A097710 -id:A097710 - cp's OEIS frontend

A097711 Column 1 of triangle A097710, in which row (n) is formed from the sums of the adjacent terms in row (n-1) of the matrix square of A097710.

1, 3, 13, 88, 951, 16691, 484490, 23701698, 1990327810, 291750344191, 75757923092106, 35286335933354828, 29791358931890967248, 45989706937220594708463, 130760311958838053647976497

Offset: 0

Paul D. Hanna, Aug 22 2004

nonn

Related to the number of tournament sequences (A008934).

G. C. Greubel, Table of n, a(n) for n = 0..84

Cf. A008934, A097710.

T[n_, k_]:= T[n, k]= If[n<0 || k>n, 0, If[n==k, 1, If[k==0, Sum[T[n-1, j]*T[j,0], {j,0,n-1}], Sum[T[n-1,j]*(T[j,k-1] +T[j,k]), {j,0,n-1}] ]]]; (* T = A097710 *)
A097711[n_]:= T[n+1,1];
Table[A097711[n], {n,0,30}] (* G. C. Greubel, Feb 21 2024 *)

@CachedFunction
def T(n, k): # T = A097710
    if n< 0 or k<0 or k>n: return 0
    elif k==n: return 1
    elif k==0: return sum(T(n-1,j)*T(j,0) for j in range(n))
    else: return sum(T(n-1, j)*(T(j, k-1)+T(j,k)) for j in range(n))
def A097711(n): return T(n+1,1)
[A097711(n) for n in range(31)] # G. C. Greubel, Feb 21 2024

A008934 Number of tournament sequences: sequences (a_1, a_2, ..., a_n) with a_1 = 1 such that a_i < a_{i+1} <= 2*a_i for all i.

Original entry on oeis.org

1, 1, 2, 7, 41, 397, 6377, 171886, 7892642, 627340987, 87635138366, 21808110976027, 9780286524758582, 7981750158298108606, 11950197013167283686587, 33046443615914736611839942, 169758733825407174485685959261, 1627880269212042994531083889564192

Offset: 0

Mauro Torelli (torelli(AT)hermes.mc.dsi.unimi.it), Jeffrey Shallit

Also number of Meeussen sequences of length n (see the Cook-Kleber reference).
Column 1 of triangle A093729. Also generated by the iteration procedure that constructs triangle A093654. - Paul D. Hanna, Apr 14 2004
a(n) is the number of sequences (u_1,u_2,...,u_n) of positive integers such that u_1=1 and u_i <= 1+ u_1+...+u_{i-1} for 2<=i<=n. For example, omitting parentheses and commas, a(3)=7 counts 111, 112, 113, 121, 122, 123, 124. The difference-between-successive-terms operator is a bijection from the title sequences to these sequences. For example, the tournament sequence (1, 2, 4, 5, 9, 16) bijects to (1,2,1,4,7). (To count tournament sequences by length, the offset should be 1.) - David Callan, Oct 31 2020

			The 7 tournament sequences of length 4 are 1234, 1235, 1236, 1245, 1246, 1247, 1248.

Alois P. Heinz, Table of n, a(n) for n = 0..85 (first 31 terms from T. D. Noe)
M. Cook and M. Kleber, Tournament sequences and Meeussen sequences, Electronic J. Comb. 7 (2000), #R44.
E. Neuwirth, Computing tournament sequence numbers efficiently..., Séminaire Lotharingien de Combinatoire, B47h (2002), 12 pp.
Mauro Torelli, Increasing integer sequences and Goldbach's conjecture, RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, 40:2 (2006), pp. 107-121.
Index entries for sequences related to tournaments

Cf. A058222, A058223, A093729, A093655.

Forms column 0 of triangle A097710.

t[n_?Negative, ] = 0; t[0, ] = 1; t[, 0] = 0; t[n, k_] /; k <= n :=  t[n, k] = t[n, k-1] - t[n-1, k] + t[n-1, 2k-1] + t[n-1, 2 k]; t[n_, k_] /; k > n :=  t[n, k] =Sum[(-1)^(j-1) Binomial[n+1, j]*t[n, k-j] , {j, 1, n+1}]; Table[t[n, 1], {n, 0, 15} ] (* Jean-François Alcover, May 17 2011, after PARI prog. *)

