cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 13 results. Next

A113108 Triangle T, read by rows, equal to the matrix square of triangle A113106, which satisfies the recurrence: A113106(n,k) = [A113106^5](n-1,k-1) + [A113106^5](n-1,k).

Original entry on oeis.org

1, 2, 1, 16, 12, 1, 440, 416, 62, 1, 43600, 48320, 10016, 312, 1, 16698560, 20765520, 5394320, 246016, 1562, 1, 26098464448, 35382716032, 10854556720, 646408320, 6116016, 7812, 1, 172513149018752, 250136469031744, 87213434633152
Offset: 0

Views

Author

Paul D. Hanna, Oct 14 2005

Keywords

Examples

			Triangle begins:
1;
2,1;
16,12,1;
440,416,62,1;
43600,48320,10016,312,1;
16698560,20765520,5394320,246016,1562,1;
26098464448,35382716032,10854556720,646408320,6116016,7812,1; ...

Crossrefs

Cf. A113106.

Programs

  • PARI
    {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^2)[n+1,k+1])}

A113110 Triangle T, read by rows, equal to the matrix cube of triangle A113106, which satisfies the recurrence: A113106(n,k) = [A113106^5](n-1,k-1) + [A113106^5](n-1,k).

Original entry on oeis.org

1, 3, 1, 33, 18, 1, 1251, 903, 93, 1, 173505, 151716, 22278, 468, 1, 94216515, 94758285, 17789766, 551778, 2343, 1, 210576669921, 235461277878, 53137278735, 2167944516, 13749903, 11718, 1, 2002383115518243, 2432344424403219
Offset: 0

Views

Author

Paul D. Hanna, Oct 14 2005

Keywords

Examples

			Triangle begins:
1;
3,1;
33,18,1;
1251,903,93,1;
173505,151716,22278,468,1;
94216515,94758285,17789766,551778,2343,1;
210576669921,235461277878,53137278735,2167944516,13749903,11718,1;

Crossrefs

Cf. A113106.

Programs

  • PARI
    {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^3)[n+1,k+1])}

A113112 Triangle T, read by rows, equal to the matrix 4th power of triangle A113106, which satisfies the recurrence: A113106(n,k) = [A113106^5](n-1,k-1) + [A113106^5](n-1,k).

Original entry on oeis.org

1, 4, 1, 56, 24, 1, 2704, 1576, 124, 1, 481376, 346624, 39376, 624, 1, 337587520, 284081376, 41686624, 979376, 3124, 1, 978162377600, 927672109184, 165184873376, 5122890624, 24434376, 15624, 1, 12088945462984960
Offset: 0

Views

Author

Paul D. Hanna, Oct 14 2005

Keywords

Examples

			Triangle begins:
1;
4,1;
56,24,1;
2704,1576,124,1;
481376,346624,39376,624,1;
337587520,284081376,41686624,979376,3124,1;
978162377600,927672109184,165184873376,5122890624,24434376,15624,1;

Crossrefs

Cf. A113106, A113108, A113110.

Programs

  • PARI
    {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^4)[n+1,k+1])}

A113114 Triangle T, read by rows, equal to the matrix 5th power of triangle A113106, which satisfies the recurrence: A113106(n,k) = [A113106^5](n-1,k-1) + [A113106^5](n-1,k).

Original entry on oeis.org

1, 5, 1, 85, 30, 1, 4985, 2435, 155, 1, 1082905, 662060, 61310, 780, 1, 930005021, 671754405, 80861810, 1528810, 3905, 1, 3306859233805, 2718081933706, 399334065655, 9987138060, 38169435, 19530, 1, 50220281721033905
Offset: 0

Views

Author

Paul D. Hanna, Oct 14 2005

Keywords

Comments

Column 0 equals A113107 shift one place left.

Examples

			Triangle begins:
1;
5,1;
85,30,1;
4985,2435,155,1;
1082905,662060,61310,780,1;
930005021,671754405,80861810,1528810,3905,1;
3306859233805,2718081933706,399334065655,9987138060,38169435,19530,1;

Crossrefs

Cf. A113106, A113108, A113110, A113112.

Programs

  • PARI
    {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^5)[n+1,k+1])}

A113104 Main diagonal of square table A113103; also, a(n) equals the n-th term in column 0 of the matrix n-th power of triangle A113106.

Original entry on oeis.org

1, 1, 16, 1251, 481376, 930005021, 9082872004032, 448356882751890343, 111655372144044770735104, 140027604270897805074354921849, 883117855371077265832943940474315776
Offset: 0

Views

Author

Paul D. Hanna, Oct 14 2005

Keywords

Crossrefs

Cf. A113103, A113106.

Programs

  • PARI
    {a(n,q=5)=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^n)[n+1,1])}

A113105 Diagonal of square table A113103; also, a(n) equals the n-th term in column 0 of the matrix (n+1)-th power of triangle A113106.

Original entry on oeis.org

1, 2, 33, 2704, 1082905, 2156566656, 21543117605345, 1081795451307347456, 273019500242348456497329, 346065491936438505902218920448, 2201645604139293737199292995777020545
Offset: 0

Views

Author

Paul D. Hanna, Oct 14 2005

Keywords

Crossrefs

Cf. A113103, A113106.

Programs

  • PARI
    {a(n,q=5)=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(if(n<1,0,(M^n)[n,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

Views

Author

Paul D. Hanna, Oct 14 2005

Keywords

Comments

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.

Examples

			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.

Links

Crossrefs

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

Programs

  • PARI
    {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])}

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

Views

Author

Paul D. Hanna, Oct 14 2005

Keywords

Comments

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.

Examples

			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.

Links

Crossrefs

Cf. A008934, A113077, A113078, A113079, A113085, A113089, A113096, A113098, A113100, A113109, A113111, A113113.

Programs

  • PARI
    {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])}

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

Views

Author

Paul D. Hanna, Oct 14 2005

Keywords

Comments

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

Examples

			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.

Crossrefs

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

Programs

  • PARI
    {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])}

Formula

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

Views

Author

Paul D. Hanna, Oct 14 2005

Keywords

Comments

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

Examples

			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.

Crossrefs

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

Programs

  • PARI
    {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])}

Formula

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)].
Showing 1-10 of 13 results. Next

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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