532,-1447482465
%N A120257 Triangle of Hankel transforms of certain binomial sums.
%C A120257 Row k is the Hankel transform of sum{j=0..n, C(k+j, j)}. Absolute value is reversal of A103905. Diagonal and sub-diagonals are essentially signed versions of the central coefficients of certain generalized Pascal-Narayana triangles (A007318, A001263, A056939, A056940, A056941).
%F A120257 T(n, k):=(cos(pi*k/2)-sin(pi*k/2))*product{j=0..n-k-1, C(2k+2+j, k+1)/C(k+1+j, j)}
%e A120257 Triangle begins
%e A120257 1,
%e A120257 2, -1,
%e A120257 3, -6, -1,
%e A120257 4, -20, -20, 1,
%e A120257 5, -50, -175, 70, 1,
%e A120257 6, -105, -980, 1764, 252, -1,
%e A120257 7, -196, -4116, 24696, 19404, -924, -1,
%e A120257 8, -336, -14112, 232848, 731808, -226512, -3432, 1
%Y A120257 Cf. A120258.
%Y A120257 Sequence in context: A093346 A115597 A103371 this_sequence A059298 A059434 A106578
%Y A120257 Adjacent sequences: A120254 A120255 A120256 this_sequence A120258 A120259 A120260
%K A120257 easy,sign,tabl
%O A120257 0,2
%A A120257 Paul Barry (pbarry(AT)wit.ie), Jun 13 2006
%I A059298
%S A059298 1,2,1,3,6,1,4,24,12,1,5,80,90,20,1,6,240,540,240,30,1,7,672,2835,
%T A059298 2240,525,42,1,8,1792,13608,17920,7000,1008,56,1,9,4608,61236,129024,
%U A059298 78750,18144,1764,72,1,10,11520,262440,860160,787500,272160,41160
%N A059298 Triangle of idempotent numbers binomial(n,k)*k^(n-k), version 2.
%D A059298 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 91, #43 and p. 135, [3i'].
%e A059298 Triangle begins 1; 0, 1; 0, 2, 1; 0, 3, 6, 1; 0, 4, 24, 12, 1; ...
%Y A059298 There are 4 versions: A059297-A059300. Diagonals give A001788, A036216, A040075, A050982, A002378, 3*A002417, etc. Row sums are A000248.
%Y A059298 Sequence in context: A115597 A103371 A120257 this_sequence A059434 A106578 A006895
%Y A059298 Adjacent sequences: A059295 A059296 A059297 this_sequence A059299 A059300 A059301
%K A059298 nonn,tabl
%O A059298 0,2
%A A059298 njas, Jan 25 2001
%I A059434
%S A059434 1,1,2,1,3,6,1,7,16,26,1,27,69,118,168,1,169,455,810,1192,1575,1,1576,
%T A059434 4343,7920,11952,16153,20355,1,20356,56864,105324,161704,222427,284726,
%U A059434 347026,1,347027,978779,1832958,2848841,3970048,5148119,6346546
%N A059434 Triangle formed when cumulative boustrophedon transform is applied to 1, 1, 1, 1, ..., read by rows in natural order.
%e A059434 1; 1,2; 6,3,1; 1,7,16,26; 168,118,69,27,1; ...
%Y A059434 Cf. A059430, A059433.
%Y A059434 Sequence in context: A103371 A120257 A059298 this_sequence A106578 A006895 A125205
%Y A059434 Adjacent sequences: A059431 A059432 A059433 this_sequence A059435 A059436 A059437
%K A059434 nonn,tabl,easy
%O A059434 0,3
%A A059434 njas, Jan 31 2001
%E A059434 More terms from Floor van Lamoen (fvlamoen(AT)hotmail.com), Oct 08 2001
%I A106578
%S A106578 1,1,1,1,2,1,3,6,2,4,48
%N A106578 First differences of indices of square-free central binomial numbers.
%C A106578 First differences of A046098. Next term >= 27929.
%Y A106578 Cf. A001405.
%Y A106578 Sequence in context: A120257 A059298 A059434 this_sequence A006895 A125205 A125206
%Y A106578 Adjacent sequences: A106575 A106576 A106577 this_sequence A106579 A106580 A106581
%K A106578 nonn
%O A106578 0,5
%A A106578 Paul Barry (pbarry(AT)wit.ie), May 09 2005
%I A006895 M0158
%S A006895 1,2,1,3,6,2,16,9,23,58,6,128,109,147,512,70,954,1233,815,4096,1650,
%T A006895 6542,13141,3243,32768,23038,42498,131072,3577,258567,272874,
%U A006895 251414,1048576
%N A006895 Parenthesized one way gives the powers of 2: (1), (2), (1+3), ..., another way the powers of 3: (1), (2+1), (3+6), ....
%Y A006895 Sequence in context: A059298 A059434 A106578 this_sequence A125205 A125206 A133904
%Y A006895 Adjacent sequences: A006892 A006893 A006894 this_sequence A006896 A006897 A006898
%K A006895 base,nonn,nice
%O A006895 0,2
%A A006895 njas, Kevin Brown [ kevin2003(AT)delphi.com ]
%I A125205
%S A125205 1,2,1,3,6,3,1,4,18,30,24,15,6,1,5,40,135,250,295,282,215,120,45,10,1,6,
%T A125205 75,420,1385,3015,4800,6365,7170,6705,5065,3009,1365,455,105,15,1,7,126,
%U A125205 1050,5355,18690,47880,96796,166890,251370,329945,373947,362292,297115
%N A125205 Triangular array T(n,k) (n>=1, 0<=k<=n(n-1)/2) giving the total number of connected components in all subgraphs (V,E') with |E'|=k of the complete labeled graph K_n=(V,E).
%F A125205 G.f.: Sum_{n,k} T(n,k)*x^n/n!*y^k=(F(x,y)-1)*exp(F(x,y)-1)=G(x,y)*ln(G(x,y)) where G(x,y)=Sum_{n=0..oo} (1+y)^(n(n-1)/2)*x^n/n!, and F(x,y)=1+ln(G(x,y)) is g.f. of A062734.
%e A125205 The array starts with
%e A125205 1
%e A125205 2, 1
%e A125205 3, 6, 3, 1
%e A125205 4, 18, 30, 24, 15, 6, 1
%e A125205 5, 40, 135, 250, 295, 282, 215, 120, 45, 10, 1
%e A125205 ...
%e A125205 T(3,1)=6 since there are three different subgraphs of K_3 with one edge, and each subgraph has two connected components.
%o A125205 (PARI) { reverse(v)=vector(length(v),i,v[length(v)+1-i]) } G=sum(n=0,6,(1+y)^(n*(n-1)/2)*x^n/n!); K=G*log(G); for(n=1,6,print(reverse(Vec(n!*polcoeff(K,n,x)))))
%Y A125205 Cf. A062734.
%Y A125205 Cf. A125206 (row-reversed version), A125207 (row sums).
%Y A125205 Sequence in context: A059434 A106578 A006895 this_sequence A125206 A133904 A094339
%Y A125205 Adjacent sequences: A125202 A125203 A125204 this_sequence A125206 A125207 A125208
%K A125205 nonn,tabf
%O A125205 1,2
%A A125205 Max Alekseyev (maxal(AT)cs.ucsd.edu), Nov 23 2006
%I A125206
%S A125206 1,1,2,1,3,6,3,1,6,15,24,30,18,4,1,10,45,120,215,282,295,250,135,40,5,1,
%T A125206 15,105,455,1365,3009,5065,6705,7170,6365,4800,3015,1385,420,75,6,1,21,
%U A125206 210,1330,5985,20349,54271,116385,204225,297115,362292,373947,329945
%N A125206 Triangular array T(n,k) (n>=1, 0<=k<=n(n-1)/2) giving the total number of connected components in all subgraphs obtained from the complete labeled graph K_n by removing k edges.
%C A125206 Row-reversed version of A125205, see A125205 for further details
%e A125206 The array starts with
%e A125206 1
%e A125206 1, 2
%e A125206 1, 3, 6, 3
%e A125206 1, 6, 15, 24, 30, 18, 4
%e A125206 1, 10, 45, 120, 215, 282, 295, 250, 135, 40, 5
%e A125206 ...
%Y A125206 Cf. A125205 (row-reversed version), A125207 (row sums).
%Y A125206 Sequence in context: A106578 A006895 A125205 this_sequence A133904 A094339 A120576
%Y A125206 Adjacent sequences: A125203 A125204 A125205 this_sequence A125207 A125208 A125209
%K A125206 nonn,tabf
%O A125206 1,3
%A A125206 Max Alekseyev (maxal(AT)cs.ucsd.edu), Nov 23 2006
%I A133904
%S A133904 1,2,1,3,6,3,13,15,19,22,11,123,375,377,381,1147,1152,576,288,144,72,36,
%T A133904 18,9,17,20,10,5,11,13,21,23,29,33,37,41,50,25,29,33,41,1683,5055,5057,
%U A133904 5063,5069,5073,5075,5085,5088,2544,1272,636,368,184,92,46,23
%N A133904 a(n)= gcd(a(n-1),n)*a(n-1) + d(n) if a(n-1) is not divisible by 2 else a(n)= a(n-1)/2, where gcd denotes common divisor, d(n) is number of divisors of n.
%Y A133904 Cf. A001222, A000005.
%Y A133904 Sequence in context: A006895 A125205 A125206 this_sequence A094339 A120576 A063707
%Y A133904 Adjacent sequences: A133901 A133902 A133903 this_sequence A133905 A133906 A133907
%K A133904 nonn
%O A133904 1,2
%A A133904 Ctibor O. Zizka (ctibor.zizka(AT)seznam.cz), Jan 07 2008
%I A094339
%S A094339 2,1,3,6,4,8,12,9,5,10,15,25,20,24,16,32,48,30,18,36,27,13,7,53,106,265,
%T A094339 159,318,212,14,107,321,214,428,642,535,35,21,181,11,33,22,23,59,70,28,
%U A094339 151,29,19,233,466,2563,699,932,40,26,38,31,61,39,49,98,42,56,50,197,17
%N A094339 Beginning with 2, least number not occurring earlier that divides the sum of all previous terms.
%C A094339 Conjecture: this is a rearrangement of natural numbers.
%C A094339 Comments from Zak Seidov (zakseidov(AT)yahoo.com), Feb 19 2005:
%C A094339 "Changing the seed produces different sequences, some of which merge into each other:
%C A094339 s2=2,1,3,6,4,8,12,9,5,10,15,25,20,24,16,32,48,30,18,36,27,13,7,53
%C A094339 s3=3,1,2,6,4,8,12,9,5,10,15,25,20,24,16,32,48,30,18,36,27,13,7,53
%C A094339 s4=4,1,5,2,3,15,6,9,45,10,20,8,16,12,13,169,26,7,53,106,265,159,18
%C A094339 s5=5,1,2,4,3,15,6,9,45,10,20,8,16,12,13,169,26,7,53,106,265,159,18
%C A094339 s6=6,1,7,2,4,5,25,10,3,9,8,16,12,18,14,20,32,24,27,81,36,15,75,30,40
%C A094339 s7=7,1,2,5,3,6,4,14,21,9,8,10,15,35,20,16,11,17,12,18,13,19,38,76,95
%C A094339 s8=8,1,3,2,7,21,6,4,13,5,10,16,12,9,39,26,14,28,32,64,20,17,51,24,18
%C A094339 s9=9,1,2,3,5,4,6,10,8,12,15,25,20,24,16,32,48,30,18,36,27,13,7,53,106
%C A094339 s10=10,1,11,2,3,9,4,5,15,6,22,8,12,18,7,19,38,95,57,114,24,16,31,17,32
%C A094339 s11=11,1,2,7,3,4,14,6,8,28,12,16,56,21,9,18,24,5,35,10,29,319,22,15,25,20,30
%C A094339 In every case one may ask if the result is a rearrangement of the natural numbers."
%e A094339 The sum of first 7 terms is 36, hence a(8) = 9 is the least divisor of 36 not occurring earlier.
%p A094339 A094339 := proc(nmax) local a,n,sprev,i; a := [2] ; while nops(a) < nmax do sprev := add(i,i=a) ; n := 1 ; while sprev mod n <> 0 or n in a do n := n+1 ; od ; a := [op(a),n] ; od ; RETURN(a) ; end: A094339(100) ; - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Apr 30 2007
%Y A094339 Cf. A094340, A094341.
%Y A094339 Sequence in context: A125205 A125206 A133904 this_sequence A120576 A063707 A118287
%Y A094339 Adjacent sequences: A094336 A094337 A094338 this_sequence A094340 A094341 A094342
%K A094339 nonn
%O A094339 1,1
%A A094339 Amarnath Murthy (amarnath_murthy(AT)yahoo.com), May 17 2004
%E A094339 Corrected and extended by R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Apr 30 2007
%I A120576
%S A120576 2,1,3,6,4,12,7,14,28,11,77,5,15,33,55,165,73,146,219,438,9,18,657,1314,
%T A120576 8,16,23,24,36,46,48,69,72,92,138,144,184,207,276,368,414,552,828,1104,
%U A120576 1656,3312,1847,12929,5541,9235,27705,19,38,3694,35093,70186,487,974
%N A120576 Irregular array where the n-th row are the divisors, not occurring earlier in the sequence, of the sum of the terms in all previous rows. a(1)=2.
%C A120576 Is this sequence a permutation of the positive integers?
%e A120576 Array begins:
%e A120576 2
%e A120576 1
%e A120576 3
%e A120576 6
%e A120576 4,12
%e A120576 7,14,28
%e A120576 Now these terms add up to 77. So row 7 is the divisors of 77 which do not occur earlier in the sequence. 1 and 7 occur in earlier rows, so row 7 is (11,77).
%t A120576 f[t_] := Flatten[Append[t, Select[Divisors[Plus @@ t], FreeQ[t, # ] &]]]; Nest[f, {2}, 14] (*Chandler*)
%Y A120576 Cf. A120577, A120578, A120579.
%Y A120576 Sequence in context: A125206 A133904 A094339 this_sequence A063707 A118287 A024930
%Y A120576 Adjacent sequences: A120573 A120574 A120575 this_sequence A120577 A120578 A120579
%K A120576 nonn,tabf
%O A120576 1,1
%A A120576 Leroy Quet (qq-quet(AT)mindspring.com), Jun 15 2006
%E A120576 Extended by Ray Chandler (rayjchandler(AT)sbcglobal.net), Jun 17 2006
%I A063707
%S A063707 2,1,3,6,4,17,2,46,8,24,5,321,6,842,12,26,48,5777,15,15126,35,167,77,
%T A063707 103681,42,16627,200,5856,234,1860497,57,4870846,2208,7602,1365,45081,
%U A063707 306,87403802,3572,51941,1927,599074577,408,1568397606,10947,80321
%N A063707 Cyclotomic polynomials Phi_n at x=phi, ceiled up (where phi = tau = (sqrt(5)+1)/2).
%H A063707 Index entries for cyclotomic polynomials, values at phi
%p A063707 with(numtheory); Phi_at_x := (n,y) -> subs(x=y,cyclotomic(n,x)); [seq(ceil(evalf(simplify(Phi_at_x(j,(sqrt(5)+1)/2)))),j=0..120)];
%Y A063707 A019320, A063703, A063705, A063708.