{T(n,k)=if(n<0,0,if(n==0,1,if(k==0,0, if(k<=n,T(n,k-1)-T(n-1,k)+T(n-1,2*k-1)+T(n-1,2*k), sum(j=1,n+1,(-1)^(j-1)*binomial(n+1,j)*T(n,k-j))))))} /*(Cook-Kleber)*/ a(n)=T(n,1)

@CachedFunction
def T(n, k):
    if n<0: return 0
    elif n==0: return 1
    elif k==0: return 0
    elif kA008934(n): return T(n,1)
[A008934(n) for n in range(31)] # G. C. Greubel, Feb 22 2024

From Paul D. Hanna, Apr 14 2004: (Start)
a(n) = A093729(n, 1).
a(n) = A093655(2^n). (End)
a(n) = A097710(n, 0). - Paul D. Hanna, Aug 24 2004
From Benedict W. J. Irwin, Nov 26 2016: (Start)
Conjecture: a(n) is given by a series of nested sums as follows:
  a(2) = Sum_{i=1..2} 1,
  a(3) = Sum_{i=1..2} Sum_{j=1..i+2} 1,
  a(4) = Sum_{i=1..2} Sum_{j=1..i+2} Sum_{k=1..i+j+2} 1,
  a(5) = Sum_{i=1..2} Sum_{j=1..i+2} Sum_{k=1..i+j+2} Sum_{l=1..i+j+k+2} 1.
(End)

A113106 Triangle T, read by rows, that satisfies the recurrence: T(n,k) = [T^5](n-1,k-1) + [T^5](n-1,k) for n>k>=0, with T(n,n)=1 for n>=0, where T^5 is the matrix 5th power of T.

Original entry on oeis.org

1, 1, 1, 5, 6, 1, 85, 115, 31, 1, 4985, 7420, 2590, 156, 1, 1082905, 1744965, 723370, 62090, 781, 1, 930005021, 1601759426, 752616215, 82390620, 1532715, 3906, 1, 3306859233805, 6024941167511, 3117415999361, 409321203715, 10025307495

Offset: 0

Paul D. Hanna, Oct 14 2005

Column 0 of the matrix power p, T^p, equals the number of 5-tournament sequences having initial term p (see A113103 for definitions).

			Triangle begins:
1;
1,1;
5,6,1;
85,115,31,1;
4985,7420,2590,156,1;
1082905,1744965,723370,62090,781,1;
930005021,1601759426,752616215,82390620,1532715,3906,1;
Matrix 4th power T^4 (A113112) begins:
1;
4,1;
56,24,1;
2704,1576,124,1;
481376,346624,39376,624,1; ...
where column 0 equals A113113.
Matrix 5th power T^5 (A113114) begins:
1;
5,1;
85,30,1;
4985,2435,155,1;
1082905,662060,61310,780,1;
930005021,671754405,80861810,1528810,3905,1; ...
where adjacent sums in row n of T^5 forms row n+1 of T.

Cf. A097710, A113084, A113095; A113103, A113107 (column 0), A113108 (T^2), A113110 (T^3), A113112 (T^4), A113112 (T^5).

{T(n,k)=local(M=matrix(n+1,n+1));for(r=1,n+1, for(c=1,r, M[r,c]=if(r==c,1,if(c>1,(M^5)[r-1,c-1])+(M^5)[r-1,c]))); return(M[n+1,k+1])}

Let GF[T] denote the g.f. of triangular matrix T. Then GF[T] = 1 + x*(1+y)*GF[T^5] and for all integer p>=1: GF[T^p] = 1 + x*Sum_{j=1..p} GF[T^(p+4*j)] + x*y*GF[T^(5*p)].

A113077 Column 3 of square table A093729; a(n) gives the number of n-th generation descendents of a node labeled (3) in the tree of tournament sequences, for n>=0.

Original entry on oeis.org

1, 3, 15, 123, 1656, 36987, 1391106, 89574978, 10036638270, 1986129275673, 703168200003336, 450303519404234922, 526421174510139860241, 1132076561237754405471033, 4507472672071759672232970720

Offset: 0

Paul D. Hanna, Oct 14 2005

nonn

Also equals column 0 of the matrix cube of triangle A097710, which satisfies the matrix recurrence: A097710(n,k) = [A097710^2](n-1,k-1) + [A097710^2](n-1,k) for n>k>=0.

			The tree of tournament sequences of descendents of a node labeled (3) begins:
[3]; generation 1: 3->[4,5,6]; generation 2: 4->[5,6,7,8],
5->[6,7,8,9,10], 6->[7,8,9,10,11,12]; ...
Then a(n) gives the number of nodes in generation n.
Also, a(n+1) = sum of labels of nodes in generation n.

