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 1-10 of 24 results. Next

A114150 Triangle, read by rows, given by the product R^2*Q^-1 = Q^3*P^-2 using triangular matrices P=A113370, Q=A113381, R=A113389.

Original entry on oeis.org

1, 4, 1, 28, 7, 1, 326, 91, 10, 1, 5702, 1722, 190, 13, 1, 136724, 43764, 4945, 325, 16, 1, 4226334, 1415799, 163705, 10751, 496, 19, 1, 161385532, 56096733, 6617605, 437723, 19896, 703, 22, 1
Offset: 0

Views

Author

Paul D. Hanna, Nov 15 2005

Keywords

Comments

Complementary to A114151, which gives R^-2*Q^3 = Q^-1*P^2.

Examples

			Triangle R^2*Q^-1 = Q^3*P^-2 begins:
1;
4,1;
28,7,1;
326,91,10,1;
5702,1722,190,13,1;
136724,43764,4945,325,16,1;
4226334,1415799,163705,10751,496,19,1; ...
Compare to P (A113370):
1;
1,1;
1,4,1;
1,28,7,1;
1,326,91,10,1;
1,5702,1722,190,13,1; ...
Thus R^2*Q^-1 = Q^3*P^-2 equals P shift left one column.

Crossrefs

Cf. A113370 (P), A113381 (Q), A113389 (R); 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); 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])); (R^2*Q^-1)[n+1,k+1]

A114151 Triangle, read by rows, given by the product R^-2*Q^3 = Q^-1*P^2 using triangular matrices P=A113370, Q=A113381, R=A113389.

Original entry on oeis.org

1, 0, 1, 0, 3, 1, 0, 15, 6, 1, 0, 136, 66, 9, 1, 0, 1998, 1091, 153, 12, 1, 0, 41973, 24891, 3621, 276, 15, 1, 0, 1166263, 737061, 110637, 8482, 435, 18, 1, 0, 40747561, 27110418, 4176549, 323874, 16430, 630, 21, 1
Offset: 0

Views

Author

Paul D. Hanna, Nov 15 2005

Keywords

Comments

Complementary to A114150, which gives R^2*Q^-1 = Q^3*P^-2.

Examples

			Triangle R^-2*Q^3 = Q^-1*P^2 begins:
1;
0,1;
0,3,1;
0,15,6,1;
0,136,66,9,1;
0,1998,1091,153,12,1;
0,41973,24891,3621,276,15,1; ...
Compare to R (A113389):
1;
3,1;
15,6,1;
136,66,9,1;
1998,1091,153,12,1;
41973,24891,3621,276,15,1; ...
Thus R^-2*Q^3 = Q^-1*P^2 equals R shift right one column.

Crossrefs

Cf. A113370 (P), A113381 (Q), A113389 (R); A114150 (R^2*Q^-1=Q^3*P^-2), A114152 (R^3*P^-1), A114153 (R^-1*P^3), A114154 (R^3*Q^-2), A114155 (Q^-2*P^3); 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^-1*P^2)[n+1,k+1]

A114154 Triangle, read by rows, given by the product R^3*Q^-2 using triangular matrices Q=A113381, R=A113389.

Original entry on oeis.org

1, 5, 1, 45, 8, 1, 635, 120, 11, 1, 12815, 2556, 231, 14, 1, 343815, 71548, 6556, 378, 17, 1, 11651427, 2508528, 233706, 13391, 561, 20, 1, 480718723, 106427700, 10069521, 579047, 23817, 780, 23, 1
Offset: 0

Views

Author

Paul D. Hanna, Nov 15 2005

Keywords

Comments

Complementary to A114155, which gives Q^-2*P^3.

Examples

			Triangle R^3*Q^-2 begins:
1;
5,1;
45,8,1;
635,120,11,1;
12815,2556,231,14,1;
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 R^3*Q^-2 equals Q shift left one column.

Crossrefs

Cf. A113394 (R^3), 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), A114155 (Q^-2*P^3); 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])); (R^3*Q^-2)[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]

A114158 Triangle, read by rows, equal to the matrix inverse of Q=A113381.

Original entry on oeis.org

1, -2, 1, 4, -5, 1, 21, -5, -8, 1, 130, 20, -32, -11, 1, 1106, 840, -260, -77, -14, 1, 10044, 24865, -2584, -1089, -140, -17, 1, -18366, 823383, -12828, -21428, -2737, -221, -20, 1, -9321125, 31847653, 1160956, -523831, -73458, -5474, -320, -23, 1
Offset: 0

Views

Author

Paul D. Hanna, Nov 15 2005

Keywords

Examples

			Triangle Q^-1 begins:
1;
-2,1;
4,-5,1;
21,-5,-8,1;
130,20,-32,-11,1;
1106,840,-260,-77,-14,1;
10044,24865,-2584,-1089,-140,-17,1;
-18366,823383,-12828,-21428,-2737,-221,-20,1; ...
Triangle Q^-2 begins:
1;
-4,1;
18,-10,1;
20,30,-16,1;
-139,255,24,-22,1;
-3945,3085,544,0,-28,1;
-99849,51015,12444,671,-42,-34,1; ...

Crossrefs

Cf. 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); A114156 (P^-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])); (Q^-1)[n+1,k+1]

A113374 Triangle, read by rows, equal to the matrix square of A113370. Also given by the product: P^2 = Q*(R^-2)*Q^3, using triangular matrices P=A113370, Q=A113381 and R=A113389.

