A113365 -id:A113365 - cp's OEIS frontend

A113366 Column 1 of triangle A113365, also equals column 1 of A113340^4.

1, 12, 138, 1830, 28805, 535004, 11568197, 287143993, 8077888153, 254672147047, 8910929460415, 343135184110984, 14435616939387951, 659276261774240232, 32504007393860850275, 1721495715845423489806, 97516667477625085469176

Offset: 0

Paul D. Hanna, Nov 09 2005

nonn

A113365 equals the matrix cube of A113350, where column 1 of A113350^3 = column 1 of A113340^4.

Cf. A113340, A113350, A113365 (=A113350^3), A113346 (column 0), A113367 (column 2); A113355 (=A113350^2), A113360 (=A113340^3).

a(n)=local(A,B);A=matrix(1,1);A[1,1]=1;for(m=2,n+2,B=matrix(m,m); for(i=1,m, for(j=1,i,if(i<3 || j==i || j>m-1,B[i,j]=1,if(j==1, B[i,1]=1,B[i,j]=(A^(2*j-1))[i-j+1,1]));));A=B); (matrix(#A,#A,r,c,if(r>=c,(A^(2*c))[r-c+1,1]))^3)[n+2,2]

A113367 Column 2 of triangle A113365, also equals column 1 of A113340^6.

Original entry on oeis.org

1, 18, 297, 5349, 109095, 2529909, 66345668, 1951815218, 63879120597, 2307569048308, 91351738972187, 3937843359099176, 183778364128820499, 9238674078250458082, 497996763423626058847, 28666454637876852704364

Offset: 0

Paul D. Hanna, Nov 09 2005

nonn

A113365 equals the matrix cube of A113350, where column 2 of A113350^3 = column 1 of A113340^6.

Cf. A113340, A113350, A113365 (=A113350^3), A113346 (column 0), A113366 (column 1).

a(n)=local(A,B);A=matrix(1,1);A[1,1]=1;for(m=2,n+3,B=matrix(m,m); for(i=1,m, for(j=1,i,if(i<3 || j==i || j>m-1,B[i,j]=1,if(j==1, B[i,1]=1,B[i,j]=(A^(2*j-1))[i-j+1,1]));));A=B); (matrix(#A,#A,r,c,if(r>=c,(A^(2*c))[r-c+1,1]))^3)[n+3,3]

A113350 Triangle Q, read by rows, such that Q^2 transforms column k of Q^2 into column k+1 of Q^2, so that column k of Q^2 equals column 0 of Q^(2*k+2), where Q^2 denotes the matrix square of Q.

Original entry on oeis.org

1, 2, 1, 5, 4, 1, 19, 22, 6, 1, 113, 166, 51, 8, 1, 966, 1671, 561, 92, 10, 1, 10958, 21510, 7726, 1324, 145, 12, 1, 156700, 341463, 129406, 23010, 2575, 210, 14, 1, 2727794, 6496923, 2572892, 471724, 53935, 4434, 287, 16, 1

Offset: 0

Paul D. Hanna, Nov 08 2005

			Triangle Q begins:
1;
2,1;
5,4,1;
19,22,6,1;
113,166,51,8,1;
966,1671,561,92,10,1;
10958,21510,7726,1324,145,12,1;
156700,341463,129406,23010,2575,210,14,1;
2727794,6496923,2572892,471724,53935,4434,287,16,1;
56306696,144856710,59525136,11198006,1305070,108593,7021,376,18,1;
Matrix square Q^2 begins:
1;
4,1;
18,8,1;
112,68,12,1;
965,712,150,16,1;
10957,9270,2184,264,20,1; ...
where Q^2 transforms column k of Q^2 into column k+1:
at k=0, [Q^2]*[1,4,18,112,965,...] = [1,8,68,712,9270,...];
at k=1, [Q^2]*[1,8,68,712,9270,...] =
[1,12,150,2184,37523,...].

Cf. A113351 (column 1), A113352 (column 2), A113353 (column 3), A113354 (column 4); A113355 (Q^2), A113365 (Q^3), A113340 (P), A113345 (P^2), A113360 (P^3).

Q(n,k)=local(A,B);A=matrix(1,1);A[1,1]=1;for(m=2,n+1,B=matrix(m,m); for(i=1,m, for(j=1,i,if(i<3 || j==i || j>m-1,B[i,j]=1,if(j==1, B[i,1]=1,B[i,j]=(A^(2*j-1))[i-j+1,1]));));A=B);(A^(2*k+2))[n-k+1,1]

Let [Q^m]_k denote column k of matrix power Q^m,
so that triangular matrix Q may be defined by
[Q]_k = [P^(2*k+2)]_0, for k>=0, where
the dual triangular matrix P = A113340 is defined by
[P]_k = [P^(2*k+1)]_0, for k>=0.
Then, amazingly, powers of P and Q satisfy:
[P^(2*j+1)]_k = [P^(2*k+1)]_j,
[P^(2*j+2)]_k = [Q^(2*k+1)]_j,
[Q^(2*j+2)]_k = [Q^(2*k+2)]_j,
for all j>=0, k>=0.
Also, we have the column transformations:
P^2 * [P]k = [P]{k+1},
P^2 * [Q]k = [Q]{k+1},
Q^2 * [P^2]k = [P^2]{k+1},
Q^2 * [Q^2]k = [Q^2]{k+1},
for all k>=0.

cp's OEIS Frontend

A113366 Column 1 of triangle A113365, also equals column 1 of A113340^4.

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A113367 Column 2 of triangle A113365, also equals column 1 of A113340^6.

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A113350 Triangle Q, read by rows, such that Q^2 transforms column k of Q^2 into column k+1 of Q^2, so that column k of Q^2 equals column 0 of Q^(2*k+2), where Q^2 denotes the matrix square of Q.

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