A113381 Triangle Q, read by rows, such that Q^3 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^(3*k+2), where Q^3 denotes the matrix cube of Q.

1, 2, 1, 6, 5, 1, 37, 45, 8, 1, 429, 635, 120, 11, 1, 7629, 12815, 2556, 231, 14, 1, 185776, 343815, 71548, 6556, 378, 17, 1, 5817106, 11651427, 2508528, 233706, 13391, 561, 20, 1, 224558216, 480718723, 106427700, 10069521, 579047, 23817, 780, 23, 1

Offset

0,2

Comments

Related matrix products are: R^3*Q^-2 (A114154), Q^-2*P^3 (A114155).

Links

Table of n, a(n) for n=0..44.

Formula

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

so that triangular matrix Q may be defined by

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

where the triangular matrix P = A113370 satisfies:

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

Define the triangular matrix R = A113389 by

[R]_k = [P^(3*k+3)]_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.

Example

Triangle Q begins:
1;
2,1;
6,5,1;
37,45,8,1;
429,635,120,11,1;
7629,12815,2556,231,14,1;
185776,343815,71548,6556,378,17,1;
5817106,11651427,2508528,233706,13391,561,20,1;
224558216,480718723,106427700,10069521,579047,23817,780,23,1;
Matrix square Q^2 (A113384) starts:
1;
4,1;
22,10,1;
212,130,16,1;
3255,2365,328,22,1;
70777,57695,8640,616,28,1; ...
Matrix cube Q^3 (A113387) starts:
1;
6,1;
48,15,1;
605,255,24,1;
11196,5630,624,33,1;
280440,159210,19484,1155,42,1; ...
where Q^3 transforms column k of Q^2 into column k+1:
at k=0, [Q^3]*[1,4,22,212,3255,...] = [1,10,130,2365,...];
at k=1, [Q^3]*[1,10,130,2365,...] = [1,16,328,8640,...].

Prog

(PARI) Q(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+2))[n-k+1, 1]

Crossrefs

Cf. A113375 (column 0), A113382 (column 1), A113383 (column 2).

Cf. A113370 (P), A113374 (P^2), A113378 (P^3), A113384 (Q^2), A113387 (Q^3), A113389 (R), A113392 (R^2), A113394 (R^3).

Cf. A114154 (R^3*Q^-2), A114155 (Q^-2*P^3).

Cf. variants: A113340, A113350.

Sequence in context: A308498 A295517 A070918 * A228175 A118980 A351385

Adjacent sequences: A113378 A113379 A113380 * A113382 A113383 A113384

Keyword

nonn,tabl

Author

Paul D. Hanna, Nov 14 2005

Status

approved