0,2

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

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.

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,...].

(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]}

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: A136124 A143491 A070918 this_sequence A118980 A090665 A021826

Adjacent sequences: A113378 A113379 A113380 this_sequence A113382 A113383 A113384

nonn,tabl

Paul D. Hanna (pauldhanna(AT)juno.com), Nov 14 2005