A113389 Triangle R, read by rows, such that R^3 transforms column k of R^3 into column k+1 of R^3, so that column k of R^3 equals column 0 of R^(3*k+3), where R^3 denotes the matrix cube of R.

1, 3, 1, 15, 6, 1, 136, 66, 9, 1, 1998, 1091, 153, 12, 1, 41973, 24891, 3621, 276, 15, 1, 1166263, 737061, 110637, 8482, 435, 18, 1, 40747561, 27110418, 4176549, 323874, 16430, 630, 21, 1, 1726907675, 1199197442, 188802141, 14813844, 751920, 28221

Offset: 0

Author

Paul D. Hanna, Nov 14 2005

Keywords

• nonn
• tabl

Comments

Related matrix products: identity R^-2*Q^3 = Q^-1*P^2 (A114151) and R^-1*P^3 (A114153).

Examples

			Triangle R begins:
1;
3,1;
15,6,1;
136,66,9,1;
1998,1091,153,12,1;
41973,24891,3621,276,15,1;
1166263,737061,110637,8482,435,18,1;
40747561,27110418,4176549,323874,16430,630,21,1;
1726907675,1199197442,188802141,14813844,751920,28221,861,24,1;
Matrix cube R^3 (A113394) starts:
1;
9,1;
99,18,1;
1569,360,27,1;
34344,9051,783,36,1;
980487,284148,26820,1368,45,1; ...
where R^3 transforms column k of R^3 into column k+1:
at k=0, [R^3]*[1,9,99,1569,...] = [1,18,360,9051,...];
at k=1, [R^3]*[1,18,360,9051,..] = [1,27,783,26820,..].
		

Crossrefs

Cf. A113379 (column 0), A113390 (column 1), A113391 (column 2).

Cf. A113370 (P), A113374 (P^2), A113378 (P^3), A113381 (Q), A113384 (Q^2), A113387 (Q^3), A113392 (R^2), A113394 (R^3).

Cf. A114151 (R^-2*Q^3 = Q^-1*P^2), A114153 (R^-1*P^3).

Cf. variants: A113340, A113350.

Programs

PARI

R(n,k)=local(A,B);A=Mat(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^(3*j-2))[i-j+1,1]));));A=B);(A^(3*k+3))[n-k+1,1]

Formula

Let [R^m]_k denote column k of matrix power R^m,

so that triangular matrix R may be defined by

[R]_k = [P^(3*k+3)]_0, k>=0,

where the triangular matrix P = A113370 satisfies:

[P]_k = [P^(3*k+1)]_0, k>=0.

Define the triangular matrix Q = A113381 by

[Q]_k = [P^(3*k+2)]_0, k>=0.

Then P, Q and R are related by:

Q^2 = R*P = R*Q*(R^-2)*Q*R = P*Q*(P^-2)*Q*P,

P^2 = Q*(R^-2)*Q^3, R^2 = Q^3*(P^-2)*Q.

Amazingly, columns in powers of P, Q, R, obey:

[P^(3*j+1)]_k = [P^(3*k+1)]_j,

[Q^(3*j+1)]_k = [P^(3*k+2)]_j,

[R^(3*j+1)]_k = [P^(3*k+3)]_j,

[Q^(3*j+2)]_k = [Q^(3*k+2)]_j,

[R^(3*j+2)]_k = [Q^(3*k+3)]_j,

[R^(3*j+3)]_k = [R^(3*k+3)]_j,

for all j>=0, k>=0.

Also, we have the column transformations:

P^3 * [P]k = [P]{k+1},

P^3 * [Q]k = [Q]{k+1},

P^3 * [R]k = [R]{k+1},

Q^3 * [P^2]k = [P^2]{k+1},

Q^3 * [Q^2]k = [Q^2]{k+1},

Q^3 * [R^2]k = [R^2]{k+1},

R^3 * [P^3]k = [P^3]{k+1},

R^3 * [Q^3]k = [Q^3]{k+1},

R^3 * [R^3]k = [R^3]{k+1},

for all k>=0.