%Y A063707 Sequence in context: A133904 A094339 A120576 this_sequence A118287 A024930 A121966
%Y A063707 Adjacent sequences: A063704 A063705 A063706 this_sequence A063708 A063709 A063710
%K A063707 nonn
%O A063707 0,1
%A A063707 Antti Karttunen Aug 03 2001
%I A118287
%S A118287 1,2,1,3,6,5,6,10,9,10,8,17,16,17,15,12,13,12,28,27,28,26,23,24,23,19,
%T A118287 20,19,21,46,45,46,44,41,42,41,37,38,37,39,30,31,30,32,35,34,35,75,74,
%U A118287 75,73,70,71,70,66,67,66,68,59,60,59,61,64,63,64,48,49,48,50,53,52,53
%N A118287 A fractal transform of the Lucas numbers: define a(1)=1, then if L(n)= 4.
%Y A050043 Sequence in context: A121966 A021472 A053225 this_sequence A113396 A112647 A057925
%Y A050043 Adjacent sequences: A050040 A050041 A050042 this_sequence A050044 A050045 A050046
%K A050043 nonn
%O A050043 1,2
%A A050043 Clark Kimberling (ck6(AT)evansville.edu)
%I A113396
%S A113396 2,1,3,6,9,1,15,7,22,27,1,25,39,31,17,23,57,31,55,69,43,67,53,33,85,99,
%T A113396 91,105,97,58,115,101,135,49,147,121,127,151,137,143,177,91,189,181,195,
%U A113396 67,79,211,225,217,203,237,151
%N A113396 Prime(n+1)^2-prime(n)^2 (mod prime(n+1)).
%e A113396 3^2-2^2=2(mod(3)); 5^2-3^2=1(mod(5); 7^2-5^2=3(mod(7)); 11^2-7^2=6(mod(11)); etc.
%Y A113396 Cf. A069482.
%Y A113396 Sequence in context: A021472 A053225 A050043 this_sequence A112647 A057925 A086964
%Y A113396 Adjacent sequences: A113393 A113394 A113395 this_sequence A113397 A113398 A113399
%K A113396 nonn
%O A113396 1,1
%A A113396 Marian Kraus (sr.dingdong(AT)gmx.de), Oct 26 2005
%I A112647
%S A112647 2,1,3,6,9,5,7,62,0,13,25,22,16,12,32,11,0,104,18,837,17,19,63,46,0,28,
%T A112647 0,116,24,58,31,2222,0,39,242,23,0,147,0,30,675,29,35,52,0,777,0,40,0,
%U A112647 435,0,42,36,41,0,91,0,67,0,65,99,0,195,110,80,53,48,124,0,243,0,70,97
%N A112647 a(n)=x is the smallest solution to Abs[sigma[x+1]-sigma[x]]=n. or zero if it is conjectured that no solution exists. In fact empirically no results were found below 10000000 but such missing of solution is rather plausible.
%C A112647 While it is known that not all m values satisfies sigma[x] = m, (see A007369), it is more difficult to determine those numbers which cannot be a difference of sigma[u]-sigma[w] for some u and w.
%e A112647 n=5: least solution is 9 because divisor-sums
%e A112647 for 9 and 9+1=10 are 13 and 13+5=18.
%t A112647 f[x_] :=Abs[DivisorSigma[1, n+1]-DivisorSigma[1, n]]; t=Table[0, {257}];Do[s=f[n];If[s<258&&t[[s]]==0, t[[s]]=n], {n, 1, 10000000}];t
%Y A112647 Cf. A000203, A007369, A112645, A112646.
%Y A112647 Sequence in context: A053225 A050043 A113396 this_sequence A057925 A086964 A076242
%Y A112647 Adjacent sequences: A112644 A112645 A112646 this_sequence A112648 A112649 A112650
%K A112647 nonn
%O A112647 1,1
%A A112647 Labos E. (labos(AT)ana.sote.hu), Sep 28 2005
%I A057925
%S A057925 2,1,3,6,9,15,27,36,51,87,135,243,315,531,735,969,1707,2679,3831,6363,
%T A057925 9663,14559,23091,33045,45999,77523,120471,177639,252717,405321,554037,
%U A057925 795831,1335087,1819527,2491233,4183467,5920215,8051991,12530967
%N A057925 Position of first occurrence of 2^n in A057925.
%Y A057925 Cf. A057923, A057924.
%Y A057925 Sequence in context: A050043 A113396 A112647 this_sequence A086964 A076242 A066203
%Y A057925 Adjacent sequences: A057922 A057923 A057924 this_sequence A057926 A057927 A057928
%K A057925 easy,nonn
%O A057925 0,1
%A A057925 Larry Reeves (larryr(AT)acm.org), Oct 03 2000
%E A057925 More terms from David W. Wilson, Jan 11, 2001.
%I A086964
%S A086964 2,1,3,6,10,4,14,8,12,5,22,9,26,7,15,20,34,30,38,16,21,11,46,24,25,13,
%T A086964 45,42,58,18,62,40,33,17,245,36,74,19,39,32,82,28,86,66,27,23,94,48,98,
%U A086964 60,51,78,106,54,110,56,57,29,118,50,122,31,147,80,130,44,134
%N A086964 a(A086957(n))=A086958(n), a(A086958(n))=A086957(n), a(A086959(n))=A086959(n), see A086956.
%C A086964 A self-inverse permutation of natural numbers: a(a(n))=n;
%C A086964 a(n) = A086962(A086963(n)) = A086963(A086962(n)).
%H A086964 Index entries for sequences that are permutations of the natural numbers
%Y A086964 Cf. A086572, A086571.
%Y A086964 Sequence in context: A113396 A112647 A057925 this_sequence A076242 A066203 A024866
%Y A086964 Adjacent sequences: A086961 A086962 A086963 this_sequence A086965 A086966 A086967
%K A086964 nonn
%O A086964 1,1
%A A086964 Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Jul 25 2003
%I A076242
%S A076242 1,2,1,3,6,10,5,8,17,19,27,31,38,35,28,39,17,17,10,38,68,63,13,55,48,4,
%T A076242 74,100,37,29,47,121,115,136,105,28,128,109,159,90,114,31,151,4,86,108,
%U A076242 81,147,149,189,185,119,231,166,88,238,197,233,64,186,258,111,128,260
%N A076242 Remainder when 3rd order prime pp[n]=A038580(n) is divided by n-th prime=A000040(n).
%F A076242 a(n)=Mod[pp(n), p(n)]=Mod[A038580(n), A000040(n)]
%Y A076242 Cf. A006450, A038580, A049090, A049203, A049202, A057809, A076240-A076243.
%Y A076242 Sequence in context: A112647 A057925 A086964 this_sequence A066203 A024866 A076058
%Y A076242 Adjacent sequences: A076239 A076240 A076241 this_sequence A076243 A076244 A076245
%K A076242 nonn
%O A076242 1,2
%A A076242 Labos E. (labos(AT)ana.sote.hu), Oct 08 2002
%I A066203
%S A066203 2,1,3,6,10,5,11,4,12,21,31,20,8,21,7,22,38,55,37,18,38,17,39,16,40,15,
%T A066203 41,14,42,13,43,74,106,73,107,72,36,73,35,74,34,75,33,76,32,77,123,170,
%U A066203 122,171,121,70,122,69,123,68,124,67,9,68,128,189,127,64,128,63,129,62
%N A066203 a(1) = 2; a(2) = 1; for n > 2, a(n) = a(n-1)-(n-1) if this is positive and has not already appeared in the sequence, otherwise a(n) = a(n-1)+(n-1).
%H A066203 Index entries for sequences related to Recaman's sequence
%Y A066203 A variant of A005132. A row of the array in A066202.
%Y A066203 Sequence in context: A057925 A086964 A076242 this_sequence A024866 A076058 A098124
%Y A066203 Adjacent sequences: A066200 A066201 A066202 this_sequence A066204 A066205 A066206
%K A066203 nonn,easy
%O A066203 1,1
%A A066203 njas, Dec 16 2001
%E A066203 More terms from Sascha Kurz (sascha.kurz(AT)uni-bayreuth.de), Mar 24 2002
%I A024866
%S A024866 1,1,2,1,3,6,10,10,14,13,18,17,23,22,29,28,36,45,54,53,63,62,73,72,84,83,
%T A024866 96,95,109,108,123,122,138,136,152,150,167,185,204,203,223,222,243,242,264,
%U A024866 263,286,285,309,308,333,332,358,357,383,381,408,406,434,432,461,459,489
%N A024866 a(n) = s(1)t(n) + s(2)t(n-1) + ... + s(k)t(n-k+1), where k = [ n/2 ], s = (natural numbers), t = A014306.
%Y A024866 Sequence in context: A086964 A076242 A066203 this_sequence A076058 A098124 A020992
%Y A024866 Adjacent sequences: A024863 A024864 A024865 this_sequence A024867 A024868 A024869
%K A024866 nonn
%O A024866 2,3
%A A024866 Clark Kimberling (ck6(AT)evansville.edu)
%I A076058
%S A076058 2,1,3,6,10,15,21,27,36,44,55,65,76,91,104,120,136,152,170,189,210,230,
%T A076058 251,275,300,324,350,377,405,433,463,496,527,560,594,629,664,702,740,
%U A076058 778,818,858,902,945,988,1032,1078,1127,1175,1224,1275,1326,1377,1430
%N A076058 First members of groups in A076062.
%Y A076058 Cf. A076059, A076060, A076061, A076062.
%Y A076058 Sequence in context: A076242 A066203 A024866 this_sequence A098124 A020992 A025110
%Y A076058 Adjacent sequences: A076055 A076056 A076057 this_sequence A076059 A076060 A076061
%K A076058 nonn
%O A076058 1,1
%A A076058 Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Oct 05 2002
%E A076058 Corrected and extended by Sascha Kurz (sascha.kurz(AT)uni-bayreuth.de), Jan 20 2003
%I A098124
%S A098124 1,0,2,1,3,6,10,15,30,54,92,160,282,492,859,1490,2570,4428,7627,13098,
%T A098124 22421,38290,65265,111018,188475,319380,540266,912397,1538371,2589858,
%U A098124 4353820,7309362,12255474,20523307,34328731,57357184,95733131,159626049
%N A098124 Number of compositions of n where the largest part is equal to the number of parts.
%F A098124 G.f.: Sum(((x^(k+1)-x)^k-(x^k-x)^k)/(x-1)^k, k=1..infinity).
%e A098124 a(7)=10 because we have 223,232,322,133,313,331,1114,1141,1411,and 4111.
%p A098124 G:=sum(((x^(k+1)-x)^k-(x^k-x)^k)/(x-1)^k,k=1..25):Gser:=series(G,x=0,45):seq(coeff(Gser,x^n),n=1..42); (Deutsch)
%Y A098124 Cf. A077229, A047993.
%Y A098124 Sequence in context: A066203 A024866 A076058 this_sequence A020992 A025110 A075346
%Y A098124 Adjacent sequences: A098121 A098122 A098123 this_sequence A098125 A098126 A098127
%K A098124 easy,nonn
%O A098124 1,3
%A A098124 Vladeta Jovovic (vladeta(AT)Eunet.yu), Sep 25 2004
%E A098124 More terms from Emeric Deutsch (deutsch(AT)duke.poly.edu), Apr 16 2005
%I A020992
%S A020992 0,2,1,3,6,10,19,35,64,118,217,399,734,1350,2483,4567,8400,
%T A020992 15450,28417,52267,96134,176818,325219,598171,1100208,2023598,
%U A020992 3721977,6845783,12591358,23159118,42596259,78346735,144102112
%N A020992 a(n)=a(n-1)+a(n-2)+a(n-3).
%F A020992 G.f.: x(2-x)/(1-x-x^2-x^3).
%Y A020992 Sequence in context: A024866 A076058 A098124 this_sequence A025110 A075346 A104529
%Y A020992 Adjacent sequences: A020989 A020990 A020991 this_sequence A020993 A020994 A020995
%K A020992 nonn
%O A020992 0,2
%A A020992 njas
%I A025110
%S A025110 1,1,2,1,3,6,11,11,18,17,29,27,45,40,66,53,87,142,230,229,372,370,600,595,
%T A025110 964,951,1540,1506,2438,2349,3802,3569,5776,5165,8358,6760,10939,17701,
%U A025110 28642,28634,46332,46311,74934,74879,121158,121014,195806,195429,316212
%N A025110 a(n) = s(1)t(n) + s(2)t(n-1) + ... + s(k)t(n-k+1), where k = [ n/2 ], s = (F(2), F(3), F(4), ...), t = A014306.
%Y A025110 Sequence in context: A076058 A098124 A020992 this_sequence A075346 A104529 A095024
%Y A025110 Adjacent sequences: A025107 A025108 A025109 this_sequence A025111 A025112 A025113
%K A025110 nonn
%O A025110 1,3
%A A025110 Clark Kimberling (ck6(AT)evansville.edu)
%I A075346
%S A075346 2,1,3,6,11,15,22,28,36,45,55,66,78,92,105,121,136,153,171,190,210,232,
%T A075346 253,276,300,324,351,378,406,435,465,496,528,561,595,630,666,704,741,
%U A075346 780,820,861,903,946,990,1035,1081,1127,1176,1225,1274,1326,1378,1431
%N A075346 Initial terms of groups in A075345.
%Y A075346 Cf. A075345, A075347.
%Y A075346 Sequence in context: A098124 A020992 A025110 this_sequence A104529 A095024 A049900
%Y A075346 Adjacent sequences: A075343 A075344 A075345 this_sequence A075347 A075348 A075349
%K A075346 nonn
%O A075346 1,1
%A A075346 Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Sep 19 2002
%E A075346 More terms from David Wasserman (dwasserm(AT)earthlink.net), Jan 16 2005
%I A104529
%S A104529 1,2,1,3,6,12,12,6,6,12,12,12,12,6,12,60,60,20,20,60,15,60,60,120,40,40,
%T A104529 40,120,120,30,15,30,60,15,60,180,180,45,180,360,360,45,90,45,90,180,
%U A104529 180,180,180,180,90,45,90,360,360,45,180,90,45,180,180,90,45,315,1260
%N A104529 Denominator of Sum_{k=1..n} 1/tau(k), where tau(k) is the number of divisors function.
%e A104529 1,3/2,2,7/3,17/6
%e A104529 a(4)=3 because 1/tau(1)+1/tau(2)+1/tau(3)+1/tau(4)=1/1+1/2+1/2+1/3=7/3.
%p A104529 with(numtheory): a:=n->denom(sum(1/tau(k),k=1..n)): seq(a(n),n=1..70);
%Y A104529 Cf. A104528.
%Y A104529 Sequence in context: A020992 A025110 A075346 this_sequence A095024 A049900 A024741
%Y A104529 Adjacent sequences: A104526 A104527 A104528 this_sequence A104530 A104531 A104532
%K A104529 frac,nonn
%O A104529 1,2
%A A104529 Emeric Deutsch (deutsch(AT)duke.poly.edu), Mar 12 2005
%I A095024
%S A095024 0,0,0,2,1,3,6,12,16,35,63,115,216,399,754,1418,2705,5077,9667,18403,
%T A095024 35047,67045,128509,246330,473457,911409,1756619,3390969,6551382,
%U A095024 12675118,24544171,47584397,92329550
%N A095024 Number of 5k+4 primes (A030433) in range ]2^n,2^(n+1)].
%H A095024 A. Karttunen and J. Moyer, C-program for computing the initial terms of this sequence
%H A095024 Index entries for sequences related to occurrences of various subsets of primes in range ]2^n,2^(n+1)]
%Y A095024 Cf. A036378.
