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 1-3 of 3 results.

A135880 Triangle P, read by rows, where column k of P^2 equals column 0 of P^(2k+2) such that column 0 of P^2 equals column 0 of P shift one place left, with P(0,0)=1.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 6, 7, 3, 1, 25, 34, 15, 4, 1, 138, 215, 99, 26, 5, 1, 970, 1698, 814, 216, 40, 6, 1, 8390, 16220, 8057, 2171, 400, 57, 7, 1, 86796, 182714, 93627, 25628, 4740, 666, 77, 8, 1, 1049546, 2378780, 1252752, 348050, 64805, 9080, 1029, 100, 9, 1, 14563135
Offset: 0

Views

Author

Paul D. Hanna, Dec 15 2007

Keywords

Comments

Amazingly, column 0 (A135881) also equals column 0 of tables A135878 and A135879, both of which have unusual recurrences seemingly unrelated to this triangle.

Examples

			Triangle P begins:
1;
1, 1;
2, 2, 1;
6, 7, 3, 1;
25, 34, 15, 4, 1;
138, 215, 99, 26, 5, 1;
970, 1698, 814, 216, 40, 6, 1;
8390, 16220, 8057, 2171, 400, 57, 7, 1;
86796, 182714, 93627, 25628, 4740, 666, 77, 8, 1;
1049546, 2378780, 1252752, 348050, 64805, 9080, 1029, 100, 9,
1;
14563135, 35219202, 19003467, 5352788, 1004176, 140908, 15855,
1504, 126, 10, 1;
where column k of P equals column 0 of R^(k+1) where R =
A135894.
Triangle Q = P^2 = A135885 begins:
1;
2, 1;
6, 4, 1;
25, 20, 6, 1;
138, 126, 42, 8, 1;
970, 980, 351, 72, 10, 1;
8390, 9186, 3470, 748, 110, 12, 1;
86796, 101492, 39968, 8936, 1365, 156, 14, 1;
1049546, 1296934, 528306, 121532, 19090, 2250, 210, 16, 1; ...
where column k of Q equals column 0 of Q^(k+1) for k>=0;
thus column k of P^2 equals column 0 of P^(2k+2).
Triangle R = A135894 begins:
1;
1, 1;
2, 3, 1;
6, 12, 5, 1;
25, 63, 30, 7, 1;
138, 421, 220, 56, 9, 1;
970, 3472, 1945, 525, 90, 11, 1;
8390, 34380, 20340, 5733, 1026, 132, 13, 1;
86796, 399463, 247066, 72030, 13305, 1771, 182, 15, 1; ...
where column k of R equals column 0 of P^(2k+1) for k>=0.
Surprisingly, column 0 is also found in triangle A135879:
1;
1,1;
2,2,1,1;
6,6,4,4,2,2,1;
25,25,19,19,13,13,9,5,5,3,1,1;
138,138,113,113,88,88,69,50,50,37,24,24,15,10,5,5,2,1; ...
and is generated by a process that seems completely unrelated.

Crossrefs

Cf. columns: A135881, A135882, A135883, A135884.
Cf. related tables: A135885 (Q=P^2), A135894 (R).
Cf. A135888 (P^3), A135891 (P^4), A135892 (P^5), A135893 (P^6).
Cf. A135898 (P^-1*R), A135899 (P*R^-1*P), A135900 (R^-1*Q).
Cf. A135878, A135879.

