A008934 -id:A008934 - cp's OEIS frontend

A016121 Number of sequences (a_1, a_2, ..., a_n) of length n with a_1 = 1 satisfying a_i <= a_{i+1} <= 2*a_i.

1, 2, 5, 17, 86, 698, 9551, 226592, 9471845, 705154187, 94285792211, 22807963405043, 10047909839840456, 8110620438438750647, 12062839548612627177590, 33226539134943667506533207, 170288915434579567358828997806, 1630770670148598007261992936663653

Offset: 0

Jeffrey Shallit

nonn

Number of n X n binary symmetric matrices with rows, considered as binary numbers, in nondecreasing order. - R. H. Hardin, May 30 2008

Also, number of (n+1) X (n+1) binary symmetric matrices with zero main diagonal and rows, considered as binary numbers, in nondecreasing order. - Max Alekseyev, Feb 06 2022

Alois P. Heinz, Table of n, a(n) for n = 0..50

Cf. A008934, A060690, A089006, A351157, A351158, A351287, A351288.

Row sums of triangle A097712.

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}]]]];
a[n_] := Sum[T[n, k], {k, 0, n}];
a /@ Range[0, 20] (* Jean-François Alcover, Oct 02 2019 *)

@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))
def A016121(n): return sum(T(n,k) for k in range(n+1))
[A016121(n) for n in range(31)] # G. C. Greubel, Feb 21 2024

a(n) = Sum_{k=0..n} A097712(n, k). - Paul D. Hanna, Aug 24 2004

Equals the binomial transform of A008934 (number of tournament sequences): a(n) = Sum_{k=0..n} C(n, k)*A008934(k). - Paul D. Hanna, Sep 18 2005

A113089 Number of 3-tournament sequences: a(n) gives the number of increasing sequences of n positive integers (t_1,t_2,...,t_n) such that t_1 = 2 and t_i = 2 (mod 2) and t_{i+1} <= 3*t_i for 1

Original entry on oeis.org

1, 2, 10, 114, 2970, 182402, 27392682, 10390564242, 10210795262650, 26494519967902114, 184142934938620227530, 3466516611360924222460082, 178346559667060145108789818842, 25264074391478558474014952210052802

Offset: 0

Paul D. Hanna, Oct 14 2005

nonn

Column 0 of triangle A113088; A113088 is the matrix square of triangle A113084, which satisfies the matrix recurrence: A113084(n,k) = [A113084^3](n-1,k-1) + [A113084^3](n-1,k). Also equals column 2 of square table A113081.

			The tree of 3-tournament sequences of even integer
descendents of a node labeled (2) begins:
[2]; generation 1: 2->[4,6];
generation 2: 4->[6,8,10,12], 6->[8,10,12,14,16,18]; ...
Then a(n) gives the number of nodes in generation n.
Also, a(n+1) = sum of labels of nodes in generation n.

T. D. Noe, Table of n, a(n) for n=0..30
M. Cook and M. Kleber, Tournament sequences and Meeussen sequences, Electronic J. Comb. 7 (2000), #R44.

Cf. A008934, A113077, A113078, A113079, A113085, A113096, A113098, A113100, A113107, A113109, A113111, A113113.

{a(n)=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^2)[n+1,1])}

A113100 Number of 4-tournament sequences: a(n) gives the number of increasing sequences of n positive integers (t_1,t_2,...,t_n) such that t_1 = 3 and t_i = 3 (mod 3) and t_{i+1} <= 4*t_i for 1

Original entry on oeis.org

1, 3, 27, 693, 52812, 12628008, 9924266772, 26507035453923, 246323730279500082, 8100479557816637139288, 954983717308947379891713642, 407790020849346203244152231395953

Offset: 0

Paul D. Hanna, Oct 14 2005

nonn

Column 0 of triangle A113099; A113099 is the matrix cube of triangle A113095, which satisfies the matrix recurrence: A113095(n,k) = [A113095^4](n-1,k-1) + [A113095^4](n-1,k). Also equals column 3 of square table A113092.

