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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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.

A113346 Column 0 of triangle A113345, also equals column 0 of A113350.

Original entry on oeis.org

1, 2, 5, 19, 113, 966, 10958, 156700, 2727794, 56306696, 1350043965, 36979531549, 1141573025172, 39272377323693, 1491452150268436, 62027842189908231, 2805631215820328992, 137199563717151509077, 7215932308408758314447
Offset: 0

Views

Author

Paul D. Hanna, Nov 08 2005

Keywords

Comments

A113345 is the matrix square of A113340.

Crossrefs

Cf. A113340, A113345, A113347 (column 1), A113348 (column 2), A113349 (column 3); A113350.

Programs

  • PARI
    a(n)=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+1,1]

A113355 Triangle T, read by rows, equal to the matrix square of triangle A113350, where T transforms column k of T into column k+1 of T.

Original entry on oeis.org

1, 4, 1, 18, 8, 1, 112, 68, 12, 1, 965, 712, 150, 16, 1, 10957, 9270, 2184, 264, 20, 1, 156699, 147174, 37523, 4912, 410, 24, 1, 2727793, 2786270, 754171, 104476, 9280, 588, 28, 1, 56306695, 61662544, 17502145, 2531004, 235025, 15672, 798, 32, 1
Offset: 0

Views

Author

Paul D. Hanna, Nov 08 2005

Keywords

Comments

Also, T transforms column k of A113340^2 into column k+1 of A113340^2. Column 0: T(n,0) = A113356(n) = A113346(n+1) - 1, where A113346 equals column 0 of triangle A113345 (=A113340^2).

Examples

			Triangle T begins:
1;
4,1;
18,8,1;
112,68,12,1;
965,712,150,16,1;
10957,9270,2184,264,20,1;
156699,147174,37523,4912,410,24,1;
2727793,2786270,754171,104476,9280,588,28,1;
56306695,61662544,17502145,2531004,235025,15672,798,32,1; ...
where T transforms column k of T into column k+1:
at k=0, [Q^2]*[1,4,18,112,965,...] = [1,8,68,712,9270,...];
at k=1, [Q^2]*[1,8,68,712,9270,...] = [1,12,150,2184,37523,...].

Crossrefs

Cf. A113340, A113350, A113356 (column 0), A113357 (column 1), A113358 (column 2), A113359 (column 3); A091351.

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]))^2)[n+1,k+1]

Formula

T(n, k) = sum_{j=0..n-k} T(n-k, j)*T(j+k-1, k-1) for n>=k>0 with T(n, 0) = A113346(n+1) - 1, for n>=0.

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]

A113347 Column 1 of triangle A113345, also equals column 0 of A113350^3.

Original entry on oeis.org

1, 6, 39, 327, 3556, 48659, 812462, 16136404, 373415239, 9900007028, 296557405704, 9921937128500, 367181525916035, 14906571298831661, 659191947156441025, 31558799717042019635, 1626968083690674214906
Offset: 0

Views

Author

Paul D. Hanna, Nov 08 2005

Keywords

Comments

A113345 is the matrix square of A113340.

Crossrefs

Cf. A113340, A113345, A113346 (column 0), A113348 (column 2), A113349 (column 3); A113350.

Programs

  • PARI
    a(n)=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^2)[n+2,2]

A113348 Column 2 of triangle A113345, also equals column 0 of A113350^5.

Original entry on oeis.org

1, 10, 105, 1315, 19875, 357860, 7547602, 183518246, 5072961513, 157525315615, 5438681986872, 206954207984234, 8613936431369952, 389602050945939891, 19038814387466399303, 1000152089409979423044, 56229083214210734799693
Offset: 0

Views

Author

Paul D. Hanna, Nov 08 2005

Keywords

Comments

A113345 is the matrix square of A113340.

Crossrefs

Cf. A113340, A113345, A113346 (column 0), A113347 (column 1), A113349 (column 3); A113350.

Programs

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

A113349 Column 3 of triangle A113345, also equals column 0 of A113350^7.

Original entry on oeis.org

1, 14, 203, 3367, 64750, 1435497, 36312626, 1036877170, 33086963196, 1169366274321, 45412092740791, 1924418011638535, 88445828358934074, 4384910640997110602, 233384463606862044134, 13278878088344760573344
Offset: 0

Views

Author

Paul D. Hanna, Nov 08 2005

Keywords

Comments

A113345 is the matrix square of A113340.

Crossrefs

Cf. A113340, A113345, A113346 (column 0), A113347 (column 1), A113348 (column 2); A113350.

Programs

  • PARI
    a(n)=local(A,B);A=matrix(1,1);A[1,1]=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^(2*j-1))[i-j+1,1]));));A=B);(A^2)[n+4,4]

A113356 Column 0 of triangle A113355, which is the matrix square of A113350.

Original entry on oeis.org

1, 4, 18, 112, 965, 10957, 156699, 2727793, 56306695, 1350043964, 36979531548, 1141573025171, 39272377323692, 1491452150268435, 62027842189908230, 2805631215820328991, 137199563717151509076, 7215932308408758314446
Offset: 0

Views

Author

Paul D. Hanna, Nov 08 2005

Keywords

Crossrefs

Cf. A113340, A113350, A113355, A113357 (column 1), A113358 (column 2), A113359 (column 3).

Programs

  • PARI
    a(n)=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]))^2)[n+1,1]

Formula

a(n) = A113346(n+1) - 1, where A113346 equals column 0 of triangle A113345 (=A113340^2).

A113357 Column 1 of triangle A113355, also equals column 0 of A113355^2.

Original entry on oeis.org

1, 8, 68, 712, 9270, 147174, 2786270, 61662544, 1568627031, 45226595865, 1460494997316, 52298603045920, 2059014449303471, 88476000281671109, 4123177399591735062, 207239886694280045429, 11179817701706220363653
Offset: 0

Views

Author

Paul D. Hanna, Nov 08 2005

Keywords

Comments

A113355 equals the matrix square of A113350, where column 1 of A113350^2 = column 0 of A113350^4.

Crossrefs

Cf. A113340, A113350, A113355, A113356 (column 0), A113358 (column 2), A113359 (column 3).

Programs

  • PARI
    a(n)=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); (matrix(#A,#A,r,c,if(r>=c,(A^(2*c))[r-c+1,1]))^2)[n+2,2]

A113358 Column 2 of triangle A113355, also equals column 0 of A113355^3.

Original entry on oeis.org

1, 12, 150, 2184, 37523, 754171, 17502145, 462930509, 13792292332, 458112945183, 16812390472566, 676432435584855, 29635374525536866, 1405425902409792025, 71770681806834337871, 3928431507732054301085, 229528875492540329214765
Offset: 0

Views

Author

Paul D. Hanna, Nov 08 2005

Keywords

Comments

A113355 equals the matrix square of A113350, where column 2 of A113350^2 = column 0 of A113350^6.

Crossrefs

Cf. A113340, A113350, A113355, A113356 (column 0), A113357 (column 1), A113359 (column 3).

Programs

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