cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A113365 Matrix cube of triangle A113350.

Original entry on oeis.org

1, 6, 1, 39, 12, 1, 327, 138, 18, 1, 3556, 1830, 297, 24, 1, 48659, 28805, 5349, 516, 30, 1, 812462, 535004, 109095, 11724, 795, 36, 1, 16136404, 11568197, 2529909, 292894, 21795, 1134, 42, 1, 373415239, 287143993, 66345668, 8117624, 643790, 36402, 1533
Offset: 0

Views

Author

Paul D. Hanna, Nov 09 2005

Keywords

Examples

			Triangle begins:
1;
6,1;
39,12,1;
327,138,18,1;
3556,1830,297,24,1;
48659,28805,5349,516,30,1;
812462,535004,109095,11724,795,36,1;
16136404,11568197,2529909,292894,21795,1134,42,1;
373415239,287143993,66345668,8117624,643790,36402,1533,48,1; ...

Crossrefs

Cf. A113340, A113350, A113346 (column 0), A113366 (column 1), A113367 (column 2); A113355 (=A113350^2), A113360 (=A113340^3).

Programs

  • PARI
    T(n,k)=local(A,B);A=matrix(1,1);A[1,1]=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^(2*j-1))[i-j+1,1]));));A=B); (matrix(#A,#A,r,c,if(r>=c,(A^(2*c))[r-c+1,1]))^3)[n+1,k+1]

Formula

Column k of A113350^3 = column 1 of A113340^(2*k+2) for k>=0.

A113368 Triangle, read by rows, given by the product Q^-1*P^2, where the triangular matrices involved are P = A113340 and Q = A113350.

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 5, 4, 1, 0, 19, 22, 6, 1, 0, 113, 166, 51, 8, 1, 0, 966, 1671, 561, 92, 10, 1, 0, 10958, 21510, 7726, 1324, 145, 12, 1, 0, 156700, 341463, 129406, 23010, 2575, 210, 14, 1, 0, 2727794, 6496923, 2572892, 471724, 53935, 4434, 287, 16, 1
Offset: 0

Views

Author

Paul D. Hanna, Nov 12 2005

Keywords

Comments

Matrix product Q^-1*P^2 = SHIFT_DOWN_RIGHT(Q). Compare to the matrix product Q^2*P^-1 = SHIFT_LEFT_UP(P), as given by triangle A113369.

Examples

			The product Q^-1*P^2 forms a triangle that begins:
1;
0,1;
0,2,1;
0,5,4,1;
0,19,22,6,1;
0,113,166,51,8,1;
0,966,1671,561,92,10,1;
0,10958,21510,7726,1324,145,12,1;
0,156700,341463,129406,23010,2575,210,14,1;
0,2727794,6496923,2572892,471724,53935,4434,287,16,1; ...
Compare Q^-1*P^2 to Q (A113350) which begins:
1;
2,1;
5,4,1;
19,22,6,1;
113,166,51,8,1;
966,1671,561,92,10,1;
10958,21510,7726,1324,145,12,1; ...

Crossrefs

Cf. A113340, A113350, A113369 (Q^2*P^-1).

Programs

  • PARI
    T(n,k)=local(A,B);A=matrix(1,1);A[1,1]=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^(2*j-1))[i-j+1,1]));));A=B);(A^(2*k))[n-k+1,1]

A113369 Triangle, read by rows, given by the product Q^2*P^-1, where the triangular matrices involved are P = A113340 and Q = A113350.

Original entry on oeis.org

1, 3, 1, 12, 5, 1, 69, 35, 7, 1, 560, 325, 70, 9, 1, 6059, 3880, 889, 117, 11, 1, 83215, 57560, 13853, 1881, 176, 13, 1, 1399161, 1030751, 258146, 36051, 3421, 247, 15, 1, 28020221, 21763632, 5633264, 805875, 77726, 5629, 330, 17, 1
Offset: 0

Views

Author

Paul D. Hanna, Nov 12 2005

Keywords

Comments

Matrix product Q^2*P^-1 = SHIFT_LEFT_UP(P). Compare to the matrix product Q^-1*P^2 = SHIFT_DOWN_RIGHT(Q), as given by triangle A113368.

Examples

			The product Q^2*P^-1 forms a triangle that begins:
1;
3,1;
12,5,1;
69,35,7,1;
560,325,70,9,1;
6059,3880,889,117,11,1;
83215,57560,13853,1881,176,13,1;
1399161,1030751,258146,36051,3421,247,15,1;
28020221,21763632,5633264,805875,77726,5629,330,17,1; ...
Compare Q^2*P^-1 to P (A113340) which begins:
1;
1,1;
1,3,1;
1,12,5,1;
1,69,35,7,1;
1,560,325,70,9,1;
1,6059,3880,889,117,11,1;
1,83215,57560,13853,1881,176,13,1; ...

Crossrefs

Cf. A113340, A113350, A113368 (Q^-1*P^2).