			The tree of 4-tournament sequences of descendents of a node labeled (3) begins:
[3]; generation 1: 3->[6,9,12]; generation 2:
6->[9,12,15,18,21,24], 9->[12,15,18,21,24,27,30,33,36],
12->[15,18,21,24,27,30,33,36,39,42,45,48]; ...
Then a(n) gives the number of nodes in generation n.
Also, a(n+1) = sum of labels of nodes in generation n.

T. D. Noe, Table of n, a(n) for n=0..30
M. Cook and M. Kleber, Tournament sequences and Meeussen sequences, Electronic J. Comb. 7 (2000), #R44.

Cf. A008934, A113077, A113078, A113079, A113085, A113089, A113096, A113098, A113107, A113109, A113111, A113113.

{a(n)=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^3)[n+1,1])}

A113113 Number of 5-tournament sequences: a(n) gives the number of increasing sequences of n positive integers (t_1,t_2,...,t_n) such that t_1 = 4 and t_i = 4 (mod 4) and t_{i+1} <= 5*t_i for 1

Original entry on oeis.org

1, 4, 56, 2704, 481376, 337587520, 978162377600, 12088945462984960, 651451173346940188160, 155573037664478034394215424, 166729581953452524706695313356800

Offset: 0

Paul D. Hanna, Oct 14 2005

nonn

Column 0 of triangle A113112; A113112 is the matrix 4th power of triangle A113106, which satisfies the matrix recurrence: A113106(n,k) = [A113106^5](n-1,k-1) + [A113106^5](n-1,k). Also equals column 4 of square table A113103.

			The tree of 5-tournament sequences of descendents
of a node labeled (4) begins:
[4]; generation 1: 4->[8,12,16,20];
generation 2: 8->[12,16,20,24,28,32,36,40],
12->[16,20,24,28,32,36,40,44,48,52,56,60],
16->[20,24,28,32,36,40,44,48,52,56,60,64,68,72,76,80],
20->[24,28,32,36,40,44,48,52,56,60,64,68,72,76,80,84,88,92,96,100];
Then a(n) gives the number of nodes in generation n.
Also, a(n+1) = sum of labels of nodes in generation n.

T. D. Noe, Table of n, a(n) for n=0..30
M. Cook and M. Kleber, Tournament sequences and Meeussen sequences, Electronic J. Comb. 7 (2000), #R44.

Cf. A008934, A113077, A113078, A113079, A113085, A113089, A113096, A113098, A113100, A113107, A113109, A113111.

{a(n)=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^4)[n+1,1])}

A113096 Number of 4-tournament sequences: a(n) gives the number of increasing sequences of n positive integers (t_1,t_2,...,t_n) such that t_1 = 1 and t_i = 1 (mod 3) and t_{i+1} <= 4*t_i for 1

Original entry on oeis.org

1, 1, 4, 46, 1504, 146821, 45236404, 46002427696, 159443238441379, 1926751765436372746, 82540801108546193896804, 12696517688186899788062326096, 7084402815778394692932546017050054

Offset: 0

Paul D. Hanna, Oct 14 2005

nonn

Equals column 0 of triangle A113095, which satisfies: A113095(n,k) = [A113095^4](n-1,k-1) + [A113095^4](n-1,k).

			The tree of 4-tournament sequences of descendents
of a node labeled (1) begins:
[1]; generation 1: 1->[4]; generation 2: 4->[7,10,13,16];
generation 3: 7->[10,13,16,19,22,25,28],
10->[13,16,19,22,25,28,31,34,37,40],
13->[16,19,22,25,28,31,34,37,40,43,46,49,52],
16->[19,22,25,28,31,34,37,40,43,46,49,52,55,58,61,64]; ...
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. A008934, A113077, A113078, A113079, A113085, A113089, A113098, A113100, A113107, A113109, A113111, A113113.

{a(n)=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+1,1])}

A113098 Number of 4-tournament sequences: a(n) gives the number of increasing sequences of n positive integers (t_1,t_2,...,t_n) such that t_1 = 2 and t_i = 2 (mod 3) and t_{i+1} <= 4*t_i for 1

Original entry on oeis.org

1, 2, 13, 242, 13228, 2241527, 1237069018, 2305369985312, 14874520949557933, 338242806223319079422, 27474512329417917714396073, 8057337874806992183898478061882, 8607002252619465665736907583406214288

Offset: 0

Paul D. Hanna, Oct 14 2005

nonn

Equals column 0 of triangle A113097 = A113095^2 (matrix square), where: A113095(n,k) = [A113095^4](n-1,k-1) + [A113095^4](n-1,k).

