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.

A185629 Column 0 of triangle A185628; also, equals row sums of triangle A185624.

Original entry on oeis.org

1, 3, 6, 16, 60, 305, 1988, 15951, 153171, 1722693, 22296396, 327270440, 5381716953, 98134088419, 1967105567802, 43024511648982, 1020235144731236, 26083065249549660, 715430961891290714, 20962974995245684878, 653654392800091507791, 21615444263990028093227
Offset: 0

Views

Author

Paul D. Hanna, Sep 07 2011

Keywords

Comments

Triangle A185628 equals the matrix cube of triangle R = A185620, which satisfies: R^3 - R^2 + I = R shifted left one column.

Crossrefs

Cf. A185620, A185628, A185630, A185631.

Programs

  • PARI
    {a(n)=local(A=Mat(1), B); for(m=1, n+1, B=A^3-A^2+A^0; A=matrix(m+1, m+1); for(i=1, m+1, for(j=1, i, if(i<2|j==i, A[i, j]=1, if(j==1, A[i, j]=1, A[i, j]=B[i-1, j-1]))))); return((A^3)[n+1, 1])}

A185630 Column 1 of triangle A185628.

Original entry on oeis.org

1, 3, 12, 55, 315, 2243, 19378, 198363, 2358860, 32060736, 491393909, 8398281337, 158541018923, 3279307650137, 73808041476439, 1796794875356409, 47063878147374980, 1320276655048586177, 39504980013582635300, 1256210676237601811305, 42313074100060933324771
Offset: 0

Views

Author

Paul D. Hanna, Sep 07 2011

Keywords

Comments

Triangle A185628 equals the matrix cube of triangle R = A185620, which satisfies: R^3 - R^2 + I = R shifted left one column.

Crossrefs

Cf. A185620, A185628, A185629, A185631.

Programs

  • PARI
    {a(n)=local(A=Mat(1), B); for(m=1, n+2, B=A^3-A^2+A^0; A=matrix(m+1, m+1); for(i=1, m+1, for(j=1, i, if(i<2|j==i, A[i, j]=1, if(j==1, A[i, j]=1, A[i, j]=B[i-1, j-1]))))); return((A^3)[n+2, 2])}

A185631 Row sums of triangle A185628.

Original entry on oeis.org

1, 4, 10, 32, 137, 766, 5382, 46022, 467206, 5519252, 74629263, 1139283601, 19411643752, 365595475542, 7548810986515, 169681672611815, 4126880020198630, 108026180912533227, 3029184083924476085, 90617948406514693702, 2881333799993441893564, 97057552263114611503090
Offset: 0

Views

Author

Paul D. Hanna, Sep 07 2011

Keywords

Comments

Triangle A185628 equals the matrix cube of triangle R = A185620, which satisfies: R^3 - R^2 + I = R shifted left one column.

Crossrefs

Cf. A185620, A185628, A185629, A185630.

Programs

  • PARI
    {a(n)=local(A=Mat(1), B); for(m=1, n+1, B=A^3-A^2+A^0; A=matrix(m+1, m+1); for(i=1, m+1, for(j=1, i, if(i<2||j==i, A[i, j]=1, if(j==1, A[i, j]=1, A[i, j]=B[i-1, j-1]))))); return(sum(k=1,n+1,(A^3)[n+1, k]))}

A185620 Triangular matrix T that satisfies: T^3 - T^2 + I = SHIFT_LEFT(T), as read by rows.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 10, 5, 1, 1, 1, 42, 27, 7, 1, 1, 1, 226, 173, 52, 9, 1, 1, 1, 1525, 1330, 442, 85, 11, 1, 1, 1, 12555, 12134, 4345, 897, 126, 13, 1, 1, 1, 123098, 129359, 49114, 10687, 1586, 175, 15, 1, 1, 1, 1408656, 1587501, 632104, 143335, 22156, 2557
Offset: 0

Views

Author

Paul D. Hanna, Feb 01 2011

Keywords

Examples

			Triangle T begins:
1;
1, 1;
1, 1, 1;
1, 3, 1, 1;
1, 10, 5, 1, 1;
1, 42, 27, 7, 1, 1;
1, 226, 173, 52, 9, 1, 1;
1, 1525, 1330, 442, 85, 11, 1, 1;
1, 12555, 12134, 4345, 897, 126, 13, 1, 1;
1, 123098, 129359, 49114, 10687, 1586, 175, 15, 1, 1;
1, 1408656, 1587501, 632104, 143335, 22156, 2557, 232, 17, 1, 1;
1, 18499835, 22127494, 9167575, 2149761, 343091, 40936, 3858, 297, 19, 1, 1; ...
Matrix square T^2 begins:
1;
2, 1;
3, 2, 1;
6, 7, 2, 1;
18, 28, 11, 2, 1;
79, 142, 66, 15, 2, 1;
463, 913, 470, 120, 19, 2, 1;
3396, 7244, 3997, 1098, 190, 23, 2, 1;
...
Matrix cube T^3 begins:
1;
3, 1;
6, 3, 1;
16, 12, 3, 1;
60, 55, 18, 3, 1;
305, 315, 118, 24, 3, 1;
1988, 2243, 912, 205, 30, 3, 1;
15951, 19378, 8342, 1995, 316, 36, 3, 1;
...
Thus T^3 - T^2 + I begins:
1;
1, 1;
3, 1, 1;
10, 5, 1, 1;
42, 27, 7, 1, 1;
226, 173, 52, 9, 1, 1;
1525, 1330, 442, 85, 11, 1, 1;
12555, 12134, 4345, 897, 126, 13, 1, 1;
...
which equals T shifted left one column.
...
ALTERNATE GENERATING FORMULA.
Let U equal T shifted up one diagonal:
1;
1, 1;
1, 3, 1;
1, 10, 5, 1;
1, 42, 27, 7, 1;
1, 226, 173, 52, 9, 1;
1, 1525, 1330, 442, 85, 11, 1;
1, 12555, 12134, 4345, 897, 126, 13, 1;
...
then U*T^2 begins:
1;
3, 1;
10, 5, 1;
42, 27, 7, 1;
226, 173, 52, 9, 1;
1525, 1330, 442, 85, 11, 1;
12555, 12134, 4345, 897, 126, 13, 1;
...
which equals U shifted left one column.

