0,5

Triangle A114150 illustrates the identity: R^2*Q^-1 = Q^3*P^-2.

See also A114152 for the matrix product: R^3*P^-1.

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

so that triangular matrix P may be defined by

[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.

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 P begins:

1;

1,1;

1,4,1;

1,28,7,1;

1,326,91,10,1;

1,5702,1722,190,13,1;

1,136724,43764,4945,325,16,1;

1,4226334,1415799,163705,10751,496,19,1;

1,161385532,56096733,6617605,437723,19896,703,22,1;

1,7378504140,2644883675,317416204,21179483,960696,33136,946,25,1;

Matrix cube P^3 (A113378) starts:

1;

3,1;

15,12,1;

136,168,21,1;

1998,3190,483,30,1;

41973,80136,13615,960,39,1; ...

where P^3 transforms column k of P into column k+1 of P:

at k=0, [P^3]*[1,1,1,1,1,...] = [1,4,28,326,5702,...];

at k=1, [P^3]*[1,4,28,326,5702,...] = [1,7,91,1722,43764,...].

(PARI) {P(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[n+1, k+1]}

Cf. A113371 (column 1), A113372 (column 2), A113373 (column 3).

Cf. A113374 (P^2), A113378 (P^3), A113381 (Q), A113384 (Q^2), A113387 (Q^3), A113389 (R), A113392 (R^2), A113394 (R^3), A114156 (P^-1).

Cf. A114150 (R^2*Q^-1=Q^3*P^-2), A114152 (R^3*P^-1).

Cf. variants: A113340, A113350.

Sequence in context: A088158 A136449 A140805 this_sequence A078536 A158390 A102602

Adjacent sequences: A113367 A113368 A113369 this_sequence A113371 A113372 A113373

nonn,tabl

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