Programs

  • PARI
    T(n,k)=local(A,B);A=matrix(1,1);A[1,1]=1;for(m=2,n+2,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^(2*j-1))[i-j+1,1]));));A=B);A[n+2,k+2]

A113340 Triangle P, read by rows, such that P^2 transforms column k of P into column k+1 of P, so that column k of P equals column 0 of P^(2*k+1), where P^2 denotes the matrix square of P.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 12, 5, 1, 1, 69, 35, 7, 1, 1, 560, 325, 70, 9, 1, 1, 6059, 3880, 889, 117, 11, 1, 1, 83215, 57560, 13853, 1881, 176, 13, 1, 1, 1399161, 1030751, 258146, 36051, 3421, 247, 15, 1, 1, 28020221, 21763632, 5633264, 805875, 77726, 5629, 330, 17, 1
Offset: 0

Views

Author

Paul D. Hanna, Nov 08 2005

Keywords

Examples

			Triangle P begins:
1;
1,1;
1,3,1;
1,12,5,1;
1,69,35,7,1;
1,560,325,70,9,1;
1,6059,3880,889,117,11,1;
1,83215,57560,13853,1881,176,13,1;
1,1399161,1030751,258146,36051,3421,247,15,1;
1,28020221,21763632,5633264,805875,77726,5629,330,17,1;
1,654110586,531604250,141487178,20661609,2023461,147810,8625,425,19,1;
Matrix square P^2 (A113345) starts:
1;
2,1;
5,6,1;
19,39,10,1;
113,327,105,14,1;
966,3556,1315,203,18,1; ...
where P^2 transforms column k of P into column k+1 of P:
at k=0, [P^2]*[1,1,1,1,1,...] = [1,3,12,69,560,...];
at k=1, [P^2]*[1,3,12,69,560,...] = [1,5,35,325,3880,...].

Crossrefs

Cf. A113341 (column 1), A113342 (column 2), A113343 (column 3), A113344 (column 4); A113345 (P^2), A113360 (P^3), A113350 (Q).

Programs

  • PARI
    P(n,k)=local(A,B);A=matrix(1,1);A[1,1]=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^(2*j-1))[i-j+1,1]));));A=B);A[n+1,k+1]

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^(2*k+1)]_0, for k>=0.
Define the dual triangular matrix Q = A113350 by
[Q]_k = [P^(2*k+2)]_0, for k>=0.
Then, amazingly, powers of P and Q satisfy:
[P^(2*j+1)]_k = [P^(2*k+1)]_j,
[P^(2*j+2)]_k = [Q^(2*k+1)]_j,
[Q^(2*j+2)]_k = [Q^(2*k+2)]_j,
for all j>=0, k>=0.
Also, we have the column transformations:
P^2 * [P]k = [P]{k+1},
P^2 * [Q]k = [Q]{k+1},
Q^2 * [P^2]k = [P^2]{k+1},
Q^2 * [Q^2]k = [Q^2]{k+1},
for all k>=0.
Further, g.f.s of P and Q satisfy:
GF(P) = 1/(1-x) + x*y*GF(Q^2*P^-1),
GF(Q^-1*P^2) = 1 + x*y*GF(Q).

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.

Original entry on oeis.org

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

Views

Author

Paul D. Hanna, Nov 14 2005

Keywords

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.

Examples

			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,...].

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.

Programs

  • 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]

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.

A113381 Triangle Q, read by rows, such that Q^3 transforms column k of Q^2 into column k+1 of Q^2, so that column k of Q^2 equals column 0 of Q^(3*k+2), where Q^3 denotes the matrix cube of Q.

Original entry on oeis.org

1, 2, 1, 6, 5, 1, 37, 45, 8, 1, 429, 635, 120, 11, 1, 7629, 12815, 2556, 231, 14, 1, 185776, 343815, 71548, 6556, 378, 17, 1, 5817106, 11651427, 2508528, 233706, 13391, 561, 20, 1, 224558216, 480718723, 106427700, 10069521, 579047, 23817, 780, 23, 1
Offset: 0

Views

Author

Paul D. Hanna, Nov 14 2005

Keywords

Comments

Related matrix products are: R^3*Q^-2 (A114154), Q^-2*P^3 (A114155).

Examples

			Triangle Q begins:
1;
2,1;
6,5,1;
37,45,8,1;
429,635,120,11,1;
7629,12815,2556,231,14,1;
185776,343815,71548,6556,378,17,1;
5817106,11651427,2508528,233706,13391,561,20,1;
224558216,480718723,106427700,10069521,579047,23817,780,23,1;
Matrix square Q^2 (A113384) starts:
1;
4,1;
22,10,1;
212,130,16,1;
3255,2365,328,22,1;
70777,57695,8640,616,28,1; ...
Matrix cube Q^3 (A113387) starts:
1;
6,1;
48,15,1;
605,255,24,1;
11196,5630,624,33,1;
280440,159210,19484,1155,42,1; ...
where Q^3 transforms column k of Q^2 into column k+1:
at k=0, [Q^3]*[1,4,22,212,3255,...] = [1,10,130,2365,...];
at k=1, [Q^3]*[1,10,130,2365,...] = [1,16,328,8640,...].