			The tree of 4-tournament sequences of descendents
of a node labeled (2) begins:
[2]; generation 1: 2->[5,8]; generation 2:
5->[8,11,14,17,20], 8->[11,14,17,20,23,26,29,32]; ...
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. A008934, A113077, A113078, A113079, A113085, A113089, A113096, A113100, A113107, A113109, A113111, A113113.

{a(n)=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^2)[n+1,1])}

A113107 Number of 5-tournament sequences: a(n) gives the number of increasing sequences of n positive integers (t_1,t_2,...,t_n) such that t_1 = 1 and t_i = 1 (mod 4) and t_{i+1} <= 5*t_i for 1

Original entry on oeis.org

1, 1, 5, 85, 4985, 1082905, 930005021, 3306859233805, 50220281721033905, 3328966349792343354865, 978820270264589718999911669, 1292724512951963810375572954693765

Offset: 0

Paul D. Hanna, Oct 14 2005

nonn

Equals column 0 of triangle A113106 which satisfies recurrence: A113106(n,k) = [A113106^5](n-1,k-1) + [A113106^5](n-1,k), where A113106^5 is the matrix 5th power.

			The tree of 5-tournament sequences of descendents
of a node labeled (1) begins:
[1]; generation 1: 1->[5]; generation 2: 5->[9,13,17,21,25]; ...
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. A008934, A113077, A113078, A113079, A113085, A113089, A113096, A113098, A113100, A113109, A113111, A113113.

{a(n)=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+1,1])}

A097710 Lower triangular matrix T, read by rows, such that row (n) is formed from the sums of adjacent terms in row (n-1) of the matrix square T^2, with T(0,0)=1.

Original entry on oeis.org

1, 1, 1, 2, 3, 1, 7, 13, 7, 1, 41, 88, 61, 15, 1, 397, 951, 781, 257, 31, 1, 6377, 16691, 15566, 6231, 1041, 63, 1, 171886, 484490, 500057, 231721, 48303, 4161, 127, 1, 7892642, 23701698, 26604323, 13843968, 3406505, 374127, 16577, 255, 1

Offset: 0

Paul D. Hanna, Aug 22 2004

Column 0 is equal to sequence A008934, which is the number of tournament sequences.

This triangle has the same row sums and first column terms as in rows 2^n, for n>=0, of triangle A093654.

			Rows of this triangle T begin:
       1;
       1,      1;
       2,      3,      1;
       7,     13,      7,      1;
      41,     88,     61,     15,     1;
     397,    951,    781,    257,    31,    1;
    6377,  16691,  15566,   6231,  1041,   63,   1;
  171886, 484490, 500057, 231721, 48303, 4161, 127, 1;
Rows of T^2 begin:
        1;
        2,        1;
        7,        6,        1;
       41,       47,       14,       1;
      397,      554,      227,      30,      1;
     6377,    10314,     5252,     979,     62,     1;
   171886,   312604,   187453,   44268,   4035,   126,   1;
  7892642, 15809056, 10795267, 3048701, 357804, 16323, 254, 1;
The sums of adjacent terms in row (n) of T^2 forms row (n+1) of T:
  T(5,0) = T^2(4,0) = 397;
  T(5,1) = T^2(4,0) + T^2(4,1) = 397 + 554 = 951;
  T(5,2) = T^2(4,1) + T^2(4,2) = 554 + 227 = 781.
Rows of matrix inverse T^(-1) begins:
   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; ...
which is a signed version of A097712.

Paul D. Hanna, Table of n, a(n) for n = 0..495, of rows 0..30 of triangle in flattened form.

Cf. A000007, A000225, A093654, A097712.

Cf. A008934 (column k=0), A093657 (row sums), A097711 (column k=1).

