A226322 Number of tilings of a 4 X n rectangle using L tetrominoes and 2 X 2 tiles.

1, 0, 3, 6, 19, 48, 141, 378, 1063, 2920, 8115, 22418, 62123, 171876, 475919, 1317250, 3646681, 10094356, 27943739, 77353070, 214129845, 592752572, 1640859689, 4542223926, 12573787053, 34806745800, 96352029241, 266721635838, 738338745535, 2043868995512

Offset: 0

Author

Alois P. Heinz, Jun 03 2013

Keywords

• nonn
• easy

Examples

			a(3) = 6:
._____.  ._____.  .___._.  ._.___.  ._____.  ._____.
| .___|  |___. |  |   | |  | |   |  |___. |  | .___|
|_|_. |  | ._|_|  |___| |  | |___|  |   |_|  |_|   |
|   | |  | |   |  | |___|  |___| |  |___| |  | |___|
|___|_|  |_|___|  |_____|  |_____|  |_____|  |_____|
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000
• Nicolas Bělohoubek and Antonín Slavík, L-Tetromino Tilings and Two-Color Integer Compositions, Univ. Karlova (Czechia, 2025). See p. 10.
• Index entries for linear recurrences with constant coefficients, signature (0,5,6,4,0,-1,0,-3,-2,-4,0,-2).

Crossrefs

Cf. A054854, A054856, A084480, A165716, A165791, A165799, A174248, A232497, A233191, A233266.

Programs

Maple

a:= n-> (Matrix(12, (i, j)-> `if`(i+1=j, 1, `if`(i=12,
    [-2, 0, -4, -2, -3, 0, -1, 0, 4, 6, 5, 0][j], 0)))^(n+8).
    <<-1, 0, 1/2, [0$5][], 1, 0, 3, 6>>)[1, 1]:
seq(a(n), n=0..40);

Mathematica

a[n_] := MatrixPower[ Table[ If[i+1 == j, 1, If[i == 12, {-2, 0, -4, -2, -3, 0, -1, 0, 4, 6, 5, 0}[[j]], 0]], {i, 1, 12}, {j, 1, 12}], n+8].{-1, 0, 1/2, 0, 0, 0, 0, 0, 1, 0, 3, 6} // First; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Dec 05 2013, after Maple *)

Formula

G.f.: (x^6+2*x^2-1) / (-2*x^12 -4*x^10 -2*x^9 -3*x^8 -x^6 +4*x^4 +6*x^3 +5*x^2-1).