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

0,2

Links

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Wikipedia, Pentomino

Index entries for linear recurrences with constant coefficients, signature (2,2,5).

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.

Example

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

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 *)

Crossrefs

Even bisection of main diagonal of A247706.

Cf. A174249, A233427, A247121.

Sequence in context: A263900 A260696 A052958 * A177792 A193235 A199102

Adjacent sequences: A247073 A247074 A247075 * A247077 A247078 A247079

Keyword

nonn

Author

Alois P. Heinz, Nov 17 2014

Status

approved