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 1-4 of 4 results.

A128181 A007318 * A128179 as infinite lower triangular matrices.

Original entry on oeis.org

1, 1, 2, 2, 4, 3, 4, 8, 9, 4, 8, 16, 21, 16, 5, 16, 32, 45, 44, 25, 6, 32, 64, 93, 104, 80, 36, 7, 64, 128, 189, 228, 210, 132, 49, 8, 128, 256, 381, 480, 495, 384, 203, 64, 9, 256, 512, 765, 988, 1095, 978, 651, 296, 81, 10
Offset: 1

Views

Author

Gary W. Adamson, Feb 17 2007

Keywords

Comments

Binomial transform of A128179.
Row sums = A058396 starting (1, 3, 9, 25, 66, 168, ...).

Examples

			First few rows of the triangle:
   1;
   1,  2;
   2,  4,  3;
   4,  8,  9,   4;
   8, 16, 21,  16,  5;
  16, 32, 45,  44, 25,  6;
  32, 64, 93, 104, 80, 36, 7;
  ...

Crossrefs

Cf. A128179, A097807, A002260, A058396.

A128590 Triangle read by rows, matrix product A128179 * A000012.

Original entry on oeis.org

1, 2, 2, 4, 3, 3, 6, 6, 4, 4, 9, 8, 8, 5, 5, 12, 12, 10, 10, 6, 6, 16, 15, 15, 12, 12, 7, 7, 20, 20, 18, 18, 14, 14, 8, 8, 25, 24, 24, 21, 21, 16, 16, 9, 9, 30, 30, 28, 28, 24, 24, 18, 18, 10, 10, 36, 35, 35, 32, 32, 27, 27, 20, 20, 11, 11, 42, 42, 40, 40, 36, 36, 30, 30, 22, 22, 12, 12
Offset: 1

Views

Author

Gary W. Adamson, Mar 11 2007

Keywords

Comments

Row sums give A000292, starting (1, 4, 10, 20, 35, 56, 84, ...).

Examples

			First few rows of the triangle are:
1;
2, 2;
4, 3, 3;
6, 6, 4, 4;
9, 8, 8, 5, 5;
12, 12, 10, 10, 6, 6;
16, 15, 15, 12, 12, 7, 7;
...
First few rows of the array are:
1,  2,  3,  4,  5,  6,  7,  8, ...
2,  3,  4,  5,  6,  7,  8,  9, ...
4,  6,  8, 10, 12, 14, 16, 18, ...
6,  8, 10, 12, 14, 16, 18, 20, ...
9, 12, 15, 18, 21, 24, 27, 30, ...
...
A(3, 4) = 10 because F(5, 3) = 1 + q^4 + q^5 + q^6 + q^10. A(4, 4) = 12 because F(6, 3) = 1 + q^4 + q^5 + q^6 + q^7 + q^10 + q^11 + q^12.

Crossrefs

Cf. A000292, A128179.

Programs

  • PARI
    {T(n, k) = (n - k + 2)\2 * ((n + k + 1)\2)} /* Michael Somos, Jun 08 2011 */

Formula

Regarded as an array by antidiagonals A(i, j) = degree in q of q-Fibonacci number F(i+2, j-1) where F(1, k) = F(2, k) = 1, F(n, k) = F(n-1, k) + q^(n+k-2) * F(n-2, k). - Michael Somos, Jun 08 2011

Extensions

a(19) = 10 inserted and more terms from Georg Fischer, Jun 08 2023

A128180 A002260 * A097807.

Original entry on oeis.org

1, -1, 2, 2, -1, 3, -2, 3, -1, 4, 3, -2, 4, -1, 5, -3, 4, -2, 5, -1, 6, 4, -3, 5, -2, 6, -1, 7, -4, 5, -3, 6, -2, 7, -1, 8, 5, -4, 6, -3, 7, -2, 8, -1, 9, -5, 6, -4, 7, -3, 8, -2, 9, -1, 10
Offset: 1

Views

Author

Gary W. Adamson, Feb 17 2007

Keywords

Comments

Row sums = A008794: (1, 1, 4, 4, 9, 9, 16, 16, ...).
Unsigned row sums = the triangular sequence, A000217: (1, 3, 6, 10, ...) by virtue of the fact that each row is a permutation of the natural numbers.
A128179 = A097807 * A002260.

Examples

			Triangle begins:
   1;
  -1,  2;
   2, -1,  3;
  -2,  3, -1,  4;
   3, -2,  4, -1,  5;
  -3,  4, -2,  5, -1,  6;
   4, -3,  5, -2,  6, -1,  7;
  ...

Crossrefs

Cf. A002260, A097807, A008794, A128179.

Programs

Formula

A002260 * A097807 as infinite lower triangular matrices.
From Franklin T. Adams-Watters, Apr 12 2011: (Start)
T(n,k) = (2k - 1 + (-1)^(n-k)*(2n+1))/4.
|T(n,k)| = (2n+1 + (-1)^(n-k)*(2k-1))/4. (End)

A128182 Binomial transform of A128180.

Original entry on oeis.org

1, 0, 2, 1, 3, 3, 2, 6, 8, 4, 4, 12, 18, 15, 5, 8, 24, 38, 40, 24, 6, 16, 48, 78, 93, 75, 35, 7, 32, 96, 158, 202, 194, 126, 48, 8, 64, 192, 318, 423, 453, 362, 196, 63, 9, 128, 384, 638, 868, 996, 914, 622, 288, 80, 10
Offset: 0

Views

Author

Gary W. Adamson, Feb 17 2007

Keywords

Comments

Row sums = 1, 2, 7, 20, 54, 140, 352, 864, 2080, 4928,

Examples

			First few rows of the triangle are:
1;
0, 2;
1, 3, 3;
2, 6, 8, 4;
4, 12, 18, 15, 5;
8, 24, 38, 40, 24, 6;
16, 48, 78, 93, 75, 35, 7;
...

Crossrefs

Cf. A128179, A128180.

Formula

A007318 * A128180 as infinite lower triangular matrices.
Showing 1-4 of 4 results.

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