Original entry on oeis.org

1, 2, 1, 6, 8, 1, 37, 84, 14, 1, 429, 1296, 252, 20, 1, 7629, 27850, 5957, 510, 26, 1, 185776, 784146, 179270, 16180, 858, 32, 1, 5817106, 27630378, 6641502, 623115, 34125, 1296, 38, 1, 224558216, 1177691946, 294524076, 28470525, 1599091, 61952
Offset: 0

Views

Author

Paul D. Hanna, Nov 14 2005

Keywords

Examples

			Triangle A113370^2 begins:
1;
2,1;
6,8,1;
37,84,14,1;
429,1296,252,20,1;
7629,27850,5957,510,26,1;
185776,784146,179270,16180,858,32,1;
5817106,27630378,6641502,623115,34125,1296,38,1;
224558216,1177691946,294524076,28470525,1599091,61952,1824,44,1;

Crossrefs

Cf. A113370, A113381, A113389; A113375 (column 0), A113376 (column 1), A113377 (column 2); A113378 (P^3), A113387 (Q^3).

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);(A^2)[n+1,k+1]

Formula

Column k of A113370^2 = column 0 of A113381^(3*k+1).

A113375 Column 0 of triangle A113374, also equals column 0 of A113381.

Original entry on oeis.org

1, 2, 6, 37, 429, 7629, 185776, 5817106, 224558216, 10362978307, 558458382528, 34504326965326, 2408502186081117, 187672037804601000, 16162473554575583148, 1525578320627987001344, 156704538246796929248712
Offset: 0

Views

Author

Paul D. Hanna, Nov 14 2005

Keywords

Crossrefs

Cf. A113374, A113376 (column 1), A113377 (column 2).

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^2)[n+1,1]

Formula

A113374 equals the matrix square of A113370, which has the property: column k of A113370^2 = column 0 of A113381^(3*k+1) for k>=0.

A113384 Triangle, read by rows, equal to the matrix square of A113381. Also given by: Q^2 = R*P = R*Q*(R^-2)*Q*R = P*Q*(P^-2)*Q*P, using triangular matrices P=A113370, Q=A113381 and R=A113389.

Original entry on oeis.org

1, 4, 1, 22, 10, 1, 212, 130, 16, 1, 3255, 2365, 328, 22, 1, 70777, 57695, 8640, 616, 28, 1, 2022897, 1798275, 284356, 21197, 994, 34, 1, 72375484, 68931064, 11358500, 875424, 42196, 1462, 40, 1, 3130502129, 3155772612, 537277044, 42499204
Offset: 0

Views

Author

Paul D. Hanna, Nov 14 2005

Keywords

Examples

			Triangle A113381^2 begins:
1;
4,1;
22,10,1;
212,130,16,1;
3255,2365,328,22,1;
70777,57695,8640,616,28,1;
2022897,1798275,284356,21197,994,34,1;
72375484,68931064,11358500,875424,42196,1462,40,1;
3130502129,3155772612,537277044,42499204,2094365,73797,2020,46,1;

Crossrefs

Cf. A113381, A113385 (column 0), A113386 (column 1); A113370, A113389.

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-1))[r-c+1,1]))^2)[n+1,k+1]

A113387 Triangle, read by rows, equal to the matrix cube of A113381.

Original entry on oeis.org

1, 6, 1, 48, 15, 1, 605, 255, 24, 1, 11196, 5630, 624, 33, 1, 280440, 159210, 19484, 1155, 42, 1, 8981460, 5584635, 731664, 46541, 1848, 51, 1, 353283128, 236051661, 32532732, 2173248, 91175, 2703, 60, 1, 16567072675, 11741443007, 1683566556
Offset: 0

Views

Author

Paul D. Hanna, Nov 14 2005

Keywords

Examples

			Triangle A113381^3 begins:
1;
6,1;
48,15,1;
605,255,24,1;
11196,5630,624,33,1;
280440,159210,19484,1155,42,1;
8981460,5584635,731664,46541,1848,51,1;
353283128,236051661,32532732,2173248,91175,2703,60,1;
16567072675,11741443007,1683566556,116647443,5086116,157760,3720,69,1;

Crossrefs

Cf. A113381, A113388 (column 0), A113389.

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-1))[r-c+1,1]))^3)[n+1,k+1]

Formula

Column k of A113381^3 = column 0 of A113389^(3*k+2) for k>=0.

A113392 Triangle, read by rows, equal to the matrix square of triangle A113389. Also given by the matrix product: R^2 = Q^3*(P^-2)*Q, using triangular matrices P=A113370, Q=A113381 and R=A113389.

Original entry on oeis.org

1, 6, 1, 48, 12, 1, 605, 186, 18, 1, 11196, 3892, 414, 24, 1, 280440, 106089, 12021, 732, 30, 1, 8981460, 3620379, 429345, 27152, 1140, 36, 1, 353283128, 149740555, 18386361, 1196910, 51445, 1638, 42, 1, 16567072675, 7316974618, 923656512
Offset: 0

Views

Author

Paul D. Hanna, Nov 14 2005

Keywords

Examples

			Triangle A113389^2 begins:
1;
6,1;
48,12,1;
605,186,18,1;
11196,3892,414,24,1;
280440,106089,12021,732,30,1;
8981460,3620379,429345,27152,1140,36,1;
353283128,149740555,18386361,1196910,51445,1638,42,1;
16567072675,7316974618,923656512,61515702,2696010,87060,2226,48,1;

Crossrefs

Cf. A113389, A113388 (column 0), A113393 (column 1).

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]))^2)[n+1,k+1]
Showing 1-10 of 24 results. Next

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