M. Cook and M. Kleber, Tournament sequences and Meeussen sequences, Electronic J. Comb. 7 (2000), #R44.

Cf. A113078, A113079.

{a(n,q=2)=local(M=matrix(n+1,n+1));for(r=1,n+1, for(c=1,r, M[r,c]=if(r==c,1,if(c>1,(M^q)[r-1,c-1])+(M^q)[r-1,c]))); return((M^3)[n+1,1])}

A113078 Number of tournament sequences: a(n) gives the number of n-th generation descendents of a node labeled (4) in the tree of tournament sequences.

Original entry on oeis.org

1, 4, 26, 274, 4721, 134899, 6501536, 537766009, 77598500096, 19821981700354, 9077118324755246, 7531446638893873684, 11423775838657143826346, 31914367054676982206368909, 165251261153335414813452988541

Offset: 0

Paul D. Hanna, Oct 14 2005

nonn

Equals column 4 of square table A093729. Also equals column 0 of the matrix 4th power of triangle A097710, which satisfies the matrix recurrence: A097710(n,k) = [A097710^2](n-1,k-1) + [A097710^2](n-1,k) for n>k>=0.

			The tree of tournament sequences of descendents of a node labeled (4) begins:
[4]; generation 1: 4->[5,6,7,8]; generation 2: 5->[6,7,8,9,10],
6->[7,8,9,10,11,12], 7->[8,9,10,11,12,13,14],
8->[9,10,11,12,13,14,15,16]; ...
Then a(n) gives the number of nodes in generation n.
Also, a(n+1) = sum of labels of nodes in generation n.

M. Cook and M. Kleber, Tournament sequences and Meeussen sequences, Electronic J. Comb. 7 (2000), #R44.

Cf. A113077, A113079, A008934, A113089, A113096, A113098, A113100, A113107, A113109, A113111, A113113.

{a(n,q=2)=local(M=matrix(n+1,n+1));for(r=1,n+1, for(c=1,r, M[r,c]=if(r==c,1,if(c>1,(M^q)[r-1,c-1])+(M^q)[r-1,c]))); return((M^4)[n+1,1])}

A113079 Number of tournament sequences: a(n) gives the number of n-th generation descendents of a node labeled (5) in the tree of tournament sequences.

Original entry on oeis.org

1, 5, 40, 515, 10810, 376175, 22099885, 2231417165, 393643922005, 123097221805100, 69087264010363930, 70321483026073531730, 130954011392485408662370, 449450774746306949114288795

Offset: 0

Paul D. Hanna, Oct 14 2005

nonn

Equals column 5 of square table A093729. Also equals column 0 of the matrix 5th power of triangle A097710, which satisfies the matrix recurrence: A097710(n,k) = [A097710^2](n-1,k-1) + [A097710^2](n-1,k) for n>k>=0.

			The tree of tournament sequences of descendents of a node labeled (5) begins:
[5]; generation 1: 5->[6,7,8,9,10]; generation 2:
6->[7,8,9,10,11,12], 7->[8,9,10,11,12,13,14],
8->[9,10,11,12,13,14,15,16], 9->[10,11,12,13,14,15,16,17,18],
10->[11,12,13,14,15,16,17,18,19,20]; ...
Then a(n) gives the number of nodes in generation n.
Also, a(n+1) = sum of labels of nodes in generation n.

M. Cook and M. Kleber, Tournament sequences and Meeussen sequences, Electronic J. Comb. 7 (2000), #R44.

Cf. A113077, A113078, A008934, A113089, A113096, A113098, A113100, A113107, A113109, A113111, A113113.

{a(n,q=2)=local(M=matrix(n+1,n+1));for(r=1,n+1, for(c=1,r, M[r,c]=if(r==c,1,if(c>1,(M^q)[r-1,c-1])+(M^q)[r-1,c]))); return((M^5)[n+1,1])}

A113084 Triangle T, read by rows, that satisfies the recurrence: T(n,k) = [T^3](n-1,k-1) + [T^3](n-1,k) for n>k>=0, with T(n,n)=1 for n>=0, where T^3 is the matrix third power of T.

