0,2

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

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.

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,..].

(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]}

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.

Sequence in context: A104990 A089463 A136231 this_sequence A038553 A135896 A134144

Adjacent sequences: A113386 A113387 A113388 this_sequence A113390 A113391 A113392

nonn,tabl

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