Programs

  • PARI
    {T(n,k)=local(P=Mat(1),R,PShR);if(n>0,for(n=0,n, PShR=matrix(#P,#P, r,c, if(r>=c,if(r==c,1,if(c==1,0,P[r-1,c-1]))));R=P*PShR; R=matrix(#P+1, #P+1, r,c, if(r>=c, if(r<#P+1,R[r,c], if(c==1,(P^2)[ #P,1],(P^(2*c-1))[r-c+1,1])))); P=matrix(#R, #R, r,c, if(r>=c, if(r<#R,P[r,c], (R^c)[r-c+1,1])))));P[n+1,k+1]}

Formula

Denote this triangle by P and define as follows.
Let [P^m]_k denote column k of matrix power P^m,
so that triangular matrix Q = A135885 may be defined by
[Q]_k = [P^(2k+2)]_0, for k>=0, such that
(1) Q = P^2 and (2) [Q]_0 = [P]_0 shifted left.
Define the dual triangular matrix R = A135894 by
[R]_k = [P^(2k+1)]_0, for k>=0.
Then columns of P may be formed from powers of R:
[P]_k = [R^(k+1)]_0, for k>=0.
Further, columns of powers of P, Q and R satisfy:
[R^(j+1)]_k = [P^(2k+1)]_j,
[Q^(j+1)]_k = [P^(2k+2)]_j,
[Q^(j+1)]_k = [Q^(k+1)]_j,
[P^(2j+2)]_k = [P^(2k+2)]_j, for all j>=0, k>=0.
Also, we have the column transformations:
R * [P]k = [P]{k+1},
Q * [Q]k = [Q]{k+1},
Q * [R]k = [R]{k+1},
P^2 * [Q]k = [Q]{k+1},
P^2 * [R]k = [R]{k+1}, for all k>=0.
Other identities include the matrix products:
P^-1*R (A135898) = P shifted right one column;
P*R^-1*P (A135899) = Q shifted down one row;
R^-1*Q (A135900) = R shifted down one row.

A135885 Triangle Q, read by rows, where column k of Q equals column 0 of Q^(k+1) and Q is equal to the matrix square of integer triangle P = A135880 such that column 0 of Q equals column 0 of P shift left.

Original entry on oeis.org

1, 2, 1, 6, 4, 1, 25, 20, 6, 1, 138, 126, 42, 8, 1, 970, 980, 351, 72, 10, 1, 8390, 9186, 3470, 748, 110, 12, 1, 86796, 101492, 39968, 8936, 1365, 156, 14, 1, 1049546, 1296934, 528306, 121532, 19090, 2250, 210, 16, 1, 14563135, 18868652, 7906598
Offset: 0

Views

Author

Paul D. Hanna, Dec 15 2007

Keywords

Examples

			Triangle Q = P^2 begins:
1;
2, 1;
6, 4, 1;
25, 20, 6, 1;
138, 126, 42, 8, 1;
970, 980, 351, 72, 10, 1;
8390, 9186, 3470, 748, 110, 12, 1;
86796, 101492, 39968, 8936, 1365, 156, 14, 1;
1049546, 1296934, 528306, 121532, 19090, 2250, 210, 16, 1;
14563135, 18868652, 7906598, 1861416, 298830, 36028, 3451, 272, 18, 1;
228448504, 308478492, 132426050, 31785380, 5193982, 637390, 62230, 5016, 342, 20, 1; ...
where column k of Q equals column 0 of Q^(k+1) for k>=0.
Related triangle P = A135880 begins:
1;
1, 1;
2, 2, 1;
6, 7, 3, 1;
25, 34, 15, 4, 1;
138, 215, 99, 26, 5, 1;
970, 1698, 814, 216, 40, 6, 1; ...
where column k of Q equals column 0 of P^(2k+2)
such that column 0 of P^2 equals column 0 of P shift left.
The matrix product P*R^-1*P = A135899 = Q (shifted down one row),
where R = A135894 begins:
1;
1, 1;
2, 3, 1;
6, 12, 5, 1;
25, 63, 30, 7, 1;
138, 421, 220, 56, 9, 1;
970, 3472, 1945, 525, 90, 11, 1; ...
in which column k of R equals column 0 of P^(2k+1).

Crossrefs

Cf. columns: A135881, A135886, A135887; related tables: A135880 (P), A135894 (R), A135891 (Q^2), A135893 (Q^3); A135898 (P^-1*R), A135899 (P*R^-1*P), A135900 (R^-1*Q).