%Y A095024 Sequence in context: A025110 A075346 A104529 this_sequence A049900 A024741 A024961
%Y A095024 Adjacent sequences: A095021 A095022 A095023 this_sequence A095025 A095026 A095027
%K A095024 nonn
%O A095024 1,4
%A A095024 Antti Karttunen (his-firstname.his-surname(AT)iki.fi), Jun 01 2004
%I A049900
%S A049900 1,2,1,3,6,12,24,43,68,159,318,631,1244,2444,4638,8350,13306,31249,
%T A049900 62498,124991,249964,499884,999518,1998110,3992826,7976984,15904776,
%U A049900 31622086,62494618,121995928,232079906,417569970,665554652
%N A049900 a(n)=a(1)+a(2)+...+a(n-1)-a(m), where m=2n-3-2^(p+1) and 2^p 5 && Mod[n, 3] == 0, 0, Block[{len = PartitionsP[n], p = Partitions[n], s = 0}, Do[ s = s + Length@ Select[FromDigits /@ Join @@@ IntegerDigits /@ Permutations@p[[i]], PrimeQ@# &], {i, len}]; s]]; Array[f, 25]
%Y A116381 Cf. A069869, A069870; not the same as A073901.
%Y A116381 Sequence in context: A024741 A024961 A125889 this_sequence A073901 A058170 A127896
%Y A116381 Adjacent sequences: A116378 A116379 A116380 this_sequence A116382 A116383 A116384
%K A116381 base,nonn
%O A116381 1,2
%A A116381 Robert G. Wilson v (rgwv(AT)rgwv.com), Feb 06 2006
%I A073901
%S A073901 0,2,1,3,7,0,29,27,0,90,236,0,761,1040,0,3959,9354,0,33005,51313,0,
%T A073901 184842,427491,0,1555470,2543382,0,9549062,21167640,0,78295483
%N A073901 Number of primes with nonzero digits and digit sum n.
%C A073901 a(3k) = 0 for all k>1. The number of candidates to consider for a(n) is 2^(n-1).
%e A073901 a(2) = 2: the two primes are 2 and 11. a(5) = 7: the primes are 5, 41, 23, 113, 131, 311 and 2111.
%t A073901 Needs["DiscreteMath`Combinatorica`"]; f[n_] := Block[{a = Partitions[n], b = {}, c = {}}, Do[b = Append[b, Permutations[ a[[i]] ]], {i, 1, Length[a]}]; b = Flatten[b, 1]; Do[c = Append[c, FromDigits[ b[[i]] ]], {i, 1, Length[b]}]; Length[ Select[c, PrimeQ[ # ] & ]]]; Table[ f[n], {n, 1, 19}] (* Or *)
%t A073901 nextodd[c_] := If[ Length[c]==2, Join[ Table[1, {c[[1]]-2}], {c[[2]]+2}], Join[ Table[1, {c[[1]]-1}], {c[[2]]+1}, Drop[c, 2]]]; a[2]=2; a[n_] := If[Mod[n, 3]==0 && n>3, 0, Module[{c, ct}, For[ c = Table[1, {n}]; ct = 0, True, c = nextodd[c], If[ PrimeQ[ FromDigits[c]], ct++ ]; If[ c[[ -1]] >= n-1, Return[ct]] ] ]]; Table[ a[n], {n, 30}]
%Y A073901 Not the same as A116381.
%Y A073901 Sequence in context: A024961 A125889 A116381 this_sequence A058170 A127896 A010757
%Y A073901 Adjacent sequences: A073898 A073899 A073900 this_sequence A073902 A073903 A073904
%K A073901 base,more,nonn
%O A073901 1,2
%A A073901 Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Aug 18 2002
%E A073901 Edited and extended by Robert G. Wilson v (rgwv(AT)rgwv.com), Sep 19 2002
%E A073901 a(20) to a(24) and alternate Mma coding from Dean Hickerson (dean(AT)math.ucdavis.edu), Sep 21 2002
%E A073901 a(25) from Robert G. Wilson v (rgwv(AT)rgwv.com), Sep 26 2002
%E A073901 a(26)-a(31) from Robert G. Wilson v (rgwv(at)rgwv.com), Nov 14 2005
%I A058170
%S A058170 1,2,1,3,7,2,10,28,33,7,82,153,261,213,37,2511,1227,2266,2989,1709,263,
%T A058170 600106,26033,24176,39176,38346,16306,2134
%N A058170 Triangle: Number of asymmetric semigroups of order n with k idempotents, considered to be equivalent when they are isomorphic or anti-isomorphic (by reversal of the operator).
%H A058170 Index entries for sequences related to semigroups
%e A058170 1; 2,1; 3,7,2; 10,28,33,7; 82,153,261,213,37; ...
%Y A058170 Row sums give A058107.
%Y A058170 Sequence in context: A125889 A116381 A073901 this_sequence A127896 A010757 A019320
%Y A058170 Adjacent sequences: A058167 A058168 A058169 this_sequence A058171 A058172 A058173
%K A058170 nonn,tabl,hard
%O A058170 1,2
%A A058170 Christian G. Bower (bowerc(AT)usa.net), Nov 19 2000
%I A127896
%S A127896 1,2,1,3,7,4,10,25,16,33,89,63,108,316,245,350,1119,943,1121,3952,3598,
%T A127896 3539,13920,13625,10971,48897,51256,33208,171287,191694,97265,598325,
%U A127896 713161,271388,2083934
%V A127896 1,-2,1,3,-7,4,10,-25,16,33,-89,63,108,-316,245,350,-1119,943,1121,-3952,3598,3539,
%W A127896 -13920,13625,10971,-48897,51256,33208,-171287,191694,97265,-598325,713161,271388,
%X A127896 -2083934
%N A127896 Expansion of 1/(1+2x+3x^2+x^3).
%C A127896 Row sums of A127895. Series reversion is A127897.
%F A127896 a(n)=sum{k=0..n, (-1)^(n-k)*C(n+2k+2,n-k)}
%Y A127896 Sequence in context: A116381 A073901 A058170 this_sequence A010757 A019320 A033640
%Y A127896 Adjacent sequences: A127893 A127894 A127895 this_sequence A127897 A127898 A127899
%K A127896 easy,sign
%O A127896 0,2
%A A127896 Paul Barry (pbarry(AT)wit.ie), Feb 04 2007
%I A010757
%S A010757 1,0,1,2,1,3,7,4,11,25,16,41,92,63,155,344,247,591,1300,967,2267,4950,
%T A010757 3785,8735,18955,14820,33775,72905,58060,130965,281403,227612,509015,
%U A010757 1089343,892926,1982269,4227273,3505386,7732659,16438345,13770404
%N A010757 Sum along upward diagonal of Pascal triangle from center.
%p A010757 seq(add(binomial(i-k,k),k=ceil(i/3)..floor(i/2)),i=0..50); # Detlef Pauly (dettodet(AT)yahoo.de), Jul 24 2003
%Y A010757 Sequence in context: A073901 A058170 A127896 this_sequence A019320 A033640 A112027
%Y A010757 Adjacent sequences: A010754 A010755 A010756 this_sequence A010758 A010759 A010760
%K A010757 nonn
%O A010757 0,4
%A A010757 R. K. Guy (rkg(AT)cpsc.ucalgary.ca)
%I A019320
%S A019320 2,1,3,7,5,31,3,127,17,73,11,2047,13,8191,43,151,257,131071,
%T A019320 57,524287,205,2359,683,8388607,241,1082401,2731,262657,
%U A019320 3277,536870911,331,2147483647,65537,599479,43691,8727391
%N A019320 Cyclotomic polynomials at x=2.
%H A019320 T. D. Noe, Table of n, a(n) for n=0..1000
%H A019320 Joerg Arndt, Fxtbook
%H A019320 Index entries for cyclotomic polynomials, values at X
%F A019320 LCM{2^k - 1, k=1..n}/LCM{2^k - 1, k=1..n-1}, n >1. - Vladeta Jovovic (vladeta(AT)Eunet.yu), Jan 20 2002
%p A019320 with(numtheory,cyclotomic); f := n->subs(x=2,cyclotomic(n,x)); seq(f(i),i=0..64);
%Y A019320 A019320[n] = A063696[n]-A063698[n] for up to n=104. Same sequence in binary: A063672. Cf. A054432, A020501.
%Y A019320 Cf. A034268.
%Y A019320 Sequence in context: A058170 A127896 A010757 this_sequence A033640 A112027 A024404
%Y A019320 Adjacent sequences: A019317 A019318 A019319 this_sequence A019321 A019322 A019323
%K A019320 nonn
%O A019320 0,1
%A A019320 Simon Plouffe (plouffe(AT)math.uqam.ca)
%I A033640
%S A033640 1,1,2,1,3,7,6,20,52,6,26,104,32,162,460,356,1438,4048,712,3588,15272,
%T A033640 5012,27460,90476,64944,300816,912472,90476,155420,611656,1067892,
%U A033640 1770024,4763360,4151704,14746316,39566064,8915064,27813084,109938548
%N A033640 Base 3 digital convolution sequence.
%e A033640 Suppose base = 3, and a(0)..a(13) are 1 1 2 1 3 7 6 20 52 6 26 104 32 162. In base 3, 14 = 112, so we convolve the last three terms with 1, 1, 2 to obtain 104*1+32*1+162*2 = 460.
%Y A033640 Sequence in context: A127896 A010757 A019320 this_sequence A112027 A024404 A077173
%Y A033640 Adjacent sequences: A033637 A033638 A033639 this_sequence A033641 A033642 A033643
%K A033640 base,nonn
%O A033640 0,3
%A A033640 David W. Wilson (davidwwilson(AT)comcast.net)
%I A112027
%S A112027 1,2,1,3,7,8,4,6,25,26,13,15,79,80,40,42,241,242,121,123,727,728,364,366,
%T A112027 2185,2186,1093,1095,6559,6560,3280,3282,19681,19682,9841,9843,59047,59048,
%U A112027 29524,29526,177145,177146,88573,88575,531439,531440,265720,265722,1594321
%N A112027 a(1)=1; then successively add 1, divide by 2, add 2, and then total up the last 4 terms.
%D A112027 Joshua Zucker, Posting to Seq Fan mailing list, Nov 24 2005
%p A112027 a[1]:=1; k:=1; for n from 1 to 16 do k:=k+1; a[k]:=a[k-1]+1; k:=k+1; a[k]:=a[k-1]/2; k:=k+1; a[k]:=a[k-1]+2; k:=k+1; a[k]:=a[k-1]+a[k-2]+a[k-3]+a[k-4]; od;
%Y A112027 Sequence in context: A010757 A019320 A033640 this_sequence A024404 A077173 A063509
%Y A112027 Adjacent sequences: A112024 A112025 A112026 this_sequence A112028 A112029 A112030
%K A112027 nonn
%O A112027 1,2
%A A112027 njas, Nov 24 2005
%E A112027 Definition found by Franklin T. Adams-Watters, Feb 01, 2006
%E A112027 More terms from njas, Feb 22 2006
%I A024404
%S A024404 1,1,0,1,2,1,3,7,10,17,33,56,102,197,359,617,1128,2182,3966,7488,14538,
%T A024404 26766,51232,95325,174742,335555,657893,1266230,2485507,4793512,8521770,
%U A024404 16519171,31580739,62246045,116079942,229064794,440508973,848388680
%N A024404 Number of products of distinct primes <= prime(n) equal to -1 (mod prime(n)).
%Y A024404 Sequence in context: A019320 A033640 A112027 this_sequence A077173 A063509 A137766
%Y A024404 Adjacent sequences: A024401 A024402 A024403 this_sequence A024405 A024406 A024407
%K A024404 nonn
%O A024404 1,5
%A A024404 David W. Wilson (davidwwilson(AT)comcast.net)
%I A077173
%S A077173 2,1,3,7,11,15,22,29,37,44,56,67,78,92,106,119,136,154,171,188,210,230,
%T A077173 252,275,299,323,349,375,402,430,460,491,522,555,588,623,658,694,731,
%U A077173 769,809,849,890,932,975,1019,1064,1110,1157,1205,1255
%N A077173 Initial terms of rows of triangle in A077172.
%Y A077173 Cf. A077172, A077174, A077175, A077176, A100918.
%Y A077173 Sequence in context: A033640 A112027 A024404 this_sequence A063509 A137766 A010758
%Y A077173 Adjacent sequences: A077170 A077171 A077172 this_sequence A077174 A077175 A077176
%K A077173 nonn
%O A077173 1,1
%A A077173 Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Nov 01 2002
%E A077173 Extended by Ray Chandler (rayjchandler(AT)sbcglobal.net), Nov 29 2004
%I A063509
%S A063509 2,1,3,7,11,29,47,123,199,322,521,843,2207,3571,9349,15127,39603,
%T A063509 64079,103682,167761,271443,1149851,3010349,4870847,12752043,20633239,
%U A063509 33385282,54018521,87403803,228826127,370248451,969323029,1568397607
%N A063509 Square-free Lucas numbers.
%o A063509 (PARI) for(n=0,30,x=fibonacci(n)+2*fibonacci(n-1); if(issquarefree(x),print(x)))
%Y A063509 Cf. A000032.
%Y A063509 Sequence in context: A112027 A024404 A077173 this_sequence A137766 A010758 A019224
%Y A063509 Adjacent sequences: A063506 A063507 A063508 this_sequence A063510 A063511 A063512
%K A063509 nonn
%O A063509 0,1
%A A063509 Jason Earls (zevi_35711(AT)yahoo.com), Aug 09 2001
%E A063509 More terms from Dean Hickerson, Aug 22 2001
%I A137766
%S A137766 1,2,1,3,7,12,1,4,9,20,1,5,49,60,11,30,1,6,71,105,13,42,1,7,351,280,97,
%T A137766 168,15,56,1,8,545,504,127,252,17,72,1,9,2561,1260,799,840,161,360,19,
%U A137766 90,1,10,4159,2310,1121,1320,199,495,21,110,1,11
%N A137766 Elements to the right of the central element in writing first the numerator and then the denominator (left to right) of Leibniz's harmonic-like triangle.
%Y A137766 Cf. A137752.
%Y A137766 Cf. A046201, A046204, A046205, A046206, A046208, A046212, A137752, A137753, A137754, A137755, A137756, A137757, A137758, A137759, A137760, A137761, A137762, A137763, A137764, A137765.
%Y A137766 Sequence in context: A024404 A077173 A063509 this_sequence A010758 A019224 A053190
%Y A137766 Adjacent sequences: A137763 A137764 A137765 this_sequence A137767 A137768 A137769
%K A137766 nonn,tabl
%O A137766 1,2
%A A137766 Mohammad K. Azarian (azarian(AT)evansville.edu), Feb 13 2008
%I A010758
%S A010758 0,0,1,2,1,3,7,14,11,25,51,97,92,189,365,674,709,1383,2587,4685,5270,
%T A010758 9955,18228,32551,38403,70954,127921,226007,276408,502415,895103,1568062,
%U A010758 1972851,3540913,6249235,10871723,13996408,24868131,43551364,75326395
%N A010758 Sum along upward diagonal of Pascal triangle from (but not including) halfway point.
%Y A010758 Sequence in context: A077173 A063509 A137766 this_sequence A019224 A053190 A135299
%Y A010758 Adjacent sequences: A010755 A010756 A010757 this_sequence A010759 A010760 A010761
%K A010758 nonn
%O A010758 0,4
%A A010758 R. K. Guy (rkg(AT)cpsc.ucalgary.ca)
%I A019224
%S A019224 0,2,1,3,7,19,36,116,191,542,1204,3076,7560,18747,48658,123404,
%T A019224 319762
%N A019224 Cycle class sequence c(n) (the number of true cycles of length n in which a certain node is included) for zeolite RTE = RUB-3 [ Si24O48 ] . 2 R.
