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

%I A226322 #29 Jan 27 2025 10:31:19
%S A226322 1,0,3,6,19,48,141,378,1063,2920,8115,22418,62123,171876,475919,
%T A226322 1317250,3646681,10094356,27943739,77353070,214129845,592752572,
%U A226322 1640859689,4542223926,12573787053,34806745800,96352029241,266721635838,738338745535,2043868995512
%N A226322 Number of tilings of a 4 X n rectangle using L tetrominoes and 2 X 2 tiles.
%H A226322 Alois P. Heinz, <a href="/A226322/b226322.txt">Table of n, a(n) for n = 0..1000</a>
%H A226322 Nicolas Bělohoubek and Antonín Slavík, <a href="https://msekce.karlin.mff.cuni.cz/~slavik/papers/L-tetromino-tilings.pdf">L-Tetromino Tilings and Two-Color Integer Compositions</a>, Univ. Karlova (Czechia, 2025). See p. 10.
%H A226322 <a href="/index/Rec#order_12">Index entries for linear recurrences with constant coefficients</a>, signature (0,5,6,4,0,-1,0,-3,-2,-4,0,-2).
%F A226322 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).
%e A226322 a(3) = 6:
%e A226322 ._____.  ._____.  .___._.  ._.___.  ._____.  ._____.
%e A226322 | .___|  |___. |  |   | |  | |   |  |___. |  | .___|
%e A226322 |_|_. |  | ._|_|  |___| |  | |___|  |   |_|  |_|   |
%e A226322 |   | |  | |   |  | |___|  |___| |  |___| |  | |___|
%e A226322 |___|_|  |_|___|  |_____|  |_____|  |_____|  |_____|
%p A226322 a:= n-> (Matrix(12, (i, j)-> `if`(i+1=j, 1, `if`(i=12,
%p A226322     [-2, 0, -4, -2, -3, 0, -1, 0, 4, 6, 5, 0][j], 0)))^(n+8).
%p A226322     <<-1, 0, 1/2, [0$5][], 1, 0, 3, 6>>)[1, 1]:
%p A226322 seq(a(n), n=0..40);
%t A226322 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 *)
%Y A226322 Cf. A054854, A054856, A084480, A165716, A165791, A165799, A174248, A232497, A233191, A233266.
%K A226322 nonn,easy
%O A226322 0,3
%A A226322 _Alois P. Heinz_, Jun 03 2013