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.

A234931 Number of tilings of a 5 X n rectangle using n pentominoes of shapes F, U, N.

Original entry on oeis.org

1, 0, 0, 0, 2, 0, 0, 0, 4, 0, 8, 0, 16, 0, 40, 0, 64, 16, 200, 96, 504, 464, 1528, 1664, 4376, 5616, 12792, 18192, 38264, 58384, 115832, 186368, 355808, 589344, 1095408, 1853664, 3383656, 5802016, 10470376, 18125280, 32461312, 56552736, 100782696, 176318464
Offset: 0

Views

Author

Alois P. Heinz, Jan 01 2014

Keywords

Examples

			a(4) = 2:
._______.   ._______.
| | ._. |   | ._. | |
| |_| |_|   |_| |_| |
|_. |_. |   | ._| ._|
| |_| | |   | | |_| |
|_____|_|   |_|_____|.

Links

Crossrefs

Cf. A174249, A233427, A234312, A247680, A247126.

Formula

G.f.: (4*x^20 +4*x^18 +8*x^16 -3*x^14 +4*x^13 -5*x^12 -2*x^11 +3*x^10 -2*x^9 +6*x^8 -2*x^7 +2*x^6 -2*x^5 -x^4 +2*x -1) / (-8*x^22 -28*x^20 -6*x^18 +8*x^17 +26*x^16 +4*x^15 +7*x^14 -8*x^13 -9*x^12 -14*x^11 +7*x^10 +2*x^9 +8*x^8 -2*x^7 +2*x^6 -6*x^5 +x^4 +2*x -1).

A077909 Expansion of 1/((1-x)*(1+x+x^2+2*x^3)).

Original entry on oeis.org

1, 0, 0, -1, 2, 0, 1, -4, 4, -1, 6, -12, 9, -8, 24, -33, 26, -40, 81, -92, 92, -161, 254, -276, 345, -576, 784, -897, 1266, -1936, 2465, -3060, 4468, -6337, 7990, -10588, 15273, -20664, 26568, -36449, 51210, -67896, 89585, -124108, 170316, -225377, 303278, -418532, 566009, -754032, 1025088
Offset: 0

Views

Author

N. J. A. Sloane, Nov 17 2002

Keywords

Comments

The absolute value of a(n) is the number of tilings of a 5 X n rectangle using n pentominoes of shapes N, U, X. |a(3)| = 1, |a(4)| = 2:
.___. ._____. ._____.
| .. | | .. | | | | ._. |
|| || || || | | || ||
|. .| , | .| .| |. |. |
| || | | | || | | |_| | |
|___| ||____| |___|_|. - Alois P. Heinz, Jan 03 2014

Links

Crossrefs

Partial sums of A077976.
Cf. A174249, A233427, A234312, A234931, A247126.

Programs

  • Maple
    a:= n-> (<<0|1|0|0>, <0|0|1|0>, <0|0|0|1>, <2|-1|0|0>>^n.
        <<1, 0, 0, -1>>)[1, 1]:
seq(a(n), n=0..60);  # Alois P. Heinz, Nov 20 2013
  • Mathematica
    CoefficientList[1/(1+x^3-2*x^4) + O[x]^60, x] (* Jean-François Alcover, Jun 08 2015, after Arkadiusz Wesolowski *)
  • PARI
    Vec( 1/((1-x)*(1+x+x^2+2*x^3)) +O(x^66)) \\ Joerg Arndt, Aug 28 2013

Formula

a(n) = (-1)^n*sum(A128099(n-2*k, n-3*k), k=0..floor(n/3)). - Johannes W. Meijer, Aug 28 2013
G.f.: 1/(1 + x^3 - 2*x^4). - Arkadiusz Wesolowski, Nov 20 2013

A247127 Number of tilings of a 5 X n rectangle using n pentominoes of shapes V, U, X, N.

Original entry on oeis.org

1, 0, 0, 1, 4, 0, 9, 8, 24, 17, 78, 64, 227, 212, 664, 699, 2004, 2220, 6033, 7196, 18112, 22859, 54882, 72560, 166251, 229284, 505632, 721421, 1540532, 2264668, 4702135, 7092742, 14376450, 22165709, 44024116, 69154334, 134973515, 215459398, 414268932
Offset: 0

Views

Author

Alois P. Heinz, Nov 19 2014

Keywords

Links

Crossrefs

Cf. A174249, A233427, A247126.

Programs

  • Maple
    gf:= -(4*x^18 +4*x^17 -8*x^16 -3*x^15 -9*x^14 +2*x^13 -3*x^12 +5*x^11 -7*x^10 +x^9 -7*x^8 -x^6 -2*x^5 -x^3+1) / (32*x^26 +32*x^25 -32*x^24 +8*x^23 -120*x^22 +12*x^21 -124*x^20 +36*x^19 -123*x^18 +35*x^17 -106*x^16 +20*x^15 -62*x^14 -23*x^13 -22*x^12 -36*x^11 +5*x^10 -18*x^9 +13*x^8 -4*x^7 +8*x^6 +2*x^5 +4*x^4 +2*x^3-1):
a:= n-> coeff(series(gf, x, n+1), x, n):
seq(a(n), n=0..50);

Formula

G.f.: see Maple program.
Showing 1-3 of 3 results.

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

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