%D A019224 G. Thimm and W. E. Klee, Zeolite cycle sequences, Zeolites, 19, pp. 422-424, 1997.
%H A019224 G. Thimm, Cycle sequences of crystal structures
%Y A019224 Sequence in context: A063509 A137766 A010758 this_sequence A053190 A135299 A092081
%Y A019224 Adjacent sequences: A019221 A019222 A019223 this_sequence A019225 A019226 A019227
%K A019224 nonn
%O A019224 3,2
%A A019224 Georg Thimm (mgeorg(AT)ntu.edu.sg)
%I A053190
%S A053190 0,1,2,1,3,7,31,69,31,100,231,562,3041,12726,28493,12726,41219,95164,
%T A053190 612203,1319570,8529623,52497308,218518855,489535018,218518855,
%U A053190 708053873,1634626601,7246560277,37867427986,82981416249,701718757978
%N A053190 Numerators in the convergents of [L_1, L_2, L_3, . . . ] where L_i is period length of the continued fraction for sqrt(i).
%e A053190 The numerator of the 7th convergent of [0, 1, 2, 0, 1, 2, 4, 2, 0, 1, 2, 2, 5, 4, 2...] is 31.
%Y A053190 Cf. A053011.
%Y A053190 Sequence in context: A137766 A010758 A019224 this_sequence A135299 A092081 A057740
%Y A053190 Adjacent sequences: A053187 A053188 A053189 this_sequence A053191 A053192 A053193
%K A053190 cofr,frac,nonn
%O A053190 1,3
%A A053190 Justin T. Miller (miller(AT)u.arizona.edu), Mar 02 2000
%E A053190 More terms from James A. Sellers (sellersj(AT)math.psu.edu), Mar 02 2000
%I A135299
%S A135299 1,1,2,1,3,8,1,4,11,32,1,5,15,43,128,1,6,20,58,171,512,1,7,26,78,229,
%T A135299 683,2048,1,8,33,104,307,912,2731,8192,1,9,41,137,411,1219,3643,10923,
%U A135299 32768,1,10,50,178,548,1630,4862,14566,43691,131072
%N A135299 Pascal's triangle, but the last element of the row is the sum of the all the previous terms.
%C A135299 T(n,n)=(4^n)/2 n>0 T(n,n)=2 sum([i=0 to n-1] T(n,i))
%F A135299 T(0,0)=1 T(n,k)=T(n-1,k-1)+T(n-1,k) if k>
%e A135299 T(2,1)=T(1,0)+T(1,1)=1+2=3
%e A135299 T(2,2)=T(0,0)+T(1,0)+T(1,1)+T(2,0)+T(2,1)=1+1+2+1+3=8
%Y A135299 Cf. A067337, A007318.
%Y A135299 Sequence in context: A010758 A019224 A053190 this_sequence A092081 A057740 A137307
%Y A135299 Adjacent sequences: A135296 A135297 A135298 this_sequence A135300 A135301 A135302
%K A135299 nonn,tabl
%O A135299 0,3
%A A135299 Jose Ramon Real (joseramonreal(AT)yahoo.es), Dec 04 2007
%I A092081
%S A092081 1,1,2,1,3,8,1,4,15,48,1,5,24,105,384,1,6,35,192,945,3840,1,7,48,315,
%T A092081 1920,10395,46080,1,8,63,480,3465,23040,135135,645120,1,9,80,693,5760,
%U A092081 45045,322560,2027025,10321920,1,10,99,960,9009,80640,675675,5160960
%N A092081 Triangle of certain double factorials.
%C A092081 This is the rectangular array A(3;m,n) := (2*n+m)!!/m!!, m >= 0, n >= 0, read by SW-NE diagonals. For n!! see A006882 (double factorials).
%H A092081 W. Lang, First 9 rows.
%F A092081 a(m, n)=(n+m)!!/(m-n)!!, 0<=n<=m, else 0, with 0!! := 1.
%Y A092081 Diagonals give: A000165 (double factorials of 2*n), A001147(n+1), A002866, A051577-83.
%Y A092081 Columns give: A000012 (powers of 1), A000027 (naturals >=2), A005563, 3*A077415, for n=0..3.
%Y A092081 Sequence in context: A019224 A053190 A135299 this_sequence A057740 A137307 A078045
%Y A092081 Adjacent sequences: A092078 A092079 A092080 this_sequence A092082 A092083 A092084
%K A092081 nonn,easy,tabl
%O A092081 0,3
%A A092081 Wolfdieter Lang (wolfdieter.lang_AT_physik_DOT_uni-karlsruhe_DOT_de), Mar 19 2004
%I A057740
%S A057740 1,1,1,0,2,1,3,8,1,15,20,0,24,1,45,80,90,144,1,105,350,630,504,210,
%T A057740 720,1,315,1232,3780,1344,5040,5760,0,0,0,0,0,0,0,2688,1,1323,5768,
%U A057740 18900,3024,37800,25920,40320,9072,0,15120,0,0,24192
%N A057740 Table T(n,k) giving number of elements of alternating group A_n with order k, n >= 1, 1<=k<=A051593(n).
%D A057740 J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson, ATLAS of Finite Groups. Oxford Univ. Press, 1985.
%e A057740 1; 1; 1,0,2; 1,3,8; 1,15,20,0,24; ...
%o A057740 (MAGMA) {* Order(g) : g in Alt(6) *};
%Y A057740 Cf. A057731, A054522, A057741, A051593, A000793.
%Y A057740 Sequence in context: A053190 A135299 A092081 this_sequence A137307 A078045 A057300
%Y A057740 Adjacent sequences: A057737 A057738 A057739 this_sequence A057741 A057742 A057743
%K A057740 nonn,tabf,easy,nice
%O A057740 1,5
%A A057740 Roger CUCULIERE (cuculier(AT)imaginet.fr), Oct 29 2000
%E A057740 More terms from njas, Nov 01 2000
%I A137307
%S A137307 1,1,1,1,2,1,3,8,4,8,1,5,18,20,48,16,32,1,7,32,56,160,112,256,64,128,1,
%T A137307 9,50,120,400,432,1120,576,1280,256,512,1,11,72,220,840,1232,3584,2816,
%U A137307 6912,2816,6144,1024,2048,1,13,98,364,1568,2912,9408,9984,26880,16640
%V A137307 1,1,-1,1,2,1,-3,-8,4,8,-1,5,18,-20,-48,16,32,1,-7,-32,56,160,-112,-256,64,128,-1,9,50,
%W A137307 -120,-400,432,1120,-576,-1280,256,512,1,-11,-72,220,840,-1232,-3584,2816,6912,-2816,
%X A137307 -6144,1024,2048,-1,13,98,-364,-1568,2912,9408,-9984,-26880,16640,39424,-13312,-28672
%N A137307 A triangular sequence of coefficients of even plus odd Chebyshev polynomials, A053120: q(x,n)=T(x,2*n-1]+T(x,2*n).
%C A137307 The row sums are all 2, and double integrations are all orthogonal except for the zero to one level.
%C A137307 This arose from an idea of Chladni Chebyshev's:
%C A137307 q(Exp[i*t],n)=T(Cos[2*Pi*t),2*n-1)+T(Sin(2*Pi*t),2*n)
%C A137307 which are strange looping spirals.
%F A137307 q(x,n)=T(x,2*n-1]+T(x,2*n).
%e A137307 {1, 1},
%e A137307 {-1, 1, 2},
%e A137307 {1, -3, -8, 4, 8},
%e A137307 {-1, 5,18, -20, -48, 16, 32},
%e A137307 {1, -7, -32, 56, 160, -112, -256, 64, 128},
%e A137307 {-1, 9, 50, -120, -400, 432, 1120, -576, -1280, 256, 512},
%e A137307 {1, -11, -72, 220, 840, -1232, -3584, 2816, 6912, -2816, -6144, 1024, 2048},
%e A137307 {-1, 13, 98, -364, -1568, 2912, 9408, -9984, -26880, 16640, 39424, -13312, -28672, 4096, 8192},
%e A137307 {1, -15, -128, 560, 2688, -6048, 21504, 28800, 84480, -70400, -180224, 92160, 212992, -61440, -131072, 16384, 32768},
%e A137307 {-1, 17, 162, -816, -4320, 11424, 44352, -71808, -228096, 239360, 658944, -452608, -1118208, 487424, 1105920, -278528, -589824, 65536, 131072},
%e A137307 {1, -19, -200, 1140, 6600, -20064, -84480, 160512, 549120, -695552, -2050048, 1770496, 4659200, -2723840, -6553600, 2490368, 5570560, -1245184, -2621440, 262144, 524288}
%t A137307 Q[x_, n_] := ChebyshevT[2*n - 1, x] + ChebyshevT[2*n, x]; Table[ExpandAll[Q[x, n]], {n, 0, 10}]; a0 = Table[CoefficientList[Q[x, n], x], {n, 0, 10}]; Flatten[a0]
%Y A137307 Cf. A053120.
%Y A137307 Sequence in context: A135299 A092081 A057740 this_sequence A078045 A057300 A076655
%Y A137307 Adjacent sequences: A137304 A137305 A137306 this_sequence A137308 A137309 A137310
%K A137307 nonn,uned,tabl
%O A137307 1,5
%A A137307 Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Apr 20 2008
%I A078045
%S A078045 1,2,1,3,8,7,7,30,37,7,104,171,53,326,721,501,872,2815,2945,1614,10189,
%T A078045 14465,1048,33795,63773,32074,99289,258909,223768,233719,975305,1189122,
%U A078045 253621,3393353,5517976,1617381,10687301,23340634,15888095,28827141,91396504
%V A078045 1,-2,1,3,-8,7,7,-30,37,7,-104,171,-53,-326,721,-501,-872,2815,-2945,-1614,10189,
%W A078045 -14465,1048,33795,-63773,32074,99289,-258909,223768,233719,-975305,1189122,
%X A078045 253621,-3393353,5517976,-1617381,-10687301,23340634,-15888095,-28827141,91396504
%N A078045 Expansion of (1-x)/(1+x+x^2-2*x^3).
%Y A078045 Sequence in context: A092081 A057740 A137307 this_sequence A057300 A076655 A101486
%Y A078045 Adjacent sequences: A078042 A078043 A078044 this_sequence A078046 A078047 A078048
%K A078045 sign
%O A078045 0,2
%A A078045 njas, Nov 17 2002
%I A057300
%S A057300 0,2,1,3,8,10,9,11,4,6,5,7,12,14,13,15,32,34,33,35,40,42,41,43,36,38,
%T A057300 37,39,44,46,45,47,16,18,17,19,24,26,25,27,20,22,21,23,28,30,29,31,48,
%U A057300 50,49,51,56,58,57,59,52,54,53,55,60,62,61,63,128,130,129,131,136,138
%N A057300 Binary counter with odd/even bit positions swapped; base 4 counter with 1's replaced by 2's and vice versa.
%C A057300 A self-inverse permutation of the integers
%C A057300 Counterexamples: a(46) = 29; a(92) = 172; binary(172) = [1, 0, 1, 0, 1, 1, 0, 0], but binary(-2*29+5*92) = binary(402) = [1, 1, 0, 0, 1, 0, 0, 1, 0]. a(45) = 30; a(91) = 167; binary(167) = [1, 0, 1, 0, 0, 1, 1, 1], but -2*30+5*91+2=397 and binary(397) = [1, 1, 0, 0, 0, 1, 1, 0, 1] - Lambert Herrgesell (zero815(AT)googlemail.com), Jan 13 2007
%H A057300 R. Stephan, Some divide-and-conquer sequences ...
%H A057300 R. Stephan, Table of generating functions
%F A057300 It is not true in general that a(2n) = -2a(n) + 5n, a(2n+1) = -2a(n) + 5n + 2 as was originally conjectured.
%Y A057300 Cf. A057301.
%Y A057300 Sequence in context: A057740 A137307 A078045 this_sequence A076655 A101486 A086606
%Y A057300 Adjacent sequences: A057297 A057298 A057299 this_sequence A057301 A057302 A057303
%K A057300 easy,nonn
%O A057300 0,2
%A A057300 Marc LeBrun (mlb(AT)well.com), Aug 24 2000
%I A076655
%S A076655 1,2,1,3,8,25,267,11408,4063825,61874069763,376863853335803168,
%T A076655 33029401504744904404673707225
%V A076655 1,2,1,-3,8,-25,267,-11408,4063825,-61874069763,376863853335803168,
%W A076655 -33029401504744904404673707225
%N A076655 Numerator of a(n), where for n > 2, a(n)=-1/a(n-1)+1/a(n-2), a(1)=1, a(2)=2.
%F A076655 a(n>2)=-1/a(n-1)+1/a(n-2), a(1)=1, a(2)=2, a(n)->-(-1)^n sqrt(2).
%e A076655 a(3)=-1/a(2)+1/a(1)=-1/2+1=1/2, therefore in the sequence, 3rd term is 1.
%Y A076655 Cf. A074935.
%Y A076655 Sequence in context: A137307 A078045 A057300 this_sequence A101486 A086606 A076112
%Y A076655 Adjacent sequences: A076652 A076653 A076654 this_sequence A076656 A076657 A076658
%K A076655 sign,frac
%O A076655 1,2
%A A076655 Zak Seidov (zakseidov(AT)yahoo.com), Oct 24 2002
%I A101486
%S A101486 1,1,2,1,3,9,1,3,17,54,1,3,18,119,378,1,3,18,134,932,2916,1,3,18,135,
%T A101486 1111,7838,24057,1,3,18,135,1133,9833,69275,208494,1,3,18,135,1134,
%U A101486 10176,90959,635279,1876446,1,3,18,135,1134,10205,95635,868827
%N A101486 Square array T(n,k), read by antidiagonals: number of labeled trees, with increments of labels along edges constrained to -1,0,1, with n nodes that have no label greater than k.
%H A101486 M. Bousquet-Melou, Limit laws for embedded trees
%F A101486 G.f. of k-th row: A(t)=B(t)*(1-C(t)^(k+1))*(1-C(t)^(k+4))/[(1-C(t)^(k+2))*(1-C(t)^(k+3))], with B(t) the g.f. of A005159 and C(t) the g.f. of A101487.
%e A101486 1,2,9,54,378,2916,24057,208494,1876446,17399772,
%e A101486 1,3,17,119,932,7838,69275,635279,5994584,57872666,
%e A101486 1,3,18,134,1111,9833,90959,868827,8504314,84866778,
%e A101486 1,3,18,135,1133,10176,95635,928442,9236144,93646430,
%e A101486 1,3,18,135,1134,10205,96191,937361,9365984,95427597,
%e A101486 1,3,18,135,1134,10206,96227,938179,9381050,95673739,
%e A101486 1,3,18,135,1134,10206,96228,938222,9382179,95697199,
%e A101486 1,3,18,135,1134,10206,96228,938223,9382229,95698688,
%Y A101486 Rows converge to A005159. First row is A000168.
