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 A113389 Triangle R, read by rows, such that R^3 transforms column k of R^3 into column k+1 of R^3, so that column k of R^3 equals column 0 of R^(3*k+3), where R^3 denotes the matrix cube of R. 24
 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 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Related matrix products: identity R^-2*Q^3 = Q^-1*P^2 (A114151) and R^-1*P^3 (A114153). LINKS FORMULA 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. EXAMPLE 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,..]. PROG (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] CROSSREFS 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 * A038553 A282629 A135896 Adjacent sequences:  A113386 A113387 A113388 * A113390 A113391 A113392 KEYWORD nonn,tabl AUTHOR Paul D. Hanna, Nov 14 2005 STATUS approved

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Last modified September 30 10:18 EDT 2020. Contains 337439 sequences. (Running on oeis4.)