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 1-9 of 9 results.

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]

A113361 Column 1 of triangle A113360, which equals the matrix cube of triangle A113340.

Original entry on oeis.org

1, 9, 81, 879, 11739, 190044, 3654814, 81947221, 2107962168, 61366149296, 1998607800064, 72112467306074, 2858551691428042, 123596917897265255, 5792708223233376744, 292682081981049699408, 15865848522184194142469
Offset: 0

Views

Author

Paul D. Hanna, Nov 08 2005

Keywords

Crossrefs

Cf. A113340, A113350, A113360 (A113340^3), A113341 (column 0), A113362 (column 2), A113363 (column 3), A113364 (column 4).

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^3)[n+2,2]

A113362 Column 2 of triangle A113360, also equals column 1 of A113340^5.

Original entry on oeis.org

1, 15, 210, 3285, 59395, 1241270, 29720808, 806720492, 24568601477, 831697990069, 31036137984664, 1267376997249262, 56270606942915489, 2701018385881136958, 139463341982980040911, 7711492696761363573725
Offset: 0

Views

Author

Paul D. Hanna, Nov 08 2005

Keywords

Comments

A113360 equals the matrix cube of triangle A113340, where column 2 of A113340^3 = column 1 of A113340^5.

Crossrefs

Cf. A113340, A113350, A113360 (A113340^3), A113341 (column 0), A113361 (column 1), A113363 (column 3), A113364 (column 4).

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^3)[n+3,3]

A113363 Column 3 of triangle A113360, also equals column 1 of A113340^7.

Original entry on oeis.org

1, 21, 399, 8127, 184436, 4695719, 133730310, 4234560596, 148077895854, 5680484146379, 237574859841676, 10771591113205720, 526750324271281348, 27655229128194306702, 1552379671658643163707, 92820542631741826797326
Offset: 0

Views

Author

Paul D. Hanna, Nov 08 2005

Keywords

Comments

A113360 equals the matrix cube of triangle A113340, where column 3 of A113340^3 = column 1 of A113340^7.

Crossrefs

Cf. A113340, A113350, A113360 (A113340^3), A113341 (column 0), A113361 (column 1), A113362 (column 2), A113364 (column 4).

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^3)[n+4,4]

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).

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.

A113341 Column 1 of triangle A113340, also equals column 0 of A113340^3.

Original entry on oeis.org

1, 3, 12, 69, 560, 6059, 83215, 1399161, 28020221, 654110586, 17494347067, 528556017365, 17830841841940, 665126088764191, 27208111182653865, 1211942062741823574, 58424831462907214924, 3031993693950136247986
Offset: 0

Views

Author

Paul D. Hanna, Nov 08 2005

Keywords

Crossrefs

Cf. A113342 (column 2), A113343 (column 3), A113344 (column 4), A113340, 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);A[n+2,2]

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 1-9 of 9 results.

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

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