A234312 Number of tilings of a 5 X n rectangle using n pentominoes of shapes L, X.

1, 0, 2, 0, 4, 2, 8, 8, 16, 24, 36, 64, 88, 160, 224, 392, 576, 960, 1472, 2368, 3728, 5888, 9376, 14720, 23488, 36896, 58752, 92544, 146944, 232064, 367680, 581632, 920448, 1457152, 2305024, 3649664, 5773312, 9140224, 14460928, 22890496, 36221184, 57327616

Offset: 0

Author

Alois P. Heinz, Dec 23 2013

Keywords

• nonn

Examples

			a(4) = 4:
._______.  ._______.  ._______.  ._______.
|_. |_. |  | ._| ._|  |_. | ._|  | ._|_. |
| | | | |  | | | | |  | | | | |  | | | | |
| | | | |  | | | | |  | | | | |  | | | | |
| |_| |_|  |_| |_| |  | |_|_| |  |_| | |_|
|___|___|  |___|___|  |___|___|  |___|___|.
a(5) = 2:
._________.  ._________.
| | ._____|  |_____. | |
| |_| |_. |  | ._| |_| |
| |_. ._| |  | |_. ._| |
|___|_| | |  | | |_|___|
|_______|_|  |_|_______|.
		

Links

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

Crossrefs

Cf. A077909, A174249, A233427, A234931, A247125, A264812.

Programs

Maple

a:= n-> (<<0|1|0|0|0>, <0|0|1|0|0>, <0|0|0|1|0>,
          <0|0|0|0|1>, <2|0|0|2|0>>^n)[5, 5]:
seq(a(n), n=0..50);

Mathematica

LinearRecurrence[{0, 2, 0, 0, 2}, {1, 0, 2, 0, 4}, 50] (* Jean-François Alcover, May 28 2019 *)

Formula

G.f.: -1/(2*x^5+2*x^2-1).

a(n) = 2*(a(n-2)+a(n-5)) for n>4, a(1)=a(3)=0, a(0)=1, a(2)=2, a(4)=4.