T[n_, k_] := T[n, k] = Which[n<0 || k>n, 0, n == k, 1, k == 0, Sum[T[n-1, j]*T[j, 0], {j, 0, n-1}], True, Sum[T[n-1, j]*T[j, k-1], {j, 0, n-1}] + Sum[T[n-1, j]*T[j, k], {j, 0, n-1}]]; Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* Jean-François Alcover, Nov 23 2016, adapted from PARI *)

/* Using Recurrence relation: */
{T(n,k) = if(n<0||k>n, 0, if(n==k,1, if(k==0, sum(j=0,n-1, T(n-1,j)*T(j,0)),  sum(j=0,n-1, T(n-1,j)*T(j,k-1)) + sum(j=0,n-1, T(n-1,j)*T(j,k));)))}
for(n=0,8, for(k=0,n, print1(T(n,k),", "));print(""))

/* Faster: using Matrix generating method: */
{T(n,k) = my(M=matrix(2,2,r,c,if(r>=c,1))); for(i=1,n,
N=matrix(#M+1,#M+1,r,c, if(r>=c, if(r<=#M,M[r,c], if(c>1,(M^2)[r-1,c-1]) + if(c<=#M,(M^2)[r-1,c])) ));
M=N;); M[n+1,k+1]}
for(n=0,10,for(k=0,n,print1(T(n,k),", "));print("")) \\ Paul D. Hanna, Nov 27 2016

@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))
flatten([[T(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Feb 21 2024

T(n, k) = T^2(n-1, k-1) + T^2(n-1, k) for n>=1 and k>1, with T(n, 1) = T^2(n-1, 1) and T(n,n) = 1 for n>=0, where T^2 is the matrix square of this triangle T.
T(n, k) = Sum_{j=0..n-1} T(n-1, j)*(T(j, k-1) + T(j,k)), with T(n, 0) = Sum_{j=0..n-1} T(n-1,j)*T(j,0), and T(n, n) = 1.
T(n, 0) = A008934(n).
T(n, 1) = A097711(n).
Sum_{k=0..n} T(n, k) = A093657(n+1) (row sums).
From G. C. Greubel, Feb 21 2024: (Start)
T(n, n-1) = A000225(n).
Sum_{k=0..n} (-1)^k*T(n, k) = A000007(n). (End)

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])}

cp's OEIS Frontend

A016121 Number of sequences (a_1, a_2, ..., a_n) of length n with a_1 = 1 satisfying a_i <= a_{i+1} <= 2*a_i.

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A113089 Number of 3-tournament sequences: a(n) gives the number of increasing sequences of n positive integers (t_1,t_2,...,t_n) such that t_1 = 2 and t_i = 2 (mod 2) and t_{i+1} <= 3*t_i for 1

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A113100 Number of 4-tournament sequences: a(n) gives the number of increasing sequences of n positive integers (t_1,t_2,...,t_n) such that t_1 = 3 and t_i = 3 (mod 3) and t_{i+1} <= 4*t_i for 1

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A113113 Number of 5-tournament sequences: a(n) gives the number of increasing sequences of n positive integers (t_1,t_2,...,t_n) such that t_1 = 4 and t_i = 4 (mod 4) and t_{i+1} <= 5*t_i for 1

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A113096 Number of 4-tournament sequences: a(n) gives the number of increasing sequences of n positive integers (t_1,t_2,...,t_n) such that t_1 = 1 and t_i = 1 (mod 3) and t_{i+1} <= 4*t_i for 1

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A113098 Number of 4-tournament sequences: a(n) gives the number of increasing sequences of n positive integers (t_1,t_2,...,t_n) such that t_1 = 2 and t_i = 2 (mod 3) and t_{i+1} <= 4*t_i for 1

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A113107 Number of 5-tournament sequences: a(n) gives the number of increasing sequences of n positive integers (t_1,t_2,...,t_n) such that t_1 = 1 and t_i = 1 (mod 4) and t_{i+1} <= 5*t_i for 1

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PARI

A097710 Lower triangular matrix T, read by rows, such that row (n) is formed from the sums of adjacent terms in row (n-1) of the matrix square T^2, with T(0,0)=1.

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