Original entry on oeis.org

1, 1, 1, 3, 4, 1, 21, 33, 13, 1, 331, 586, 294, 40, 1, 11973, 23299, 13768, 2562, 121, 1, 1030091, 2166800, 1447573, 333070, 22569, 364, 1, 218626341, 490872957, 361327779, 97348117, 8466793, 200931, 1093, 1, 118038692523, 280082001078

Offset: 0

Paul D. Hanna, Oct 14 2005

Column 0 of the matrix power p, T^p, equals the number of 3-tournament sequences having initial term p.

			Triangle T begins:
1;
1,1;
3,4,1;
21,33,13,1;
331,586,294,40,1;
11973,23299,13768,2562,121,1;
1030091,2166800,1447573,333070,22569,364,1; ...
Matrix square T^2 (A113088) begins:
1;
2,1;
10,8,1;
114,118,26,1;
2970,3668,1108,80,1;
182402,257122,96416,9964,242,1; ...
where column 0 equals A113089.
Matrix cube T^3 (A113090) begins:
1;
3,1;
21,12,1;
331,255,39,1;
11973,11326,2442,120,1;
1030091,1136709,310864,22206,363,1; ...
where adjacent sums in row n of T^3 forms row n+1 of T.

Cf. A113081; A097710, A113095, A113106; A113085 (column 0), A113088 (T^2), A113087 (row sums).

{T(n,k)=local(M=matrix(n+1,n+1));for(r=1,n+1, for(c=1,r, M[r,c]=if(r==c,1,if(c>1,(M^3)[r-1,c-1])+(M^3)[r-1,c]))); return(M[n+1,k+1])}

Let GF[T] denote the g.f. of triangular matrix T. Then GF[T] = 1 + x*(1+y)*GF[T^3] and for all integer p>=1: GF[T^p] = 1 + x*Sum_{j=1..p} GF[T^(p+2*j)] + x*y*GF[T^(3*p)].

A113095 Triangle T, read by rows, that satisfies the recurrence: T(n,k) = [T^4](n-1,k-1) + [T^4](n-1,k) for n>k>=0, with T(n,n)=1 for n>=0, where T^4 is the matrix 4th power of T.

Original entry on oeis.org

1, 1, 1, 4, 5, 1, 46, 66, 21, 1, 1504, 2398, 978, 85, 1, 146821, 255113, 122914, 14962, 341, 1, 45236404, 84425001, 46001193, 7046354, 235122, 1365, 1, 46002427696, 91159696960, 54661544301, 9933169553, 432627794, 3738738, 5461, 1

Offset: 0

Paul D. Hanna, Oct 14 2005

Column 0 of the matrix power p, T^p, equals the number of 4-tournament sequences having initial term p (see A113092 for definitions).

			Triangle T begins:
  1;
  1,1;
  4,5,1;
  46,66,21,1;
  1504,2398,978,85,1;
  146821,255113,122914,14962,341,1;
  45236404,84425001,46001193,7046354,235122,1365,1; ...
Matrix third power T^3 (A113099) begins:
  1;
  3,1;
  27,15,1;
  693,513,63,1;
  52812,47619,8289,255,1; ...
 where column 0 equals A113100.
Matrix 4th power T^4 (A113101) begins:
  1;
  4,1;
  46,20,1;
  1504,894,84,1;
  146821,108292,14622,340,1;
  45236404,39188597,6812596,233758,1364,1; ...
 where adjacent sums in row n of T^4 forms row n+1 of T.

Cf. A097710, A113084, A113106; A113092, A113096 (column 0), A113097 (T^2), A113099 (T^3), A113101 (T^4).

{T(n,k)=local(M=matrix(n+1,n+1));for(r=1,n+1, for(c=1,r, M[r,c]=if(r==c,1,if(c>1,(M^4)[r-1,c-1])+(M^4)[r-1,c]))); return(M[n+1,k+1])}

Let GF[T] denote the g.f. of triangular matrix T. Then GF[T] = 1 + x*(1+y)*GF[T^4] and for all integer p>=1: GF[T^p] = 1 + x*Sum_{j=1..p} GF[T^(p+3*j)] + x*y*GF[T^(4*p)].

