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.

Showing 1-8 of 8 results.

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]

A113359 Column 3 of triangle A113355, also equals column 0 of A113355^4.

Original entry on oeis.org

1, 16, 264, 4912, 104476, 2531004, 69265724, 2122120824, 72160283026, 2702008172582, 110631977612048, 4922281897250776, 236665779016591350, 12236187035970192634, 677311496213007409312, 39980910968200568816168
Offset: 0

Views

Author

Paul D. Hanna, Nov 08 2005

Keywords

Comments

A113355 equals the matrix square of A113350, where column 3 of A113350^2 = column 0 of A113350^8.

Crossrefs

Cf. A113340, A113350, A113355, A113356 (column 0), A113357 (column 1), A113358 (column 2).

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); (matrix(#A,#A,r,c,if(r>=c,(A^(2*c))[r-c+1,1]))^2)[n+4,4]

A113350 Triangle Q, read by rows, such that Q^2 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^(2*k+2), where Q^2 denotes the matrix square of Q.

Original entry on oeis.org

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, 156700, 341463, 129406, 23010, 2575, 210, 14, 1, 2727794, 6496923, 2572892, 471724, 53935, 4434, 287, 16, 1
Offset: 0

Views

Author

Paul D. Hanna, Nov 08 2005

Keywords

Examples

			Triangle Q 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;
156700,341463,129406,23010,2575,210,14,1;
2727794,6496923,2572892,471724,53935,4434,287,16,1;
56306696,144856710,59525136,11198006,1305070,108593,7021,376,18,1;
Matrix square Q^2 begins:
1;
4,1;
18,8,1;
112,68,12,1;
965,712,150,16,1;
10957,9270,2184,264,20,1; ...
where Q^2 transforms column k of Q^2 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. A113351 (column 1), A113352 (column 2), A113353 (column 3), A113354 (column 4); A113355 (Q^2), A113365 (Q^3), A113340 (P), A113345 (P^2), A113360 (P^3).

Programs

  • PARI
    Q(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+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^(2*k+2)]_0, for k>=0, where
the dual triangular matrix P = A113340 is defined by
[P]_k = [P^(2*k+1)]_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.

A113394 Triangle, read by rows, equal to the matrix cube of triangle A113389.

Original entry on oeis.org

1, 9, 1, 99, 18, 1, 1569, 360, 27, 1, 34344, 9051, 783, 36, 1, 980487, 284148, 26820, 1368, 45, 1, 34930455, 10865358, 1089126, 59250, 2115, 54, 1, 1502349459, 494019714, 51784137, 2946456, 110715, 3024, 63, 1, 76058669082, 26168502684
Offset: 0

Views

Author

Paul D. Hanna, Nov 14 2005

Keywords

Examples

			Triangle A113389^3 begins:
1;
9,1;
99,18,1;
1569,360,27,1;
34344,9051,783,36,1;
980487,284148,26820,1368,45,1;
34930455,10865358,1089126,59250,2115,54,1;
1502349459,494019714,51784137,2946456,110715,3024,63,1;
76058669082,26168502684,2840586075,167137110,6510780,185589,4095,72,1;

Crossrefs

Cf. A113389, A113395 (column 0); recurrence: A091351, A113355.

Programs

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

Formula

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

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.

A113366 Column 1 of triangle A113365, also equals column 1 of A113340^4.

Original entry on oeis.org

1, 12, 138, 1830, 28805, 535004, 11568197, 287143993, 8077888153, 254672147047, 8910929460415, 343135184110984, 14435616939387951, 659276261774240232, 32504007393860850275, 1721495715845423489806, 97516667477625085469176
Offset: 0

Views

Author

Paul D. Hanna, Nov 09 2005

Keywords

Comments

A113365 equals the matrix cube of A113350, where column 1 of A113350^3 = column 1 of A113340^4.

Crossrefs

Cf. A113340, A113350, A113365 (=A113350^3), A113346 (column 0), A113367 (column 2); A113355 (=A113350^2), A113360 (=A113340^3).

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]))^3)[n+2,2]
Showing 1-8 of 8 results.

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

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