%Y A101486 Sequence in context: A078045 A057300 A076655 this_sequence A086606 A076112 A122454
%Y A101486 Adjacent sequences: A101483 A101484 A101485 this_sequence A101487 A101488 A101489
%K A101486 nonn,tabl
%O A101486 0,3
%A A101486 Ralf Stephan, Jan 21 2005
%I A086606
%S A086606 1,1,2,1,3,9,1,4,14,32,1,5,20,55,140,1,6,27,86,243,630,1,7,35,126,392,
%T A086606 1099,2870,1,8,44,176,598,1808,5048,13256,1,9,54,237,873,2835,8433,
%U A086606 23454,61389,1,10,65,310,1230,4272,13495,39640,109400,286710,1,11,77
%N A086606 Triangle, read by rows, where the n-th row is the first n terms of the n-th self-convolution of the sequence formed by flattening this triangle.
%F A086606 The first n terms of the n-th self-convolution forms the n-th row: A={1, _1, 2, _1, 3, 9, _1, 4, 14, 32, _1, 5, 20, 55, 140, ...}; A^2={1, 2, _5, 6, 12, 28, 33, 52, 67, 164, 217, 210, 275, ...}; A^3={1, 3, 9, _16, 33, 72, 125, 222, 330, 646, 1089, 1602, ...}; A^4={1, 4, 14, 32, _73, 164, 334, 660, 1152, 2184, 3960, ...}; A^5={1, 5, 20, 55, 140, _336, 755, 1625, 3195, 6315, 12112, ...}; ...
%e A086606 Rows begin:
%e A086606 {1},
%e A086606 {1,2},
%e A086606 {1,3,9},
%e A086606 {1,4,14,32},
%e A086606 {1,5,20,55,140},
%e A086606 {1,6,27,86,243,630},
%e A086606 {1,7,35,126,392,1099,2870},
%e A086606 {1,8,44,176,598,1808,5048,13256}, ...
%Y A086606 Cf. A086607 (main diagonal), A086608 (row sums).
%Y A086606 Sequence in context: A057300 A076655 A101486 this_sequence A076112 A122454 A083782
%Y A086606 Adjacent sequences: A086603 A086604 A086605 this_sequence A086607 A086608 A086609
%K A086606 nonn
%O A086606 1,3
%A A086606 Paul D. Hanna (pauldhanna(AT)juno.com), Jul 23 2003
%I A076112
%S A076112 1,1,2,1,3,9,1,4,16,64,1,5,25,125,625,1,6,36,216,1296,7776,1,7,49,343,
%T A076112 2401,16807,117649,1,8,64,512,4096,32768,262144,2097152,1,9,81,729,
%U A076112 6561,59049,531441,4782969,43046721,1,10,100,1000,10000,100000,1000000
%N A076112 Triangle (read by rows) in which the n-th row contains first n terms of n geometric progression with first term 1 and common ratio (n-1).
%e A076112 1; 1,2; 1,3,9; 1,4,16,64; 1,5,25,125,625; 1,6,36,216,1296,7776; ...
%Y A076112 Cf. A000169, A076113.
%Y A076112 Sequence in context: A076655 A101486 A086606 this_sequence A122454 A083782 A076240
%Y A076112 Adjacent sequences: A076109 A076110 A076111 this_sequence A076113 A076114 A076115
%K A076112 easy,nonn
%O A076112 1,3
%A A076112 Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Oct 09 2002
%E A076112 More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Apr 20 2003
%I A122454
%S A122454 1,2,1,3,9,1,4,24,18,24,1,5,50,100,100,150,50,1,6,90,225,150,300,1200,
%T A122454 300,300,675,90,1,7,147,441,735,735,3675,2450,3675,1225,7350,3675,735,
%U A122454 2205,147,1,8,224,784,1568,980,1568,9408,15680,11760,15680,3920,29400
%N A122454 A triangle with shape A000041 defined by sequence A098546 times sequence A036040.
%C A122454 Shape sequence for A122454 is A000041 which counts numeric partitions.
%F A122454 A122454(n) = A098546(n) times A036040(n).
%e A122454 A098546(n) begins 1 2 1 3 3 1 4 6 6 4 1 ...
%e A122454 A036040(n) begins 1 1 1 1 3 1 1 4 3 6 1 ...
%e A122454 so the present sequence begins 1 2 1 3 9 1 4 24 18 24 1 ...
%p A122454 sortAbrSteg := proc(L1,L2) local i ; if nops(L1) < nops(L2) then RETURN(true) ; elif nops(L2) < nops(L1) then RETURN(false) ; else for i from 1 to nops(L1) do if op(i,L1) < op(i,L2) then RETURN(false) ; fi ; od ; RETURN(true) ; fi ; end: A098546 := proc(n,k) local prts,m ; prts := combinat[partition](n) ; prts := sort(prts, sortAbrSteg) ; if k <= nops(prts) then m := nops(op(k,prts)) ; binomial(n,m) ; else 0 ; fi ; end: M3 := proc(L) local n,k,an,resul; n := add(i,i=L) ; resul := factorial(n) ; for k from 1 to n do an := add(1-min(abs(j-k),1),j=L) ; resul := resul/ (factorial(k))^an /factorial(an) ; od ; end: A036040 := proc(n,k) local prts,m ; prts := combinat[partition](n) ; prts := sort(prts, sortAbrSteg) ; if k <= nops(prts) then M3(op(k,prts)) ; else 0 ; fi ; end: A122454 := proc(n,k) A098546(n,k)*A036040(n,k) ; end: for n from 1 to 10 do for k from 1 to combinat[numbpart](n) do a:=A122454(n,k) ; printf("%d, ",a) ; od; od ; - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jul 17 2007
%Y A122454 Cf. A122455.
%Y A122454 Sequence in context: A101486 A086606 A076112 this_sequence A083782 A076240 A099644
%Y A122454 Adjacent sequences: A122451 A122452 A122453 this_sequence A122455 A122456 A122457
%K A122454 easy,nonn,tabf
%O A122454 1,2
%A A122454 Alford Arnold (Alford1940(AT)aol.com), Sep 18 2006
%E A122454 More terms from R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jul 17 2007
%I A083782
%S A083782 1,1,2,1,3,9,2,4,6,16
%N A083782 n-th row of the following triangle contains n distinct natural numbers such that every sum of n-1 of them +1 is a prime,n >1, with a(1) = 1 by convention. Sequence contains the triangle by rows.
%e A083782 1
%e A083782 1 2
%e A083782 1 3 9
%e A083782 2 4 10 16
%e A083782 ...
%Y A083782 Cf. A083783, A083784.
%Y A083782 Sequence in context: A086606 A076112 A122454 this_sequence A076240 A099644 A058113
%Y A083782 Adjacent sequences: A083779 A083780 A083781 this_sequence A083783 A083784 A083785
%K A083782 more,nonn,tabl
%O A083782 1,3
%A A083782 Amarnath Murthy and Meenakshi Srikanth (amarnath_murthy(AT)yahoo.com), May 07 2003
%I A076240
%S A076240 1,2,1,3,9,2,8,10,14,22,3,9,15,19,23,29,41,39,63,69,2,6,16,16,24,42,48,
%T A076240 52,54,52,74,84,88,102,114,122,134,152,156,166,168,1,7,13,19,23,31,71,
%U A076240 71,73,73,65,77,91,79,91,109,115,125,137,149,155,185,197,203,197,235
%N A076240 Remainder when 2nd order prime pp[n]=A006450(n) is divided by n-th prime=A000040(n).
%F A076240 a(n)=Mod[pp(n), p(n)]=Mod[A006450(n), A000040(n)]
%Y A076240 Cf. A006450, A038580, A049090, A049203, A049202, A057809, A076240-A076243.
%Y A076240 Sequence in context: A076112 A122454 A083782 this_sequence A099644 A058113 A126009
%Y A076240 Adjacent sequences: A076237 A076238 A076239 this_sequence A076241 A076242 A076243
%K A076240 nonn
%O A076240 1,2
%A A076240 Labos E. (labos(AT)ana.sote.hu), Oct 08 2002
%I A099644
%S A099644 1,2,1,3,9,2,8,11,23,23,5,9,15,19,17,31,31,41,41,55,2,14,16,16,31,49,54,
%T A099644 52,61,59,109,111,107,117,121,164,166,169,171,181,11,23,41,35,29,29,77,
%U A099644 71,77,71,89,83,95,107,113,125,155,149,167,173,185,185,203,197,203,209
%N A099644 a[n]=Mod[q(n),PrimePi[q(n)]]=Mod[A001359(n), A000720[A001359(n))] where q(n) is the n-th lesser-twin-prime. A004648 restricted to lesser twins.
%C A099644 Sequence display diagram similar to that of A004648.
%e A099644 n=9: p(26)=101 is the 9th lesser-twin-prime,
%e A099644 a(9)-Mod[p(26),26]=Mod[101,26]=23=a(9).
%Y A099644 Cf. A001359, A004648.
%Y A099644 Sequence in context: A122454 A083782 A076240 this_sequence A058113 A126009 A091858
%Y A099644 Adjacent sequences: A099641 A099642 A099643 this_sequence A099645 A099646 A099647
%K A099644 nonn
%O A099644 1,2
%A A099644 Labos E. (labos(AT)ana.sote.hu), Nov 04 2004
%I A058113
%S A058113 1,2,1,3,9,3,11,37,52,12,112,224,434,383,68,4398,1984,3976,5568,3256,
%T A058113 502,1181185,47622,44581,74851,74604,31881,4158
%N A058113 Triangle: Number of asymmetric semigroups of order n with k idempotents.
%H A058113 Index entries for sequences related to semigroups
%e A058113 1; 2,1; 3,9,3; 11,37,52,12; 112,224,434,383,68; ...
%Y A058113 Row sums give A058104. Main diagonal: A058115. Column 1: A058114.
%Y A058113 Sequence in context: A083782 A076240 A099644 this_sequence A126009 A091858 A070165
%Y A058113 Adjacent sequences: A058110 A058111 A058112 this_sequence A058114 A058115 A058116
%K A058113 nonn,tabl,hard
%O A058113 1,2
%A A058113 Christian G. Bower (bowerc(AT)usa.net), Nov 09 2000
%E A058113 Updated Feb 19 2001
%I A126009
%S A126009 0,2,1,3,9,10,8,12,6,4,5,11,7,17,18,19,20,13,14,15,16,21
%N A126009 Self-inverse permutation of integers induced when A106485 is rectricted to A126011.
%C A126009 The Scheme-program given cannot in practice compute this further than n=21, as A106485(A126011(22))=36893488147419103232. However, the further terms could be deduced by other means. This sequence is permutation of nonnegative integers because combinatorial games form a group under (game) addition, and each game has a well-defined, unique negative.
%H A126009 A. Karttunen, Scheme-program for computing this sequence.
%H A126009 Index entries for sequences that are permutations of the natural numbers
%F A126009 a(n) = A126013(A106485(A126011(n))).
%Y A126009 Sequence in context: A076240 A099644 A058113 this_sequence A091858 A070165 A119928
%Y A126009 Adjacent sequences: A126006 A126007 A126008 this_sequence A126010 A126011 A126012
%K A126009 nonn
%O A126009 0,2
%A A126009 Antti Karttunen (His-Firstname.His-Surname(AT)gmail.com), Jan 02 2007
%I A091858
%S A091858 1,2,1,3,10,5,8,2,9,1,22,7,4,40,29,16,23,60,48,51,49,24,74,40,15,85,3,
%T A091858 41,16,42,119,43,51,73,14,23,150,49,104,20,128,44,185,66,146,159,178,
%U A091858 150,44,51,48,4,134,143,118,143,242,141,149,108,115,205,306,196,292,242
%N A091858 n! mod prime(n).
%C A091858 No term is 0. For positions of 1's see A067999.
%t A091858 A091858[n_] := Block[{k = p = 1), m = Prime[n]}, While[p = Mod[p k, m]; k < n, k++ ]; p]; Table[ f[ n], {n, 66}] (from Robert G. Wilson v Mar 16 2004)
%Y A091858 Sequence in context: A099644 A058113 A126009 this_sequence A070165 A119928 A122050
%Y A091858 Adjacent sequences: A091855 A091856 A091857 this_sequence A091859 A091860 A091861
%K A091858 nonn
%O A091858 1,2
%A A091858 Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Mar 13 2004
%E A091858 Corrected and extended by Robert G. Wilson v (rgwv(AT)rgwv.com), Mar 16 2004
%I A070165
%S A070165 1,2,1,3,10,5,16,8,4,2,1,4,2,1,5,16,8,4,2,1,6,3,10,5,16,8,4,2,1,7,22,
%T A070165 11,34,17,52,26,13,40,20,10,5,16,8,4,2,1,8,4,2,1,9,28,14,7,22,11,34,17,
%U A070165 52,26,13,40,20,10,5,16,8,4,2,1,10,5,16,8,4,2,1,11,34,17,52,26,13
%N A070165 Triangle read by rows giving trajectory of n in Collatz problem.
%C A070165 n-th row has A006577(n) entries.
%H A070165 T. D. Noe, Rows n=1..100 of triangle, flattened
%H A070165 Eric Weisstein's World of Mathematics, Collatz Problem
%H A070165 Index entries for sequences related to 3x+1 (or Collatz) problem
%e A070165 1; 2,1; 3,10,5,16,8,4,2,1; 4,2,1; 5,16,8,4,2,1; ...
%Y A070165 Cf. A006667.
%Y A070165 Sequence in context: A058113 A126009 A091858 this_sequence A119928 A122050 A081323
%Y A070165 Adjacent sequences: A070162 A070163 A070164 this_sequence A070166 A070167 A070168
%K A070165 nonn,easy,tabf
%O A070165 1,2
%A A070165 Eric Weisstein (eric(AT)weisstein.com), Apr 23, 2002
%I A119928
%S A119928 2,1,3,11,10,2,1,1,2,5,2,4,23,45,1,3,9,4,3,1,2,1,24,13,1,1,1,1,2,1,1,1,
%T A119928 1,2,3,1,2,51,2,7,2,1198400,1,3,2,10,1,11,13,1,2,1,4,1,10,3,13,1,1,1,2,
%U A119928 13,1,1,122,148,1,48,3,1,46,1,1,1,4,2,1,5,4,1,2,1,1,8,1,8,5,1,7,1,2,1,1
%N A119928 Continued fraction expansion of the value of Minkowski's question mark function at Khinchin's constant (A002210).
%H A119928 Index entries for Minkowski's question mark function
%H A119928 Index entries for sequences related to Minkowski's question mark function
%t A119928 ContinuedFraction[(cf = ContinuedFraction[Khinchin, 80(*arbitrary precision*)]; IntegerPart[Khinchin] + Sum[(-1)^(k)/2^(Sum[cf[[i]], {i, 2, k}] - 1), {k, 2, Length[cf]}])]
%Y A119928 Cf. A119929.
%Y A119928 Sequence in context: A126009 A091858 A070165 this_sequence A122050 A081323 A007447
%Y A119928 Adjacent sequences: A119925 A119926 A119927 this_sequence A119929 A119930 A119931
%K A119928 cofr,nonn
%O A119928 0,1
%A A119928 Joseph Biberstine (jrbibers(AT)indiana.edu), May 29 2006; corrected Jun 04 2006
%I A122050
%S A122050 0,1,2,1,3,11,53,317,2216,17717,159400,1593683,17528297,210321847,
%T A122050 2734024611,38274750871,574103734768,9185449434441,156149906360886,
%U A122050 2810660039745077,53401966651421695,1068030147578999459
%N A122050 An n-th level recursion:a(n) = (n-1)a(n - 1) - a(n - 4).
%F A122050 a(n) = (n-1)a(n - 1) -a(n - 4)
%t A122050 a[0] = 0; a[1] = 1; a[2] = 2; a[3] = 1; a[n_] := a[n] = (n - 1)*a[n - 1] - a[n - 4] Table[a[n], {n, 0, 30}]
%Y A122050 Cf. A122022.
