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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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.

A113379 Column 0 of triangle A113378, also equals column 0 of A113389.

Original entry on oeis.org

1, 3, 15, 136, 1998, 41973, 1166263, 40747561, 1726907675, 86421647389, 5002021986418, 329382745551946, 24351172588548270, 1999205882982496161, 180613538916429940159, 17817366508243503227269
Offset: 0

Views

Author

Paul D. Hanna, Nov 14 2005

Keywords

Crossrefs

Cf. A113378, A113370, A113389, A113380 (column 1).

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^3)[n+1,1]

Formula

A113378 equals the matrix cube of A113370, which has the property: column k of A113370^3 = column 0 of A113389^(3*k+1) for k>=0.

A113388 Column 0 of triangle A113387, also equals column 0 of A113389^2.

Original entry on oeis.org

1, 6, 48, 605, 11196, 280440, 8981460, 353283128, 16567072675, 905357065354, 56632746126107, 3997082539456084, 314584709388906568, 27340439653453247728, 2602372304420672868499, 269388182085308601450047
Offset: 0

Views

Author

Paul D. Hanna, Nov 14 2005

Keywords

Crossrefs

Cf. A113381, A113387, A113389.

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

Formula

A113387 equals the matrix cube of A113381, which has the property: column k of A113381^3 = column 0 of A113389^(3*k+2) for k>=0.

A113380 Column 1 of triangle A113378, also equals column 0 of A113389^4.

Original entry on oeis.org

1, 12, 168, 3190, 80136, 2553162, 99579994, 4624245724, 250138459808, 15488221792442, 1082305443525010, 84364431201000877, 7264439969560330768, 685338322012632405151, 70341947440289270101707
Offset: 0

Views

Author

Paul D. Hanna, Nov 14 2005

Keywords

Crossrefs

Cf. A113378, A113370, A113389, A113379 (column 0).

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^3)[n+2,2]

Formula

A113378 equals the matrix cube of A113370, which has the property: column k of A113370^3 = column 0 of A113389^(3*k+1) 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.

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.

A114156 Triangle, read by rows, equal to the matrix inverse of P=A113370.

Original entry on oeis.org

1, -1, 1, 3, -4, 1, 6, 0, -7, 1, -8, 38, -21, -10, 1, -501, 692, -119, -60, -13, 1, -13623, 14910, -420, -735, -117, -16, 1, -409953, 401802, 22911, -12470, -2080, -192, -19, 1, -14544683, 13278520, 1577527, -255570, -51064, -4424, -285, -22, 1
Offset: 0

Views

Author

Paul D. Hanna, Nov 15 2005

Keywords

Examples

			Triangle P^-1 begins:
1;
-1,1;
3,-4,1;
6,0,-7,1;
-8,38,-21,-10,1;
-501,692,-119,-60,-13,1;
-13623,14910,-420,-735,-117,-16,1;
-409953,401802,22911,-12470,-2080,-192,-19,1; ...
Triangle P^-2 begins:
1;
-2,1;
10,-8,1;
-9,28,-14,1;
-177,160,28,-20,1;
-2307,1366,455,10,-26,1;
-38874,15982,8666,660,-26,-32,1; ...

Crossrefs

Cf. A114157 (column 0), A113370 (P), A113381 (Q), A113389 (R); A114150 (R^2*Q^-1=Q^3*P^-2), A114151 (R^-2*Q^3=Q^-1*P^2), A114152 (R^3*P^-1), A114153 (R^-1*P^3), A114154 (R^3*Q^-2), A114155 (Q^-2*P^3); A114158 (Q^-1), A114159 (R^-1).

Programs

  • PARI
    T(n,k)=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,k+1]

A114155 Triangle, read by rows, given by the product Q^-2*P^3 using triangular matrices P=A113370, Q=A113381.

Original entry on oeis.org

1, -1, 1, 3, 2, 1, 6, 6, 5, 1, -8, 37, 45, 8, 1, -501, 429, 635, 120, 11, 1, -13623, 7629, 12815, 2556, 231, 14, 1, -409953, 185776, 343815, 71548, 6556, 378, 17, 1, -14544683, 5817106, 11651427, 2508528, 233706, 13391, 561, 20, 1
Offset: 0

Views

Author

Paul D. Hanna, Nov 15 2005

Keywords

Comments

Complementary to A114154, which gives R^3*Q^-2. Column 0 equals column 0 of P^-1 (A114157).

Examples

			Triangle Q^-2*P^3 begins:
1;
-1,1;
3,2,1;
6,6,5,1;
-8,37,45,8,1;
-501,429,635,120,11,1;
-13623,7629,12815,2556,231,14,1;
-409953,185776,343815,71548,6556,378,17,1; ...
Compare to Q (A113381):
1;
2,1;
6,5,1;
37,45,8,1;
429,635,120,11,1;
7629,12815,2556,231,14,1;...
Thus Q^-2*P^3 shift left one column equals Q.

Crossrefs

Cf. A114157 (column 0), A113370 (P), A113381 (Q), A113389 (R); A114150 (R^2*Q^-1=Q^3*P^-2), A114151 (R^-2*Q^3=Q^-1*P^2), A114152 (R^3*P^-1), A114153 (R^-1*P^3), A114154 (R^3*Q^-2); A114156 (P^-1), A114158 (Q^-1), A114159 (R^-1).

Programs

  • PARI
    T(n,k)=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); Q=matrix(#P,#P,r,c,if(r>=c,(P^(3*c-1))[r-c+1,1])); R=matrix(#P,#P,r,c,if(r>=c,(P^(3*c))[r-c+1,1])); (Q^-2*P^3)[n+1,k+1]
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