A113370 Triangle P, read by rows, such that P^3 transforms column k of P into column k+1 of P, so that column k of P equals column 0 of P^(3*k+1), where P^3 denotes the matrix cube of P.

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

Offset

0,5

Comments

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

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

Links

Table of n, a(n) for n=0..47.

Formula

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.

Example

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,...].

Prog

(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]

Crossrefs

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: A136449 A209427 A140805 * A078536 A173918 A174412

Adjacent sequences: A113367 A113368 A113369 * A113371 A113372 A113373

Keyword

nonn,tabl

Author

Paul D. Hanna, Nov 14 2005

Status

approved