%Y A122050 Sequence in context: A091858 A070165 A119928 this_sequence A081323 A007447 A095852
%Y A122050 Adjacent sequences: A122047 A122048 A122049 this_sequence A122051 A122052 A122053
%K A122050 nonn
%O A122050 1,3
%A A122050 Roger Bagula (rlbagulatftn(AT)yahoo.com), Sep 13 2006
%I A081323
%S A081323 2,1,3,11,322,1149851,425730551631123,
%T A081323 208406472252232726621841472637412401,
%U A081323 18490864749804416780204061487408593066264011288598603441079481989361240762271783
%N A081323 L(P(n)), where L(n) = Lucas numbers A000032, P(n) = Pell numbers A000129.
%t A081323 l[n_] := l[n] = l[n - 1] + l[n - 2]; l[0] = 2; l[1] = 1; p[n_] := p[n] = 2p[n - 1] + p[n - 2]; p[0] = 0; p[1] = 1; Table[l[p[n]], {n, 0, 8}]
%Y A081323 Cf. A081322.
%Y A081323 Sequence in context: A070165 A119928 A122050 this_sequence A007447 A095852 A000618
%Y A081323 Adjacent sequences: A081320 A081321 A081322 this_sequence A081324 A081325 A081326
%K A081323 easy,nonn
%O A081323 0,1
%A A081323 Mario Catalani (mario.catalani(AT)unito.it), Mar 18 2003
%I A007447 M0159
%S A007447 2,1,3,12,59,354,2535,21190,202731,2183462,26130441,343956264,
%T A007447 4938891841,76827253854,1287026203647,23100628140676,
%U A007447 442271719973507,8996704216880580,193776558133638811
%V A007447 2,-1,3,-12,59,-354,2535,-21190,202731,-2183462,26130441,-343956264,
%W A007447 4938891841,-76827253854,1287026203647,-23100628140676,
%X A007447 442271719973507,-8996704216880580,193776558133638811
%N A007447 Logarithm of e.g.f. for primes.
%Y A007447 Sequence in context: A119928 A122050 A081323 this_sequence A095852 A000618 A132950
%Y A007447 Adjacent sequences: A007444 A007445 A007446 this_sequence A007448 A007449 A007450
%K A007447 sign
%O A007447 1,1
%A A007447 njas
%E A007447 Signs from Christian G. Bower (bowerc(AT)usa.net), Nov 15 1998
%I A095852
%S A095852 1,2,1,3,16,1,4,81,512,1,5,256,19683,65536,1,6,625,262144,43046721,
%T A095852 33554432,1,7,1296,1953125,4294967296,847288609443,68719476736,1,8,2401,
%U A095852 10077696,152587890625,1125899906842624,150094635296999121
%N A095852 Triangle read by rows: T(n,k) = (n-k+1)^(k^2), n>=1, 1<=k<=n.
%Y A095852 Sequence in context: A122050 A081323 A007447 this_sequence A000618 A132950 A106169
%Y A095852 Adjacent sequences: A095849 A095850 A095851 this_sequence A095853 A095854 A095855
%K A095852 easy,nonn,tabl
%O A095852 1,2
%A A095852 Herman Jamke (hermanjamke(AT)fastmail.fm), Jul 10 2004
%I A000618 M0160 N0063
%S A000618 2,1,3,16,380,1227756,400507805615570
%N A000618 Nondegenerate Boolean functions of n variables.
%D A000618 S. Muroga, Threshold Logic and Its Applications. Wiley, NY, 1971, p. 38, Table 2.3.2. - Row 12.
%D A000618 J. Sklansky, General synthesis of tributary switching networks, IEEE Trans. Elect. Computers, 12 (1963), 464-469.
%H A000618 Index entries for sequences related to Boolean functions
%Y A000618 Sequence in context: A081323 A007447 A095852 this_sequence A132950 A106169 A014015
%Y A000618 Adjacent sequences: A000615 A000616 A000617 this_sequence A000619 A000620 A000621
%K A000618 nonn,nice,easy,more
%O A000618 0,1
%A A000618 njas
%I A132950
%S A132950 1,1,2,1,3,18,1,4,48,1728,1,5,100,8000,2560000,1,6,180,27000,20250000,
%T A132950 75937500000,1,7,294,74088,112021056,1016255020032,55316793250381824,1,
%U A132950 8,448,175616,481890304,9256148959232,1244544764462497792
%N A132950 Generalization of an a(n)=3*2^n*a(n-1) as 3=(m+1) and 2=m To give general term: t(n,m)=a(n)=(m+1)^n*m^(n*(n-1)/2) ( here n taken first).
%C A132950 From the ratio: a[2*n+1]/a[n]=(p/q)^(2*n)/(1/q)^(2*n+1)=q*p^(2*n) where p/q+1/q=1 or q=p+1 to give a[2*n+1]=(p+1)*p^(2*n)*a[2*n) Substitution of 2*n+1=m gives: a[m]=(p+1)*p^(m-1)*a[m] The general term is: a[n]=(p+1)^n*p(n*n-1)/2) Tha generalizes to the triangular sequence: t{n,m]=(m+1)^n*m^(n*(n-1)/2) There are a sequence of integer sequences. The row sums are: Table[If[m == 0, 1, (m + 1)^n*m^(n*(n - 1)/2)], {n, 0, m}]], {m, 0, 10}]; {1, 3, 22, 1781, 2568106, 75957777187, 55317809617497302, 1171356820508008315371465, 832644723581477539857134797829266, 22528399597273938808766298802728163594239911, 25937424603357947693143588829487172771562610524642332222}
%F A132950 If m==0,t(n,0)=1 else t(n,m)=a(n)=(m+1)^n*m^(n*(n-1)/2)
%e A132950 {1},
%e A132950 {1, 2},
%e A132950 {1, 3, 18},
%e A132950 {1, 4, 48, 1728},
%e A132950 {1, 5, 100, 8000, 2560000},
%e A132950 {1, 6, 180, 27000, 20250000, 75937500000},
%e A132950 {1, 7, 294, 74088, 112021056, 1016255020032, 55316793250381824}
%t A132950 a = Table[Table[If[m == 0, 1, (m + 1)^n*m^(n*(n - 1)/2)], {n, 0, m}], {m, 0, 10}]; Flatten[a]
%Y A132950 Sequence in context: A007447 A095852 A000618 this_sequence A106169 A014015 A108353
%Y A132950 Adjacent sequences: A132947 A132948 A132949 this_sequence A132951 A132952 A132953
%K A132950 nonn,tabl,uned
%O A132950 1,3
%A A132950 Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Nov 19 2007
%I A106169
%S A106169 1,2,1,3,19,1,1020
%N A106169 Number of inequivalent codes attaining highest minimal Hamming distance of any Type (4_II)^H+ even Hermitian additive self-dual code over GF(4) of length 2n.
%C A106169 The minimal distance of these codes is (so far) 2,2,4,4,4,6.
%D A106169 C. Bachoc and P. Gaborit, On extremal additive F_4 codes of length 10 to 18, in International Workshop on Coding and Cryptography (Paris, 2001), Electron. Notes Discrete Math. 6 (2001), 10 pp.
%D A106169 P. Gaborit, W. C. Huffman, J.-L. Kim and V. S. Pless, On additive GF(4) codes, in Codes and Association Schemes (Piscataway, NJ, 1999), A. Barg and S. Litsyn, eds., Amer. Math. Soc., Providence, RI, 2001, pp. 135-149.
%D A106169 G. Hoehn, Self-dual codes over the Kleinian four-group, Math. Ann. 327 (2003), 227-255.
%D A106169 W. C. Huffman, On the classification and enumeration of self-dual codes, Finite Fields Applic. 11 (2005), 451-490.
%D A106169 W. C. Huffman, Additive self-dual codes over F_4 with an automorphism of odd prime order, Adv. Math. Commun., 1 (2007), 357-398.
%H A106169 A. R. Calderbank, E. M. Rains, P. W. Shor and N. J. A. Sloane, Quantum error correction via codes over GF(4), IEEE Trans. Inform. Theory, 44 (1998), 1369-1387.
%H A106169 G. Nebe, E. M. Rains and N. J. A. Sloane, Self-Dual Codes and Invariant Theory, Springer, Berlin, 2006.
%H A106169 E. M. Rains and N. J. A. Sloane, Self-dual codes, pp. 177-294 of Handbook of Coding Theory, Elsevier, 1998 (Abstract, pdf, ps).
%Y A106169 Cf. A105687.
%Y A106169 Cf. also A105674, A105675, A105676, A105677, A105678, A016729, A066016, A105681, A105682.
%Y A106169 Sequence in context: A095852 A000618 A132950 this_sequence A014015 A108353 A059333
%Y A106169 Adjacent sequences: A106166 A106167 A106168 this_sequence A106170 A106171 A106172
%K A106169 nonn
%O A106169 1,2
%A A106169 njas, May 09 2005
%I A014015
%S A014015 2,1,3,19,983,1140455,25739184407616,687786653376698575362597850,
%T A014015 531547061991816754123214108000546228669815660470108834
%N A014015 Alternating Egyptian fraction expansion of e-2.
%H A014015 Index entries for sequences related to Egyptian fractions
%Y A014015 Sequence in context: A000618 A132950 A106169 this_sequence A108353 A059333 A106485
%Y A014015 Adjacent sequences: A014012 A014013 A014014 this_sequence A014016 A014017 A014018
%K A014015 nonn
%O A014015 0,1
%A A014015 Simon Plouffe (plouffe(AT)math.uqam.ca)
%I A108353
%S A108353 2,1,3,20,756,178200
%N A108353 For each nonnegative integer n, a(n) is the smallest positive integer j whose primal code characteristic is n, that is, the smallest j such that A108352(j) = n.
%C A108353 Suggested by Antti Karttunen.
%e A108353 Writing (prime(i))^j as i:j, we have the following table:
%e A108353 Primal Functions and Functional Digraphs for a(0) to a(5)
%e A108353 ` ` ` 2 = 1:1 ` ` ` ` ` ` || 1 -> 1 (infinite loop)
%e A108353 ` ` ` 1 = { } ` ` ` ` ` ` || 1
%e A108353 ` ` ` 3 = 2:1 ` ` ` ` ` ` || 2 -> 1
%e A108353 ` ` `20 = 1:2 3:1 ` ` ` ` || 3 -> 1 -> 2
%e A108353 ` ` 756 = 1:2 2:3 4:1 ` ` || 4 -> 1 -> 2 -> 3
%e A108353 `178200 = 1:3 2:4 3:2 5:1 || 5 -> 1 -> 3 -> 2 -> 4
%Y A108353 Cf. A106177, A108352, A108370, A108371, A108372, A108373, A108374, A111801.
%Y A108353 Sequence in context: A132950 A106169 A014015 this_sequence A059333 A106485 A126008
%Y A108353 Adjacent sequences: A108350 A108351 A108352 this_sequence A108354 A108355 A108356
%K A108353 nonn
%O A108353 0,1
%A A108353 Jon Awbrey (jawbrey(AT)att.net), Jun 17 2005, extended Aug 20 2005
%I A059333
%S A059333 2,1,3,23,5,2,2,73,1,2,3,52,2,1,3,227,5,14,2,44,1,5,2,232,1,2,1,4,5,66,
%T A059333 2,1669,1,1,7,92,2,1,3,344,4,6,3,1,11,10,2,976,3,22,9,2,2,10,11,328,1,
%U A059333 5,3,4,9,13,9,3581,3,6,2,4,7,10,3,952,8,2,1,4,4,3,3,944,15
%N A059333 For 0<=A, 0<=B, n is an A-almost prime; m is a B-almost prime, k = n+m, k is a C-almost prime; a(n) = smallest number m such that A+B=C.
%e A059333 E.g. [ n=2 (A=1), m=1 (B=0), k=n+m=3 (C=A+B=1), so a(2)=m=1 ]; [ n=4 (A=2), m=23 (B=1), k=n+m=27 (C=A+B=3), so a(4)=m=23 ]
%Y A059333 Sequence in context: A106169 A014015 A108353 this_sequence A106485 A126008 A096098
%Y A059333 Adjacent sequences: A059330 A059331 A059332 this_sequence A059334 A059335 A059336
%K A059333 easy,nonn
%O A059333 1,1
%A A059333 Naohiro Nomoto (6284968128(AT)geocities.co.jp), Jan 26 2001
%I A106485
%S A106485 0,2,1,3,32,34,33,35,16,18,17,19,48,50,49,51,8,10,9,11,40,42,41,43,24,
%T A106485 26,25,27,56,58,57,59,4,6,5,7,36,38,37,39,20,22,21,23,52,54,53,55,12,14,
%U A106485 13,15,44,46,45,47,28,30,29,31,60,62,61,63,128,130,129,131,160,162
%N A106485 CGT-tree negating involution of nonnegative integers.
%C A106485 This involution negates game trees used in the combinatorial game theory, when they are encoded in the way explained in A106486.
%C A106485 Cycles are confined into ranges [a(n),a(n+1)[, where a(0)=0, and a(n+1)=2^(2*a(n)), i.e. the ranges are [0,0], [1,3], [4,255], [256,(2^512)-1], ...
%H A106485 Index entries for sequences that are permutations of the natural numbers
%o A106485 (Scheme:) (define (A106485 n) (let loop ((n n) (i 0) (s 0)) (cond ((zero? n) s) ((odd? n) (loop (/ (- n 1) 2) (1+ i) (+ s (if (even? i) (expt 2 (+ 1 (* 2 (A106485 (/ i 2))))) (expt 2 (* 2 (A106485 (/ (- i 1) 2)))))))) (else (loop (/ n 2) (1+ i) s)))))
%Y A106485 A057300 is a "shallow" version which just swaps the left and right options of the game tree, but does not reflect the subtrees themselves. Cf. A106486-A106487.
%Y A106485 Sequence in context: A014015 A108353 A059333 this_sequence A126008 A096098 A096097
%Y A106485 Adjacent sequences: A106482 A106483 A106484 this_sequence A106486 A106487 A106488
%K A106485 nonn
%O A106485 0,2
%A A106485 Antti Karttunen (His-Firstname.His-Surname(AT)iki.fi), May 21 2005
%I A126008
%S A126008 0,2,1,3,32,34,33,35,16,18,17,19,48,50,49,51,8,10,9,11,40,42,41,43,24,
%T A126008 26,25,27,56,58,57,59,4,6,5,7,36,38,37,39,20,22,21,23,52,54,53,55,12,
%U A126008 14,13,15,44,46,45,47,28,30,29,31,60,62,61,63,512,514,513,515,544,546
%N A126008 Involution of nonnegative integers: composition of involutions A057300 and A126007.
%H A126008 Index entries for sequences that are permutations of the natural numbers
%Y A126008 a(n) = A057300(A126007(n)) = A126007(A057300(n)). The first 64 terms are identical with A106485.
%Y A126008 Sequence in context: A108353 A059333 A106485 this_sequence A096098 A096097 A016585
%Y A126008 Adjacent sequences: A126005 A126006 A126007 this_sequence A126009 A126010 A126011
%K A126008 nonn,base
%O A126008 0,2
%A A126008 Antti Karttunen (His-Firstname.His-Surname(AT)gmail.com), Jan 02 2007
%I A096098
%S A096098 2,1,3,71,7,21,599,173,11,23,161,49,13,9,131,19,33,17,1489,331,3989,69,
%T A096098 3097350956401900335673788279883089441874368101,349387,5651,443,29,51
%N A096098 a(1) = 2, a(2) = 1; for n >= 3, a(n) = least number not included earlier that divides the concatenation of all previous terms.