Programs

  • PARI
    {T(n,k)=local(P=Mat(1),R,PShR);if(n>0,for(i=0,n, PShR=matrix(#P,#P, r,c, if(r>=c,if(r==c,1,if(c==1,0,P[r-1,c-1]))));R=P*PShR; R=matrix(#P+1, #P+1, r,c, if(r>=c, if(r<#P+1,R[r,c], if(c==1,(P^2)[ #P,1],(P^(2*c-1))[r-c+1,1])))); P=matrix(#R, #R, r,c, if(r>=c, if(r<#R,P[r,c], (R^c)[r-c+1,1])))));(P^2)[n+1,k+1]}

Formula

See formulas relating triangles P, Q and R, in entry A135880.

A135893 Triangle, read by rows, equal to P^6, where triangle P = A135880; also equals Q^3 where Q = P^2 = A135885.

Original entry on oeis.org

1, 6, 1, 42, 12, 1, 351, 132, 18, 1, 3470, 1554, 270, 24, 1, 39968, 20260, 4089, 456, 30, 1, 528306, 294218, 65874, 8436, 690, 36, 1, 7906598, 4745522, 1147662, 161576, 15075, 972, 42, 1, 132426050, 84534154, 21710680, 3277148, 334390, 24486, 1302
Offset: 0

Views

Author

Paul D. Hanna, Dec 15 2007

Keywords

Comments

Triangle P = A135880 is defined by: column k of P^2 equals column 0 of P^(2k+2) such that column 0 of P^2 equals column 0 of P shift left.

Examples

			Triangle P^6 = Q^3 begins:
1;
6, 1;
42, 12, 1;
351, 132, 18, 1;
3470, 1554, 270, 24, 1;
39968, 20260, 4089, 456, 30, 1;
528306, 294218, 65874, 8436, 690, 36, 1;
7906598, 4745522, 1147662, 161576, 15075, 972, 42, 1;
132426050, 84534154, 21710680, 3277148, 334390, 24486, 1302, 48, 1;
2457643895, 1652665714, 445574768, 70977244, 7732100, 617100, 37149, 1680, 54, 1;
where P = A135880 begins:
1;
1, 1;
2, 2, 1;
6, 7, 3, 1;
25, 34, 15, 4, 1;
138, 215, 99, 26, 5, 1;
970, 1698, 814, 216, 40, 6, 1; ...
and Q = P^2 = A135885 begins:
1;
2, 1;
6, 4, 1;
25, 20, 6, 1;
138, 126, 42, 8, 1;
970, 980, 351, 72, 10, 1;
8390, 9186, 3470, 748, 110, 12, 1; ...
where column k of Q = column 0 of Q^(k+1).

Crossrefs

Cf. A135887 (column 0); A135880 (P), A135885 (Q=P^2), A135891 (Q^2).

Programs

  • PARI
    {T(n,k)=local(P=Mat(1),R,PShR);if(n>0,for(i=0,n, PShR=matrix(#P,#P, r,c, if(r>=c,if(r==c,1,if(c==1,0,P[r-1,c-1]))));R=P*PShR; R=matrix(#P+1, #P+1, r,c, if(r>=c, if(r<#P+1,R[r,c], if(c==1,(P^2)[ #P,1],(P^(2*c-1))[r-c+1,1])))); P=matrix(#R, #R, r,c, if(r>=c, if(r<#R,P[r,c], (R^c)[r-c+1,1])))));(P^6)[n+1,k+1]}

Formula

Column k of Q^3 = column 2 of Q^(k+1) for k>=0 where triangle Q = P^2 = A135885; column 0 of Q^3 = column 2 of Q; column 1 of Q^3 = column 2 of Q^2.
Showing 1-3 of 3 results.

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

Content is available under The OEIS End-User License Agreement.