A093729 Square table T, read by antidiagonals, where T(n,k) gives the number of n-th generation descendents of a node labeled (k) in the tree of tournament sequences.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 2, 2, 1, 0, 7, 7, 3, 1, 0, 41, 41, 15, 4, 1, 0, 397, 397, 123, 26, 5, 1, 0, 6377, 6377, 1656, 274, 40, 6, 1, 0, 171886, 171886, 36987, 4721, 515, 57, 7, 1, 0, 7892642, 7892642, 1391106, 134899, 10810, 867, 77, 8, 1, 0, 627340987, 627340987, 89574978, 6501536, 376175, 21456, 1351, 100, 9, 1

Offset: 0

Paul D. Hanna, Apr 14 2004; revised Oct 14 2005

Column 1, of array T and antidiagonals, equals A008934, which is the number of tournament sequences.

A tournament sequence is an increasing sequence of positive integers (t_1,t_2,...) such that t_1 = 1 and t_{i+1} <= 2*t_i, where integer k>1.

			Array begins:
  1,      1,       1,       1,       1,      1,      1,     1,     1, ...],
  0,      1,       2,       3,       4,      5,      6,     7,     8, ...],
  0,      2,       7,      15,      26,     40,     57,    77,   100, ...],
  0,      7,      41,     123,     274,    515,    867,  1351,  1988, ...],
  0,     41,     397,    1656,    4721,  10810,  21456, 38507, 64126, ...],
  0,    397,    6377,   36987,  134899, 376175, 880032, .................],
  0,   6377,  171886, 1391106, 6501536, ...],
  0, 171886, 7892642, .....................];
Antidiagonals begin as:
  1;
  0,      1;
  0,      1,      1;
  0,      2,      2,     1;
  0,      7,      7,     3,    1;
  0,     41,     41,    15,    4,   1;
  0,    397,    397,   123,   26,   5,   1;
  0,   6377,   6377,  1656,  274,  40,   6,   1;
  0, 171886, 171886, 36987, 4721, 515,  57,   7,   1;

G. C. Greubel, Antidiagonals n = 0..50, flattened
M. Cook and M. Kleber, Tournament sequences and Meeussen sequences, Electronic J. Comb. 7 (2000), #R44.
Michael Somos, A functional power series equation, Mathematics StackExchange answer.

Cf. A008934, A097710, A113080, A113081, A113092, A113103.

Cf. A008934 (column k=1 of array and antidiagonals), A093730 (antidiagonal row sums).

t[n_?Negative, ] = 0; t[0, ] = 1; t[n_, k_] /; k <= n := t[n, k] = t[n, k - 1] - t[n-1, k] + t[n - 1, 2 k - 1] + t[n - 1, 2 k]; t[n_, k_] := t[n, k] = Sum[(-1)^(j - 1)*Binomial[n + 1, j]*t[n, k - j], {j, 1, n + 1}]; Flatten[Table[t[i - k, k - 1], {i, 10}, {k, i}]] (* Jean-François Alcover, May 31 2011, after PARI prog. *)

{T(n,k)=if(n<0,0,if(n==0,1,if(k==0,0, if(k<=n,T(n,k-1)-T(n-1,k)+T(n-1,2*k-1)+T(n-1,2*k), sum(j=1,n+1, (-1)^(j-1)*binomial(n+1,j)*T(n,k-j))))))}

{a(n, m) = my(A=1); for(k=1, n, A = (A - q^k * r * subst( subst(A, q, q^2), r, r^2)) / (1-q)); subst(subst(A, r, q^(m-1)), q, 1)}; /* Michael Somos, Jun 19 2017 */

@CachedFunction
def T(n, k):
    if n<0: return 0
    elif n==0: return 1
    elif k==0: return 0
    elif kA093729(n,k): return T(n-k,k)
flatten([[A093729(n,k) for k in range(n+1)] for n in range(16)]) # G. C. Greubel, Feb 22 2024

T(0, k)=1 for k>=0, T(n, 0)=0 for n>=1; else T(n, k) = T(n, k-1) - T(n-1, k) + T(n-1, 2*k-1) + T(n-1, 2*k) for k<=n; else T(n, k) = Sum_{j=1..n+1} (-1)^(j-1)*C(n+1, j)*T(n, k-j) for k>n (Cook-Kleber).
Column k of T equals column 0 of the matrix k-th power of triangle A097710, which satisfies the matrix recurrence: A097710(n, k) = [A097710^2](n-1, k-1) + [A097710^2](n-1, k) for n>k>=0.
Sum_{k=0..n} T(n-k, k) = A093730(n) (antidiagonal row sums).