%C A096098 Conjecture:(1) Every concatenation is squarefree. (2) This is a rearrangement of the squarefree numbers not divisible by 5.
%C A096098 The concatenations are not always square-free, since a(12)=49 and a(14)=9. There are no more even numbers in the sequence since odd a(n) => odd concatenation => odd a(n+1). Conjecture:(3) the sequence for n>=2 is a permutation of the positive integers not divisible by 2 or 5. a(29) is probably 479470832244949, in which case the sequence continues 479470832244949, 661, 1129, 1873, 181. - Martin Fuller (martin_n_fuller(AT)btinternet.com), Nov 21 2007
%e A096098 a(6) = 21 as 213717 = 3*7*10177.
%Y A096098 Cf. A096097.
%Y A096098 Sequence in context: A059333 A106485 A126008 this_sequence A096097 A016585 A127192
%Y A096098 Adjacent sequences: A096095 A096096 A096097 this_sequence A096099 A096100 A096101
%K A096098 base,more,nonn
%O A096098 1,1
%A A096098 Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Jun 24 2004
%E A096098 More terms from R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Aug 03 2007
%E A096098 a(23)-a(26) from njas, Nov 10 2007
%E A096098 Corrected and extended by Martin Fuller (martin_n_fuller(AT)btinternet.com), Nov 21 2007
%I A096097
%S A096097 2,1,3,71,7,10177,2100001,101770000001,4603,13,107,4013,23,
%T A096097 3097349301044927552199565217412468305904367
%N A096097 a(1) = 2, a(2) = 1; for n >= 3, a(n) = least prime not included earlier that divides the concatenation of all previous terms.
%C A096097 Conjecture:(1) Every concatenation is squarefree. (2) This is a rearrangement of the noncomposite numbers other than 5.
%e A096097 a(4) = 71 as 213 = 3*71.
%Y A096097 Cf. A096098.
%Y A096097 Sequence in context: A106485 A126008 A096098 this_sequence A016585 A127192 A122141
%Y A096097 Adjacent sequences: A096094 A096095 A096096 this_sequence A096098 A096099 A096100
%K A096097 base,more,nonn
%O A096097 1,1
%A A096097 Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Jun 24 2004
%I A016585
%S A016585 2,1,4,0,0,6,6,1,6,3,4,9,6,2,7,0,7,7,0,8,3,2,3,0,2,4,9,6,4,1,4,9,4,
%T A016585 9,9,6,7,5,1,2,7,0,2,8,7,8,2,2,5,4,8,9,5,3,3,1,7,6,5,5,7,7,2,8,3,8,
%U A016585 8,8,9,8,9,5,3,8,3,1,2,3,6,5,6,5,3,0,6,2,3,1,5,4,1,0,4,1,0,8,4,2,6
%N A016585 Decimal expansion of ln(17/2).
%Y A016585 Sequence in context: A126008 A096098 A096097 this_sequence A127192 A122141 A091604
%Y A016585 Adjacent sequences: A016582 A016583 A016584 this_sequence A016586 A016587 A016588
%K A016585 nonn,cons
%O A016585 1,1
%A A016585 njas
%I A127192
%S A127192 1,2,1,4,0,1,5,2,0,1,8,0,0,0,1,8,4,2,0,0,1,12,0,0,0,0,0,1,8,4,2,0,0,1,
%T A127192 12,0,0,0,0,0,1,12,5,0,2,0,0,0,1,16,0,4,0,0,0,0,0,1,16,8,0,0,2,0,0,0,0,
%U A127192 1
%N A127192 Triangle, square of A054523; row sums = A018804.
%C A127192 Row sums = A018804: (1, 3, 5, 8, 9, 15,...), Sum of gcd(k,n) for 1<= k <= n. Left column = A029935: (1, 2, 4, 5, 8, 8, 12, 12,...). A127192 * d(n) = d(n) * n, or A127192 * A000005 = A038040 = (1, 4, 6, 12, 10, 24, 14,...).
%F A127192 Triangle, Square of A054523 By columns, parsed terms of A029935 interspersed with (k-1) zeros.
%e A127192 First few rows of the triangle are:
%e A127192 1;
%e A127192 2, 1;
%e A127192 4, 0, 1;
%e A127192 5, 2, 0, 1;
%e A127192 8, 0, 0, 0, 1;
%e A127192 8, 4, 2, 0, 0, 1;
%e A127192 12, 0, 0, 0, 0, 0, 1;
%e A127192 12, 5, 0, 2, 0, 0, 0, 1;
%e A127192 16, 0, 4, 0, 0, 0, 0, 0, 1;
%e A127192 ...
%Y A127192 Cf. A029935, A018804, A054523, A000005, A038040.
%Y A127192 Sequence in context: A096098 A096097 A016585 this_sequence A122141 A091604 A137629
%Y A127192 Adjacent sequences: A127189 A127190 A127191 this_sequence A127193 A127194 A127195
%K A127192 nonn,tabl
%O A127192 1,2
%A A127192 Gary W. Adamson (qntmpkt(AT)yahoo.com), Jan 07 2007
%I A122141
%S A122141 1,1,2,1,4,0,1,6,4,0,1,8,12,0,2,1,10,24,8,4,0,1,12,40,32,6,8,0,1,14,60,
%T A122141 80,24,24,0,0,1,16,84,160,90,48,24,0,0,1,18,112,280,252,112,96,0,4,2,1,
%U A122141 20,144,448,574,312,240,64,12,4,0,1,22,180,672,1136,840,544,320,24,30,8
%N A122141 Array: T(d,n) = number of ways of writing n as a sum of d squares, read along diagonals.
%H A122141 Index entries for sequences related to sums of squares
%F A122141 T(1,n)=A000122(n); T(2,n)=A004018(n); T(3,n)=A005875(n); T(4,n)=A000118(n); T(5,n)=A000132(n); T(6,n)=A000141(n); T(7,n)=A008451(n); T(8,n)=A000143(n); T(9,n)=A008452(n); T(10,n)=A000144(n); T(d,0)=1; T(d,1)=A005843(d); T(d,2)=A046092(d).
%e A122141 Array T(d,n) with rows d=1,2,3... and columns n=0,1,2,3.. reads
%e A122141 1 2 0 0 2 0 0 0 0 2 0 ...
%e A122141 1 4 4 0 4 8 0 0 4 4 8 ...
%e A122141 1 6 12 8 6 24 24 0 12 30 24 ...
%e A122141 1 8 24 32 24 48 96 64 24 104 144 ...
%e A122141 1 10 40 80 90 112 240 320 200 250 560 ...
%e A122141 1 12 60 160 252 312 544 960 1020 876 1560 ...
%e A122141 1 14 84 280 574 840 1288 2368 3444 3542 4424 ...
%e A122141 1 16 112 448 1136 2016 3136 5504 9328 12112 14112 ...
%e A122141 1 18 144 672 2034 4320 7392 12672 22608 34802 44640 ...
%e A122141 1 20 180 960 3380 8424 16320 28800 52020 88660 129064
%p A122141 T := proc(d,n) local i,cnts ; cnts := 0 ; for i from -trunc(sqrt(n)) to trunc(sqrt(n)) do if n-i^2 >= 0 then if d > 1 then cnts := cnts+T(d-1,n-i^2) ; elif n-i^2 = 0 then cnts := cnts+1 ; fi ; fi ; od ; RETURN(cnts) ; end: for diag from 1 to 14 do for n from 0 to diag-1 do d := diag-n ; printf("%d,",T(d,n)) ; od ; od;
%Y A122141 Sequence in context: A096097 A016585 A127192 this_sequence A091604 A137629 A087569
%Y A122141 Adjacent sequences: A122138 A122139 A122140 this_sequence A122142 A122143 A122144
%K A122141 nonn,tabl
%O A122141 1,3
%A A122141 R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Oct 29 2006
%I A091604
%S A091604 1,2,1,4,0,1,6,4,0,1,10,4,2,0,1,17,8,4,2,0,1,25,12,7,2,2,0,1,38,25,9,7,
%T A091604 2,2,0,1,57,33,19,8,5,2,2,0,1,84,57,27,16,8,5,2,2,0,1,120,81,46,22,15,6,
%U A091604 5,2,2,0,1,174,129,68,41,19,15,6,5,2,2,0,1,243,182,107,56,36,18,13,6
%N A091604 Matrix square of triangle A091602.
%e A091604 1; 2,1; 4,0,1; 6,4,0,1; 10,4,2,0,1; ...
%Y A091604 Row sums: A091610. Column 1: A091611.
%Y A091604 Sequence in context: A016585 A127192 A122141 this_sequence A137629 A087569 A048614
%Y A091604 Adjacent sequences: A091601 A091602 A091603 this_sequence A091605 A091606 A091607
%K A091604 nonn,tabl
%O A091604 1,2
%A A091604 Christian G. Bower (bowerc(AT)usa.net), Jan 23 2004
%I A137629
%S A137629 1,2,1,4,0,1,7,2,0,1,11,2,0,0,1,18,4,2,0,0,1,26,4,2,0,0,0,1,39,9,2,2,0,
%T A137629 0,0,1,55,9,4,2,0,0,0,0,1,79,16,4,2,2,0,0,0,0,1,106,18,6,2,2,0,0,0,0,0,
%U A137629 1,150,29,9,4,2,2,0,0,0,0,0,1
%N A137629 Triangle read by rows, A026794^2.
%C A137629 Row sums = A137630: (1, 3, 5, 10, 14, 25, 33, 53, 71, 104,...). Left border = A137631: (1, 2, 4, 7, 11, 18, 26, 39, 55, 79,...).
%F A137629 Square of the partition triangle
%e A137629 First few rows of the triangle are:
%e A137629 1;
%e A137629 2, 1;
%e A137629 4, 0, 1;
%e A137629 7, 2, 0, 1;
%e A137629 11, 2, 0, 0, 1;
%e A137629 18, 4, 2, 0, 0, 1;
%e A137629 26, 4, 2, 0, 0, 0, 1;
%e A137629 39, 9, 2, 2, 0, 0, 0, 1;
%e A137629 ...
%Y A137629 Cf. A016794, A137630, A137631.
%Y A137629 Sequence in context: A127192 A122141 A091604 this_sequence A087569 A048614 A001442
%Y A137629 Adjacent sequences: A137626 A137627 A137628 this_sequence A137630 A137631 A137632
%K A137629 nonn,tabl
%O A137629 1,2
%A A137629 Gary W. Adamson (qntmpkt(AT)yahoo.com), Jan 31 2008
%I A087569
%S A087569 0,0,1,0,1,0,2,1,4,0,2,0,7,1,1,0,1,0,2,1,1,0,5,1
%N A087569 Erroneous version of A092928.
%Y A087569 Sequence in context: A122141 A091604 A137629 this_sequence A048614 A001442 A059781
%Y A087569 Adjacent sequences: A087566 A087567 A087568 this_sequence A087570 A087571 A087572
%K A087569 dead
%O A087569 2,7
%I A048614
%S A048614 0,0,0,1,1,2,1,4,0,3,0,4,0,0,2,1,3,1,2,3,10,0,4,7,4,3,2,1,2,18,0,0,2,2,
%T A048614 17,0,1,0,2,6,9,3,1,1,1,8,3,2,0,15,1,4,1,1,7,7,0,4,0,4,3,4,0,1,0,1,7,2,
%U A048614 5,1,5,18,2,5,4,3,1,5,1,18,0,12,2,8,0,1,4,2,0,0,5,0,4,1,1,1,9,10,4,2,6
%N A048614 Number of primes between successive pairs of twin primes.
%H A048614 T. D. Noe, Table of n, a(n) for n=1..10000
%F A048614 a(n) = A027833(n)-2 for n>1. - T. D. Noe, Feb 28 2007
%e A048614 a(8)= 4 because between the 8th and 9th pairs of twins there are 4 primes: (71,73) 79 83 89 97 (101,103).
%Y A048614 Cf. A001223, A027833.
%Y A048614 Sequence in context: A091604 A137629 A087569 this_sequence A001442 A059781 A087664
%Y A048614 Adjacent sequences: A048611 A048612 A048613 this_sequence A048615 A048616 A048617
%K A048614 nonn,easy
%O A048614 1,6
%A A048614 Den Roussel (DenRoussel(AT)webtv.net)
%E A048614 Corrected and extended by Labos E. (labos(AT)ana.sote.hu), Mar 23 2000
%I A001442
%S A001442 1,1,0,2,1,4,0,3,0,24,0,44,4,31,0,77,0,29,0,3,0,55,0
%N A001442 G-symmetric Costas arrays of order n that are inequivalent under dihedral group.
%D A001442 CRC Handbook of Combinatorial Designs, C. Colbourn and J. Dinitz Eds., 1996, IV.7: Costas Arrays by Herbert Taylor (IV.7.6, page 259, Table 2.29).
%Y A001442 Cf. A008404, A001440, A001441, A008403.
%Y A001442 Sequence in context: A137629 A087569 A048614 this_sequence A059781 A087664 A120112
%Y A001442 Adjacent sequences: A001439 A001440 A001441 this_sequence A001443 A001444 A001445
%K A001442 nonn,nice
%O A001442 1,4
%A A001442 njas
%I A059781
%S A059781 0,0,0,0,0,1,0,1,0,0,0,1,0,1,0,0,0,2,1,4,0,4,1,2,0,0,0,0,1,0,0,0,1,0,0,
%T A059781 0,0,1,0,1,1,1,0,1,1,1,0,1,0,0,0,2,0,0,1,0,0,0,1,0,0,2,0,0,0,3,2,4,1,3,
%U A059781 4,3,0,3,4,3,1,4,2,3,0,0,0,0,2,1,1,1,8,0,0,0,8,1,1,1,2,0,0,0,0,1,0,1,0
%N A059781 Triangle T(n,k) giving exponent of power of 2 dividing entry (n,k) of trinomial triangle A027907.
%D A059781 B. A. Bondarenko, Generalized Pascal Triangles and Pyramids (in Russian), FAN, Tashkent, 1990, ISBN 5-648-00738-8. English translation published by Fibonacci Association, Santa Clara Univ., Santa Clara, CA, 1993; see p. 117.
%e A059781 0; 0,0,0; 0,1,0,1,0; ...
%p A059781 with(numtheory): T := proc(i,j) option remember: if i >= 0 and j=0 then RETURN(1) fi: if i >= 0 and j=2*i then RETURN(1) fi: if i >= 1 and j=1 then RETURN(i) fi: if i >= 1 and j=2*i-1 then RETURN(i) fi: T(i-1,j-2)+T(i-1,j-1)+T(i-1,j): end: for i from 0 to 20 do for j from 0 to 2*i do if T(i,j) mod 2 = 1 then printf(`%d,`,0) else printf(`%d,`, ifactors(T(i,j))[2,1,2] ) fi: od:od:
%Y A059781 Sequence in context: A087569 A048614 A001442 this_sequence A087664 A120112 A103977
%Y A059781 Adjacent sequences: A059778 A059779 A059780 this_sequence A059782 A059783 A059784
%K A059781 nonn,easy,tabf
%O A059781 0,18
%A A059781 njas, Feb 22 2001
%E A059781 More terms and Maple code from James A. Sellers (sellersj(AT)math.psu.edu), Feb 22 2001
%I A087664
%S A087664 0,2,1,4,0,5,2,9,0,1,1,1,0,3,3,2,0,4,1,2,0,6,2,7,0,1,1,1,0,5,4,4,0,2,1,
%T A087664 3,0,8,2,7,0,1,1,1,0,2,3,6,0,3,1,2,0,2,2,2,0,1,1,1,0,4,5,5,0,2,1,6,0,5,
%U A087664 2,4,0,1,1,1,0,4,3,2,0,5,1,2,0,3,2,3,0,1,1,1,0,4,4,6,0,2,1,3,0,3,2,3,0
%N A087664 Consider recurrence b(0) = n/4, b(n) = b(n-1)*floor(b(n-1)); sequence gives number of steps to reach an integer, or -1 if no integer is ever reached.
