A247076 Number of tilings of a 5 X 2n rectangle using 2n pentominoes of shape P.

1, 2, 6, 20, 62, 194, 612, 1922, 6038, 18980, 59646, 187442, 589076, 1851266, 5817894, 18283700, 57459518, 180575906, 567489348, 1783428098, 5604714422, 17613731780, 55354032894, 173959101458, 546694927604, 1718078222594, 5399341807686, 16968314698580

Offset: 0

Author

Alois P. Heinz, Nov 17 2014

Keywords

• nonn

Examples

			a(2) = 6:
._______. ._______. ._______. ._______. ._______. ._______.
|   |   | |   |   | |   |   | |   |   | |   ._| | | |_.   |
| ._| ._| |_. |_. | | ._|_. | |_. | ._| |___|   | |   |___|
|_| |_| | | |_| |_| |_| | |_| | |_|_| | |   |___| |___|   |
|   |   | |   |   | |   |   | |   |   | | ._|   | |   |_. |
|___|___| |___|___| |___|___| |___|___| |_|_____| |_____|_| .
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000
• Wikipedia, Pentomino
• Index entries for linear recurrences with constant coefficients, signature (2,2,5).

Crossrefs

Even bisection of main diagonal of A247706.

Cf. A174249, A233427, A247121.

Programs

Maple

a:= proc(n) option remember; `if`(n<4, [1, 2, 6, 20][n+1],
       2*a(n-1) +2*a(n-2) +5*a(n-3))
    end:
seq(a(n), n=0..40);

Mathematica

Join[{1}, LinearRecurrence[{2, 2, 5}, {2, 6, 20}, 40]] (* Jean-François Alcover, May 29 2018 *)

Formula

G.f.: (x-1)*(x^2+x+1)/(5*x^3+2*x^2+2*x-1).

a(n) = 2*a(n-1)+2*a(n-2)+5*a(n-3) for n>3, a(0)=1; a(1)=2, a(2)=6, a(3)=20.