A097712 Lower triangular matrix T, read by rows, such that T(n,0) = 1 and T(n,k) = T(n-1,k) + T^2(n-1,k-1) for k>0, where T^2 is the matrix square of T.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 8, 7, 1, 1, 25, 44, 15, 1, 1, 111, 346, 208, 31, 1, 1, 809, 4045, 3720, 912, 63, 1, 1, 10360, 77351, 99776, 35136, 3840, 127, 1, 1, 236952, 2535715, 4341249, 2032888, 308976, 15808, 255, 1, 1, 9708797, 145895764, 319822055, 189724354, 37329584, 2608864, 64256, 511, 1

Offset: 0

Paul D. Hanna, Aug 24 2004

This triangle has the same row sums and first column terms as in rows 2^n, for n>=0, of triangle A093662.

			T(5,1) = T(4,1) + T^2(4,0) = 25 + 86 = 111.
T(5,2) = T(4,2) + T^2(4,1) = 44 + 302 = 346.
T(5,3) = T(4,3) + T^2(4,2) = 15 + 193 = 208.
Rows of T begin:
  1;
  1,      1;
  1,      3,       1;
  1,      8,       7,       1;
  1,     25,      44,      15,       1;
  1,    111,     346,     208,      31,      1;
  1,    809,    4045,    3720,     912,     63,     1;
  1,  10360,   77351,   99776,   35136,   3840,   127,   1;
  1, 236952, 2535715, 4341249, 2032888, 308976, 15808, 255, 1;
Rows of T^2 begin:
       1;
       2,       1;
       5,       6,       1;
      17,      37,      14,       1;
      86,     302,     193,      30,      1;
     698,    3699,    3512,     881,     62,     1;
    9551,   73306,   96056,   34224,   3777,   126,   1;
  226592, 2458364, 4241473, 1997752, 305136, 15681, 254, 1;
Column 0 of T^2 forms A016121.
Row sums of T^2 form the first differences of A016121.

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A016121 (row sums), A093662, A097710, A097713.

T[n_, k_] := T[n, k] = If[n < 0 || k > n, 0, If[n == k, 1, If[k == 0, 1, T[n - 1, k] + Sum[T[n - 1, j] T[j, k - 1], {j, 0, n - 1}]]]];
Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Oct 02 2019 *)

T(n,k)=if(n<0 || k>n,0,if(n==k,1,if(k==0,1, T(n-1,k)+sum(j=0,n-1,T(n-1,j)*T(j,k-1));)))

@CachedFunction
def T(n,k): # T = A097712
    if k<0 or k>n: return 0
    elif k==0 or k==n: return 1
    else: return T(n-1,k) + sum(T(n-1,j)*T(j,k-1) for j in range(n))
flatten([[T(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Feb 20 2024

T(n, k) = T(n-1, k) + Sum_{j=0..n-1} T(n-1, j)*T(j, k-1), with T(n, 0) = T(n, n) = 1.
T(n, 1) = A097713(n-1), n >= 1.
Sum_{k=0..n} T(n, k) = A016121(n) (row sums).

cp's OEIS Frontend

A097711 Column 1 of triangle A097710, in which row (n) is formed from the sums of the adjacent terms in row (n-1) of the matrix square of A097710.

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A008934 Number of tournament sequences: sequences (a_1, a_2, ..., a_n) with a_1 = 1 such that a_i < a_{i+1} <= 2*a_i for all i.

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A113106 Triangle T, read by rows, that satisfies the recurrence: T(n,k) = [T^5](n-1,k-1) + [T^5](n-1,k) for n>k>=0, with T(n,n)=1 for n>=0, where T^5 is the matrix 5th power of T.

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A113077 Column 3 of square table A093729; a(n) gives the number of n-th generation descendents of a node labeled (3) in the tree of tournament sequences, for n>=0.

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A113078 Number of tournament sequences: a(n) gives the number of n-th generation descendents of a node labeled (4) in the tree of tournament sequences.

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A113079 Number of tournament sequences: a(n) gives the number of n-th generation descendents of a node labeled (5) in the tree of tournament sequences.

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A113084 Triangle T, read by rows, that satisfies the recurrence: T(n,k) = [T^3](n-1,k-1) + [T^3](n-1,k) for n>k>=0, with T(n,n)=1 for n>=0, where T^3 is the matrix third power of T.

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A113095 Triangle T, read by rows, that satisfies the recurrence: T(n,k) = [T^4](n-1,k-1) + [T^4](n-1,k) for n>k>=0, with T(n,n)=1 for n>=0, where T^4 is the matrix 4th power of T.

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