%C A087664 It is conjectured that an integer is always reached if the initial value is >= 2.
%H A087664 J. C. Lagarias and N. J. A. Sloane, Approximate squaring (pdf, ps), Experimental Math., 13 (2004), 113-128.
%Y A087664 Cf. A087665 (integer reached), A087667 and A087668 (records), A057016.
%Y A087664 Sequence in context: A048614 A001442 A059781 this_sequence A120112 A103977 A109883
%Y A087664 Adjacent sequences: A087661 A087662 A087663 this_sequence A087665 A087666 A087667
%K A087664 nonn
%O A087664 8,2
%A A087664 njas, Sep 27 2003
%E A087664 More terms from John W. Layman (layman(AT)math.vt.edu), Sep 27 2003
%I A120112
%S A120112 1,1,2,1,4,0,6,1,2,0,10,0,12,0,0,1,16,0,18,0,0,0,22,0,4,0,2,0,28,0,30,1,
%T A120112 0,0,0,0,36,0,0,0,40,0,42,0,0,0,46,0,6,0,0,0,52,0,0,0,0,0,58,0,60
%V A120112 1,-1,-2,-1,-4,0,-6,-1,-2,0,-10,0,-12,0,0,-1,-16,0,-18,0,0,0,-22,0,-4,0,-2,0,-28,0,-30,
%W A120112 -1,0,0,0,0,-36,0,0,0,-40,0,-42,0,0,0,-46,0,-6,0,0,0,-52,0,0,0,0,0,-58,0,-60
%N A120112 Row sums of number triangle A120111.
%C A120112 The last row of the columns in tables A133232 and A133233 are given by this sequence via the formula: if nTable of n, a(n) for n=1..2000
%H A033880 Eric Weisstein's World of Mathematics, Abundance
%p A033880 with(numtheory); n->sigma(n) - 2*n;
%Y A033880 Equals -A033879. Cf. A005100.
%Y A033880 Sequence in context: A120112 A103977 A109883 this_sequence A033879 A033883 A106316
%Y A033880 Adjacent sequences: A033877 A033878 A033879 this_sequence A033881 A033882 A033883
%K A033880 sign,nice
%O A033880 1,3
%A A033880 njas
%E A033880 Definition corrected Jul 04 2005
%I A033879
%S A033879 1,1,2,1,4,0,6,1,5,2,10,4,12,4,6,1,16,3,18,2,10,8,22,12,19,10,14,
%T A033879 0,28,12,30,1,18,14,22,19,36,16,22,10,40,12,42,4,12,20,46,28,41,
%U A033879 7,30,6,52,12,38,8,34,26,58,48,60,28,22,1,46,12,66,10,42,4,70,51
%V A033879 1,1,2,1,4,0,6,1,5,2,10,-4,12,4,6,1,16,-3,18,-2,10,8,22,-12,19,10,14,
%W A033879 0,28,-12,30,1,18,14,22,-19,36,16,22,-10,40,-12,42,4,12,20,46,-28,41,
%X A033879 7,30,6,52,-12,38,-8,34,26,58,-48,60,28,22,1,46,-12,66,10,42,-4,70,-51
%N A033879 Deficiency of n, or 2n - (sum of divisors of n).
%C A033879 Records for the sequence of the absolute values are in A075728, and the indices of these records in A074918. - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Mar 02 2007
%D A033879 R. K. Guy, Unsolved Problems in Number Theory, Section B2.
%H A033879 T. D. Noe, Table of n, a(n) for n=1..2000
%p A033879 with(numtheory); n->2*n-sigma(n);
%Y A033879 Cf. A005101. Equals -A033880.
%Y A033879 Cf. A074918, A075728.
%Y A033879 Sequence in context: A103977 A109883 A033880 this_sequence A033883 A106316 A126707
%Y A033879 Adjacent sequences: A033876 A033877 A033878 this_sequence A033880 A033881 A033882
%K A033879 sign,nice
%O A033879 1,3
%A A033879 njas
%E A033879 Definition corrected Jul 04 2005
%I A033883
%S A033883 1,1,2,1,4,0,6,1,5,2,10,12,4,6,1,16,18,10,8,22,19,10,14,
%T A033883 0,28,30,1,18,14,22,36,16,22,40,42,4,12,20,46,41,7,30,6,
%U A033883 52,38,34,26,58,60,28,22,1,46,66,10,42,70,72,34,26,12,58
%N A033883 Deficiency of the deficient or perfect numbers: m = 2n - sigma(n) for n such that m >= 0.
%Y A033883 Sequence in context: A109883 A033880 A033879 this_sequence A106316 A126707 A057458
%Y A033883 Adjacent sequences: A033880 A033881 A033882 this_sequence A033884 A033885 A033886
%K A033883 nonn
%O A033883 0,3
%A A033883 njas
%I A106316
%S A106316 0,1,2,1,4,0,6,2,1,4,10,4,12,8,12,2,16,12,18,16,20,16,22,12,13,20,1,0,
%T A106316 28,24,30,3,3,28,9,15,36,32,5,10,40,6,42,12,36,40,46,12,33,21,9,18,52,
%U A106316 18,4,32,11,52,58,48,60,56,3,3,8,30,66,30,15,58,70,12,72,68,3,36,20,42
%N A106316 Remainder of the harmonic residue of n when divided by n.
%t A106316 RemainderOfHarmonicResidue[n_]=Mod[Mod[n*DivisorSigma[0, n], DivisorSigma[1, n]], n]
%Y A106316 Cf. A106315, A106317.
%Y A106316 Sequence in context: A033880 A033879 A033883 this_sequence A126707 A057458 A011017
%Y A106316 Adjacent sequences: A106313 A106314 A106315 this_sequence A106317 A106318 A106319
%K A106316 nonn
%O A106316 1,3
%A A106316 George J. Schaeffer (gschaeff(AT)andrew.cmu.edu), Apr 29 2005
%I A126707
%S A126707 1,0,2,1,4,0,6,2,5,1,11,0,13,2,5,4,17,1,19,3,9,5,23,2,19,7,15,6,29,0,31,
%T A126707 9,16,10,21,3,37,12,20,8,41,2,43,12,18,15,47,5,41,10,26,14,53,5,36,13,
%U A126707 30,19,59,2,61,20,28,21,44,6,67,21,37,10,71,9,73,25,29,23,56,7,79,16,45
%N A126707 a(n) = number of composites coprime to n and <= the n-th composite.
%t A126707 c = Select[Range[120], # != 1 && ! PrimeQ[ # ] &];Table[Length[Select[Take[c, n], GCD[ #, n] == 1 &]], {n, Length[c]}] (*Chandler*)
%Y A126707 Cf. A002808.
%Y A126707 Sequence in context: A033879 A033883 A106316 this_sequence A057458 A011017 A077954
%Y A126707 Adjacent sequences: A126704 A126705 A126706 this_sequence A126708 A126709 A126710
%K A126707 nonn
%O A126707 1,3
%A A126707 Leroy Quet (qq-quet(AT)mindspring.com), Feb 11 2007
%E A126707 Extended by Ray Chandler (rayjchandler(AT)sbcglobal.net), Feb 17 2007
%I A057458
%S A057458 1,2,1,4,0,6,3,2,2,8,1,12,2,2,4,16,2,10,5,2,8,14,2,14,2,8,9,16,0,20,11,
%T A057458 4,4,14,2,18,16,8,7,28,2,32,6,6,10,24,5,14,14,8,21,42,0,22,8,14,12,22,
%U A057458 4,24,18,14,20,14,2,44,14,14,10,34,5,46,20,4,18,38,4,38,15,10,16,46,2
%N A057458 Number of k, 1 <= k <= n, where {k (n+1-k) + 1} is prime.
%e A057458 For n = 7, 2*6 + 1, 4*4 + 1, and 6*2 + 1 are prime, so a(7) = 3.
%Y A057458 Sequence in context: A033883 A106316 A126707 this_sequence A011017 A077954 A077979
%Y A057458 Adjacent sequences: A057455 A057456 A057457 this_sequence A057459 A057460 A057461
%K A057458 nonn
%O A057458 1,2
%A A057458 Leroy Quet (qq-quet(AT)mindspring.com), Sep 26 2000
%I A011017
%S A011017 2,1,4,0,6,9,5,1,4,2,9,2,8,0,7,2,3,2,6,5,4,6,7,9,6,3,0,0,0,6,5,1,3,
%T A011017 6,7,8,1,7,6,6,5,1,3,5,5,5,6,8,8,4,0,8,1,4,2,0,9,6,5,7,8,4,1,6,0,0,
%U A011017 3,5,2,8,9,2,9,4,8,2,4,9,2,7,2,2,0,1,1,8,2,8,8,4,8,8,2,1,7,8,6,1,6
%N A011017 Decimal expansion of 4th root of 21.
%Y A011017 Sequence in context: A106316 A126707 A057458 this_sequence A077954 A077979 A122161
%Y A011017 Adjacent sequences: A011014 A011015 A011016 this_sequence A011018 A011019 A011020
%K A011017 nonn,cons
%O A011017 1,1
%A A011017 njas
%I A077954
%S A077954 1,1,1,2,1,4,0,7,3,11,10,15,24,16,49,7,89,26,145,108,208,279,245,595,174,
%T A077954 1119,176,1888,1121,2831,3185,3598,7137,3244,13920,295,24301,10971,37926,
%U A077954 35567,51256,84464,53615,171287,20407,309366,97265,501060,386224,713161
%V A077954 1,1,-1,-2,1,4,0,-7,-3,11,10,-15,-24,16,49,-7,-89,-26,145,108,-208,-279,245,595,-174,
%W A077954 -1119,-176,1888,1121,-2831,-3185,3598,7137,-3244,-13920,-295,24301,10971,-37926,
%X A077954 -35567,51256,84464,-53615,-171287,20407,309366,97265,-501060,-386224,713161
%N A077954 Expansion of 1/(1-x+2*x^2-x^3).
%F A077954 a(0)=1, a(1)=1, a(2)=-1, a(n)=a(n-1)-2*a(n-2)+a(n-3) for n>=3 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Sep 15 2006
%Y A077954 Sequence in context: A126707 A057458 A011017 this_sequence A077979 A122161 A067164
%Y A077954 Adjacent sequences: A077951 A077952 A077953 this_sequence A077955 A077956 A077957
%K A077954 sign
%O A077954 0,4
%A A077954 njas, Nov 17 2002
%I A077979
%S A077979 1,1,1,2,1,4,0,7,3,11,10,15,24,16,49,7,89,26,145,108,208,279,245,595,174,
%T A077979 1119,176,1888,1121,2831,3185,3598,7137,3244,13920,295,24301,10971,37926,
%U A077979 35567,51256,84464,53615,171287,20407,309366,97265,501060,386224,713161
%V A077979 1,-1,-1,2,1,-4,0,7,-3,-11,10,15,-24,-16,49,7,-89,26,145,-108,-208,279,245,-595,-174,
%W A077979 1119,-176,-1888,1121,2831,-3185,-3598,7137,3244,-13920,295,24301,-10971,-37926,
%X A077979 35567,51256,-84464,-53615,171287,20407,-309366,97265,501060,-386224,-713161
%N A077979 Expansion of 1/(1+x+2*x^2+x^3).
%Y A077979 Sequence in context: A057458 A011017 A077954 this_sequence A122161 A067164 A140505
%Y A077979 Adjacent sequences: A077976 A077977 A077978 this_sequence A077980 A077981 A077982
%K A077979 sign
%O A077979 0,4
%A A077979 njas, Nov 17 2002
%I A122161
%S A122161 1,2,1,4,0,9,5,23,24,65,90,196,311,613,1039,1954,3419,6288,11172,
%T A122161 20329,36385,65871,118312,213669,384422,693448,1248623,2251097,4054895,
%U A122161 7308466,13167159,23729196,42755048,77046281,138827181,250164695,450772776
%V A122161 1,-2,1,-4,0,-9,-5,-23,-24,-65,-90,-196,-311,-613,-1039,-1954,-3419,-6288,-11172,
%W A122161 -20329,-36385,-65871,-118312,-213669,-384422,-693448,-1248623,-2251097,-4054895,
%X A122161 -7308466,-13167159,-23729196,-42755048,-77046281,-138827181,-250164695,-450772776
%N A122161 Steinbach 3 X 3 minus the Indentity matrix to give a new vector matrix Markov with a Steinbach characteristic polynomial of: -1 + 2 x + x^2 - x^3.
%C A122161 Remember? 1/(1-x)=Sum[x^n,{n,0,Infinitity}] So to try with the Steinbach field: (I-A[i,j])^(-1)=Sun[A[i,j]^n,{n,0,Infinity}] It doesn't appear it shoulsd be finite? But I-A[i,j] is finite--> zero? {{1,0,0}, {{1,1,1}, {{0,-1,-1}, {0,1,0}, {1,1,0}, {-1,0,0}, {0,0,1}} - 1,0,0}}= { -1,0,1}}
%D A122161 P. Steinbach, Golden fields: a case for the heptagon, Math. Mag. 70 (1997), no. 1, 22-31.
%F A122161 M = {{0, -1, -1}, {-1, 0, 0}, {-1, 0, 1}}; v[1] = {1, 1, 1}; v[n_] := v[n] = M.v[n - 1]; a(n) = v[n][[1]]
%t A122161 M = {{0, -1, -1}, {-1, 0, 0}, {-1, 0, 1}}; v[1] = {1, 1, 1}; v[n_] := v[n] = M.v[n - 1]; a1 = Table[v[n][[1]], {n, 1, 50}]
%Y A122161 Cf. A046854. Cf. A046854. Cf. A007700, A059455. Cf. A065941.
%Y A122161 Sequence in context: A011017 A077954 A077979 this_sequence A067164 A140505 A117971
%Y A122161 Adjacent sequences: A122158 A122159 A122160 this_sequence A122162 A122163 A122164
%K A122161 uned,sign
%O A122161 1,2
%A A122161 Gary Adamson and Roger Bagula (qntmpkt(AT)yahoo.com), Oct 17 2006
%I A067164
%S A067164 0,1,0,2,1,4,0,9,7,10,8,21,4,34,29,43,37,68,35,82,81,106,76,125,48,166,
%T A067164 153,185,163,234,207,283,268,365,329,404,271,492,467,540,520,646,544,
%U A067164 719,654,791,741,978,778,1071,1019,1161,1108,1320,1137,1490,1415,1638
%N A067164 A067152(n)/n.
%D A067164 B. Poonen and M. Rubinstein, Number of Intersection Points Made by the Diagonals of a Regular Polygon, SIAM J. Discrete Mathematics, Vol. 11, pp. 135-156.
%H A067164 Sascha Kurz, m-gons in regular n-gons
%H A067164 B. Poonen and M. Rubinstein, The number of intersection points made by the diagonals of a regular polygon, SIAM J. on Discrete Mathematics, Vol. 11, No. 1, 135-156 (1998).
%H A067164