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.

A104447 Column 1 of triangular matrix A104446.

Original entry on oeis.org

1, 2, 5, 13, 39, 139, 587, 2897, 16462, 106301, 771313, 6228073, 55494336, 541651873, 5753940704, 66147591142, 818802488476, 10864622564915, 153914784829775, 2319599022540318, 37068215129072522, 626279667948552452
Offset: 0

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Author

Paul D. Hanna, Mar 08 2005

Keywords

Comments

A104446 equals the square of triangular matrix A104445, read by rows, where X=A104445 satisfies: SHIFT_LEFT_UP(X) = X^2 - X + I.

Crossrefs

Programs

  • PARI
    a(n)=local(A=Mat(1),B);for(m=1,n+1,B=A^2-A+A^0; A=matrix(m+1,m+1);for(i=1,m+1, for(j=1,i, if(i<2 || j==i,A[i,j]=1,if(j==1,A[i,j]=1,A[i,j]=B[i-1,j-1]))))); return((A^2)[n+2,2])

Formula

a(n) = A091352(n-1) + A091353(n-1).

A104448 Row sums of triangle A104446.

Original entry on oeis.org

1, 3, 6, 13, 33, 101, 372, 1624, 8263, 48285, 320031, 2380114, 19675986, 179314868, 1788473424, 19398149629, 227510745445, 2871040422932, 38810001746171, 559745948482030, 8582882169611759, 139467832061599433
Offset: 0

Views

Author

Paul D. Hanna, Mar 08 2005

Keywords

Comments

A104446 equals the square of triangular matrix A104445, read by rows, where X=[A104445] satisfies: SHIFT_LEFT_UP(X) = X^2 - X + I.

Crossrefs

Programs

  • PARI
    a(n)=local(A=Mat(1),B);for(m=1,n+1,B=A^2-A+A^0; A=matrix(m+1,m+1);for(i=1,m+1, for(j=1,i, if(i<2 || j==i,A[i,j]=1,if(j==1,A[i,j]=1,A[i,j]=B[i-1,j-1]))))); return(sum(k=1,n+1,(A^2)[n+1,k]))

Formula

a(n) = A091352(n) + A091352(n-1) for n>0.

A104445 Triangular matrix T, read by rows, that satisfies: SHIFT_LEFT_UP(T) = T^2 - T + I, or, equivalently: T(n+1,k+1) = [T^2](n,k) - T(n,k) + [T^0](n,k) for n>=k>=0, with T(0,0)=1.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 4, 3, 1, 1, 1, 9, 9, 4, 1, 1, 1, 24, 30, 16, 5, 1, 1, 1, 77, 115, 70, 25, 6, 1, 1, 1, 295, 510, 344, 135, 36, 7, 1, 1, 1, 1329, 2602, 1908, 805, 231, 49, 8, 1, 1, 1, 6934, 15133, 11904, 5325, 1616, 364, 64, 9, 1, 1, 1, 41351, 99367, 83028, 39001
Offset: 0

Views

Author

Paul D. Hanna, Mar 07 2005

Keywords

Comments

Surprisingly, SHIFT_UP(T) = A091351, or T(n+1,k) = A091351(n,k) for n>=k>=0, where column k of A091351 equals column 0 of A091351^(k+1) for k>=0.

Examples

			Rows begin:
1;
1,1;
1,1,1;
1,2,1,1;
1,4,3,1,1;
1,9,9,4,1,1;
1,24,30,16,5,1,1;
1,77,115,70,25,6,1,1;
1,295,510,344,135,36,7,1,1;
1,1329,2602,1908,805,231,49,8,1,1;
1,6934,15133,11904,5325,1616,364,64,9,1,1; ...
		

Crossrefs

Cf. A091351, A104446 (matrix square); columns form: A091352, A091353, A091354.

Programs

  • PARI
    T(n,k)=if(n
    				

Formula

T(n, k) = Sum_{j=0..n-k-1} T(n-k, j)*T(j+k, k-1) for n>k>0 with T(n, 0)=T(n, n)=1 (n>=0).
Showing 1-3 of 3 results.