Crossrefs

Cf. columns: A185621, A185622, A185623; A185624 (T^2), A185628 (T^3).
Cf. variants: A104445, A185641.

Programs

  • PARI
    {T(n, k)=local(A=Mat(1), B); for(m=1, n, B=A^3-A^2+A^0;
A=matrix(m+1, m+1); for(i=1, m+1, for(j=1, i, if(i<2|j==i, A[i, j]=1,
if(j==1, A[i, j]=1, A[i, j]=B[i-1, j-1]))))); return(A[n+1, k+1])}

Formula

Recurrence: T(n+1,k+1) = [T^3](n,k) - [T^2](n,k) + [T^0](n,k) for n>=k>=0, with T(n,0)=1 for n>=0.
Let U equal T shifted up one diagonal; then U*T^2 equals U shifted left one column.

A185624 Triangle, read by rows, equal to the matrix square of triangle A185620.

Original entry on oeis.org

1, 2, 1, 3, 2, 1, 6, 7, 2, 1, 18, 28, 11, 2, 1, 79, 142, 66, 15, 2, 1, 463, 913, 470, 120, 19, 2, 1, 3396, 7244, 3997, 1098, 190, 23, 2, 1, 30073, 69004, 40079, 11587, 2122, 276, 27, 2, 1, 314037, 771359, 466448, 140092, 26707, 3638, 378, 31, 2, 1, 3796561, 9933242, 6208551, 1921122
Offset: 0

Views

Author

Paul D. Hanna, Sep 07 2011

Keywords

Examples

			 Triangle begins:
1;
2, 1;
3, 2, 1;
6, 7, 2, 1;
18, 28, 11, 2, 1;
79, 142, 66, 15, 2, 1;
463, 913, 470, 120, 19, 2, 1;
3396, 7244, 3997, 1098, 190, 23, 2, 1;
30073, 69004, 40079, 11587, 2122, 276, 27, 2, 1;
314037, 771359, 466448, 140092, 26707, 3638, 378, 31, 2, 1;
3796561, 9933242, 6208551, 1921122, 377495, 53149, 5742, 496, 35, 2, 1; ...
This triangle equals the matrix square, R^2, of triangle R = A185620, which begins:
1;
1, 1;
1, 1, 1;
1, 3, 1, 1;
1, 10, 5, 1, 1;
1, 42, 27, 7, 1, 1;
1, 226, 173, 52, 9, 1, 1;
1, 1525, 1330, 442, 85, 11, 1, 1;
1, 12555, 12134, 4345, 897, 126, 13, 1, 1; ...
where R^3 - R^2 + I equals R shifted left one column.

Crossrefs

Cf. A185620, A185625, A185626, A185627, A185628.

A185625 Column 0 of triangle A185624; also, equals row sums of triangle A185620.

Original entry on oeis.org

1, 2, 3, 6, 18, 79, 463, 3396, 30073, 314037, 3796561, 52332869, 812013163, 14029926801, 267461387125, 5581705418715, 126657160962835, 3106644875467396, 81943115662983569, 2313753122033185373, 69654973411269008826, 2227679407193350385775
Offset: 0

Views

Author

Paul D. Hanna, Sep 07 2011

Keywords

Comments

Triangle A185624 equals the matrix square of triangle R = A185620, which satisfies: R^3 - R^2 + I = R shifted left one column.

Crossrefs

Cf. A185620, A185624, A185626, A185627, A185628.

A185626 Column 1 of triangle A185624.

Original entry on oeis.org

1, 2, 7, 28, 142, 913, 7244, 69004, 771359, 9933242, 145148660, 2376879719, 43167728519, 861914166441, 18779307089740, 443624137266205, 11299168270825627, 308780443524256478, 9014690419440269085, 280081388252523565158, 9229143937723290061482
Offset: 0

Views

Author

Paul D. Hanna, Sep 07 2011

Keywords

Comments

Triangle A185624 equals the matrix square of triangle R = A185620, which satisfies: R^3 - R^2 + I = R shifted left one column.

Crossrefs

Cf. A185620, A185624, A185625, A185627, A185628.

A185627 Column 2 of triangle A185624.

Original entry on oeis.org

1, 2, 11, 66, 470, 3997, 40079, 466448, 6208551, 93297013, 1565360082, 29047075343, 591275224311, 13110484254998, 314726620750589, 8135993235767295, 225426087495479680, 6666553071325468948, 209645759506867904384, 6987310422457948266944, 246075579251657940919613
Offset: 0

Views

Author

Paul D. Hanna, Sep 07 2011

Keywords

Comments

Triangle A185624 equals the matrix square of triangle R = A185620, which satisfies: R^3 - R^2 + I = R shifted left one column.

Crossrefs

Cf. A185620, A185624, A185625, A185626, A185628.
Showing 1-8 of 8 results.

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