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 1-3 of 3 results.

A113102 Column 1 of triangle A113101.

Original entry on oeis.org

1, 20, 894, 108292, 39188597, 45157269264, 172969059719500, 2268555493366383236, 104065414379018585400528, 16965802042440759287849083708, 9953305892540385313201626908460431
Offset: 1

Views

Author

Paul D. Hanna, Oct 14 2005

Keywords

Crossrefs

Cf. A113101.

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^4)[r-1,c-1])+(M^4)[r-1,c]))); return(if(n<1,0,(M^4)[n+1,2]))}

Extensions

Offset changed to 1 by Georg Fischer, Jun 20 2022

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)].

A113099 Triangle T, read by rows, equal to the matrix cube of triangle A113095, which satisfies the recurrence: A113095(n,k) = [A113095^4](n-1,k-1) + [A113095^4](n-1,k).

Original entry on oeis.org

1, 3, 1, 27, 15, 1, 693, 513, 63, 1, 52812, 47619, 8289, 255, 1, 12628008, 13176189, 2920527, 131841, 1023, 1, 9924266772, 11586274263, 3078907929, 181929087, 2101761, 4095, 1, 26507035453923, 33825995695125, 10365262415703
Offset: 0

Views

Author

Paul D. Hanna, Oct 14 2005

Keywords

Examples

			Triangle begins:
1;
3,1;
27,15,1;
693,513,63,1;
52812,47619,8289,255,1;
12628008,13176189,2920527,131841,1023,1;
9924266772,11586274263,3078907929,181929087,2101761,4095,1; ...

Crossrefs

Cf. A113095, A113098, A113101.

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^3)[n+1,k+1])}
Showing 1-3 of 3 results.

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