Crossrefs

Cf. A113375 (column 0), A113382 (column 1), A113383 (column 2).
Cf. A113370 (P), A113374 (P^2), A113378 (P^3), A113384 (Q^2), A113387 (Q^3), A113389 (R), A113392 (R^2), A113394 (R^3).
Cf. A114154 (R^3*Q^-2), A114155 (Q^-2*P^3).
Cf. variants: A113340, A113350.

Programs

  • PARI
    Q(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+2))[n-k+1,1]

Formula

Let [Q^m]_k denote column k of matrix power Q^m,
so that triangular matrix Q may be defined by
[Q]_k = [P^(3*k+2)]_0, k>=0,
where the triangular matrix P = A113370 satisfies:
[P]_k = [P^(3*k+1)]_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.

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.

Original entry on oeis.org

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
Offset: 0

Views

Author

Paul D. Hanna, Nov 14 2005

Keywords

Comments

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

Examples

			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,..].

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.

Programs

  • 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]

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.

A113345 Matrix square of triangle A113340.

Original entry on oeis.org

1, 2, 1, 5, 6, 1, 19, 39, 10, 1, 113, 327, 105, 14, 1, 966, 3556, 1315, 203, 18, 1, 10958, 48659, 19875, 3367, 333, 22, 1, 156700, 812462, 357860, 64750, 6867, 495, 26, 1, 2727794, 16136404, 7547602, 1435497, 160005, 12199, 689, 30, 1
Offset: 0

Views

Author

Paul D. Hanna, Nov 08 2005

Keywords

Comments

Transforms column k of triangle A113340 into column k+1 of A113340. Also transforms column k of triangle A113350 into column k+1 of A113350.

Examples

			Matrix square A113340^2 starts:
1;
2,1;
5,6,1;
19,39,10,1;
113,327,105,14,1;
966,3556,1315,203,18,1;
10958,48659,19875,3367,333,22,1;
156700,812462,357860,64750,6867,495,26,1;
2727794,16136404,7547602,1435497,160005,12199,689,30,1; ...

Crossrefs

Cf. A113340, A113346 (column 0), A113347 (column 1), A113348 (column 2), A113349 (column 3); A113350.

Programs

  • PARI
    T(n,k)=local(A,B);A=matrix(1,1);A[1,1]=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^(2*j-1))[i-j+1,1]));));A=B);(A^2)[n+1,k+1]

A113360 Matrix cube of triangle A113340.

Original entry on oeis.org

1, 3, 1, 12, 9, 1, 69, 81, 15, 1, 560, 879, 210, 21, 1, 6059, 11739, 3285, 399, 27, 1, 83215, 190044, 59395, 8127, 648, 33, 1, 1399161, 3654814, 1241270, 184436, 16245, 957, 39, 1, 28020221, 81947221, 29720808, 4695719, 442890, 28479, 1326, 45, 1
Offset: 0

Views

Author

Paul D. Hanna, Nov 08 2005

Keywords

Examples

			Triangle begins:
1;
3,1;
12,9,1;
69,81,15,1;
560,879,210,21,1;
6059,11739,3285,399,27,1;
83215,190044,59395,8127,648,33,1;
1399161,3654814,1241270,184436,16245,957,39,1;
28020221,81947221,29720808,4695719,442890,28479,1326,45,1; ...

Crossrefs

Cf. A113340, A113350, A113361 (column 1), A113362 (column 2), A113363 (column 3), A113364 (column 4); A113345 (A113340^2).

Programs

  • PARI
    T(n,k)=local(A,B);A=matrix(1,1);A[1,1]=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^(2*j-1))[i-j+1,1]));));A=B);(A^3)[n+1,k+1]

A113364 Column 4 of triangle A113360, also equals column 1 of A113360^3.

Original entry on oeis.org

1, 27, 648, 16245, 442890, 13269069, 437085705, 15779515035, 621549547326, 26584365342579, 1228849871633643, 61116983784901476, 3257058909552139683, 185287271910574343301, 11212875913429712533737, 719555515466643129103760
Offset: 0

Views

Author

Paul D. Hanna, Nov 09 2005

Keywords

Comments

A113360 equals the matrix cube of triangle A113340, where column 4 of A113340^3 = column 1 of A113340^9.

Crossrefs

Cf. A113340, A113350, A113360 (A113340^3), A113341 (column 0), A113361 (column 1), A113362 (column 2), A113363 (column 3).

Programs

  • PARI
    a(n)=local(A,B); A=matrix(1,1);A[1,1]=1;for(m=2,n+5,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^(2*j-1))[i-j+1,1]));));A=B);(A^3)[n+5,5]
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