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.

A108484 a(n) = Sum_{k=0..floor(n/2)} binomial(2n-2k,2k) * 3^(n-k).

Original entry on oeis.org

1, 1, 4, 19, 55, 220, 793, 2845, 10480, 37963, 138259, 503608, 1831969, 6669865, 24276892, 88362451, 321640831, 1170726484, 4261339801, 15510894949, 56458080328, 205502135851, 748007984827, 2722677076336, 9910284168961
Offset: 0

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Author

Paul Barry, Jun 04 2005

Keywords

Comments

In general, Sum_{k=0..floor(n/2)} C(2n-2k,2k)a^k*b^(n-k) has expansion (1-bx-abx^2)/(1-2bx-(2ab-b^2)x^2-2ab^2*x^3+(ab)^2*x^4).

Links

Crossrefs

Cf. A108485, A108486, A108487.

Formula

G.f.: (1-x-3x^2)/(1-2x-5x^2-6x^3+9x^4).
a(n) = 2a(n-1)+5a(n-2)+6a(n-3)-9a(n-4).

A375276 Expansion of 1/sqrt(1 - 4*x - 8*x^3 + 4*x^4).

Original entry on oeis.org

1, 2, 6, 24, 92, 360, 1448, 5888, 24144, 99744, 414432, 1729920, 7249088, 30476416, 128487552, 543014912, 2299764992, 9758138880, 41473582592, 176530905088, 752401603584, 3210723420160, 13716154361856, 58653842276352, 251049168687104, 1075442759868416
Offset: 0

Views

Author

Seiichi Manyama, Aug 09 2024

Keywords

Crossrefs

Cf. A108485.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(1/sqrt(1-4*x-8*x^3+4*x^4))
  • PARI
    a(n) = sum(k=0, n\2, 2^(n-k)*binomial(n-k, k)^2);

Formula

n * a(n) = 2*(2*n-1)*a(n-1) + 4*(2*n-3)*a(n-3) - 4*(n-2)*a(n-4).
a(n) = Sum_{k=0..floor(n/2)} 2^(n-k) * binomial(n-k,k)^2.

A374512 Number of ways to tile a 3 X n board with 2 X 2 and 3 X 3 staircase tiles.

Original entry on oeis.org

1, 0, 2, 4, 6, 16, 32, 64, 140, 288, 600, 1264, 2632, 5504, 11520, 24064, 50320, 105216, 219936, 459840, 961376, 2009856, 4201984, 8784896, 18366144, 38397440, 80275840, 167829248, 350873728, 733556736, 1533616128, 3206266880, 6703206656, 14014111744
Offset: 0

Views

Author

Greg Dresden and Shaolun Han, Jul 09 2024

Keywords

Comments

Here are the 2 X 2 and 3 X 3 staircase tiles, both of which can be rotated as desired:
_
| |
| | | |
|_| |___|.
This is a natural generalization of A127864, which counts the number of ways to tile a 2 X n board with 1 X 1 and 2 X 2 staircase tiles.

Examples

			Here is one of the a(6)=32 ways to tile the 3 X 6 board:
   ___________
  | |_  |    _|
  |   |_|  _| |
  |_____|_|___|.

Links

Crossrefs

Cf. A002605, A127864, A108485.

Programs

  • Mathematica
    LinearRecurrence[{0, 2, 4, 2}, {1, 0, 2, 4}, 50]

Formula

a(n) = 2*a(n-2) + 4*a(n-3) + 2*a(n-4).
a(2*n) = A108485(n).
a(2*n+3) = 4*Sum_{k=0..n} a(2*k)*A002605(n+1-k).
G.f.: 1/(1 - 2*x^2 - 4*x^3 - 2*x^4).
Showing 1-3 of 3 results.

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

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