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A113370
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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.
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28
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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
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,5
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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.
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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.
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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,...].
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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]}
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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: A088158 A136449 A140805 * A078536 A173918 A174412
Adjacent sequences: A113367 A113368 A113369 * A113371 A113372 A113373
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KEYWORD
| nonn,tabl
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AUTHOR
| Paul D. Hanna (pauldhanna(AT)juno.com), Nov 14 2005
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