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

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

Author

N. J. A. Sloane, Nov 17 2002

Keywords

• sign
• easy

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

• Alois P. Heinz, Table of n, a(n) for n = 0..1000
• Wikipedia, Pentomino
• Index entries for linear recurrences with constant coefficients, signature (0,0,-1,2)

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