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.

Previous Showing 11-20 of 28 results. Next

A113371 Column 1 of triangle A113370, also equals column 0 of A113370^4.

Original entry on oeis.org

1, 4, 28, 326, 5702, 136724, 4226334, 161385532, 7378504140, 394404094270, 24193638303234, 1677962100799727, 129990908689219749, 11135889051081255518, 1046032727101344902679, 106966601176207198837104
Offset: 0

Views

Author

Paul D. Hanna, Nov 13 2005

Keywords

Comments

Column k of A113370 = column 0 of A113370^(3*k+1) for k>=0.

Crossrefs

Cf. A113370, A113372 (column 2), A113373 (column 3).

Programs

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

A113372 Column 2 of triangle A113370, also equals column 0 of A113370^7.

Original entry on oeis.org

1, 7, 91, 1722, 43764, 1415799, 56096733, 2644883675, 145131435225, 9107198292451, 644373208531066, 50814103000624929, 4423148359685316443, 421540670702940409866, 43680252604560889074884
Offset: 0

Views

Author

Paul D. Hanna, Nov 13 2005

Keywords

Comments

Column k of A113370 = column 0 of A113370^(3*k+1) for k>=0.

Crossrefs

Cf. A113370, A113371 (column 1), A113373 (column 3).

Programs

  • PARI
    a(n)=local(A,B);A=Mat(1);for(m=2,n+3,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+3,3]

A113373 Column 3 of triangle A113370, also equals column 0 of A113370^10.

Original entry on oeis.org

1, 10, 190, 4945, 163705, 6617605, 317416204, 17677189426, 1123660618048, 80411135580587, 6405386721320372, 562632770359355432, 54061761393693393203, 5643864294810397451552, 636388356169482970183598
Offset: 0

Views

Author

Paul D. Hanna, Nov 13 2005

Keywords

Comments

Column k of A113370 = column 0 of A113370^(3*k+1) for k>=0.

Crossrefs

Cf. A113370, A113371 (column 1), A113372 (column 2).

Programs

  • PARI
    a(n)=local(A,B);A=Mat(1);for(m=2,n+4,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+4,4]

A113382 Column 1 of triangle A113381, also equals column 0 of A113370^5.

Original entry on oeis.org

1, 5, 45, 635, 12815, 343815, 11651427, 480718723, 23489845779, 1330745268401, 85944092769721, 6242138253088466, 504185328302302736, 44867722807185829082, 4364538423763543903228, 460969199012824227856506
Offset: 0

Views

Author

Paul D. Hanna, Nov 14 2005

Keywords

Crossrefs

Cf. A113381, A113375 (column 0), A113383 (column 2), A113370.

Programs

  • PARI
    a(n)=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^5)[n+1,1]

Formula

Column k of A113381 = column 0 of A113370^(3*k+2) for k>=0.

A113383 Column 2 of triangle A113381, also equals column 0 of A113370^8.

Original entry on oeis.org

1, 8, 120, 2556, 71548, 2508528, 106427700, 5323786728, 307710142888, 20222341451124, 1491479257952300, 122128352186849366, 11002901720698439826, 1082337197005046142588, 115485905212456384697750
Offset: 0

Views

Author

Paul D. Hanna, Nov 14 2005

Keywords

Crossrefs

Cf. A113381, A113375 (column 0), A113382 (column 1), A113370.

Programs

  • PARI
    a(n)=local(A,B);A=Mat(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^(3*j-2))[i-j+1,1]));));A=B);(A^8)[n+1,1]

Formula

Column k of A113381 = column 0 of A113370^(3*k+2) for k>=0.

A113390 Column 1 of triangle A113389, also equals column 0 of A113370^6.

Original entry on oeis.org

1, 6, 66, 1091, 24891, 737061, 27110418, 1199197442, 62240034172, 3718021355407, 251730371459590, 19076604651022143, 1601423150451641820, 147628858305489901288, 14834881996161804192069
Offset: 0

Views

Author

Paul D. Hanna, Nov 14 2005

Keywords

Crossrefs

Cf. A113389, A113378 (column 0), A113391 (column 2), A113370.

Programs

  • PARI
    a(n)=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^6)[n+1,1]

Formula

Column k of A113389 = column 0 of A113370^(3*k+3) for k>=0.

A113391 Column 2 of triangle A113389, also equals column 0 of A113370^9.

Original entry on oeis.org

1, 9, 153, 3621, 110637, 4176549, 188802141, 9981491997, 605817292893, 41590997891929, 3190816992548889, 270817573670371995, 25214094974302894695, 2556615042094813435491, 280570514270855698070535
Offset: 0

Views

Author

Paul D. Hanna, Nov 14 2005

Keywords

Crossrefs

Cf. A113389, A113378 (column 0), A113390 (column 1), A113370.

Programs

  • PARI
    a(n)=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^9)[n+1,1]

Formula

Column k of A113389 = column 0 of A113370^(3*k+3) for k>=0.

A114157 Column 0 of triangle A114156, which is the matrix inverse of A113370.

Original entry on oeis.org

1, -1, 3, 6, -8, -501, -13623, -409953, -14544683, -607055209, -29421219966, -1632715832341, -102423931253271, -7182065357683744, -557479490745141406, -47499193522110824640, -4410100445892863617679, -443346294340660384160358
Offset: 0

Views

Author

Paul D. Hanna, Nov 15 2005

Keywords

Crossrefs

Cf. A114156, A113370.

Programs

  • PARI
    a(n)=local(P,Q,R,W);P=Mat(1);for(m=2,n+1,W=matrix(m,m); for(i=1,m, for(j=1,i,if(i<3 || j==i || j>m-1,W[i,j]=1,if(j==1, W[i,1]=1,W[i,j]=(P^(3*j-2))[i-j+1,1]));));P=W); (P^-1)[n+1,1]

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.
Previous Showing 11-20 of 28 results. Next

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.