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A242636 Number of tilings of a 4 X n rectangle using tetrominoes of shapes L, Z, O.

%I A242636 #17 Jan 27 2025 10:23:27
%S A242636 1,0,3,12,23,94,289,842,2771,8510,26411,83122,258199,805914,2517287,
%T A242636 7846960,24490017,76416244,238387767,743840496,2320800841,7240890040,
%U A242636 22592311143,70488834118,219928631821,686190651342,2140948175385,6679872756528,20841562274863
%N A242636 Number of tilings of a 4 X n rectangle using tetrominoes of shapes L, Z, O.
%H A242636 Alois P. Heinz, <a href="/A242636/b242636.txt">Table of n, a(n) for n = 0..1000</a>
%H A242636 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 A242636 Wikipedia, <a href="https://en.wikipedia.org/wiki/Tetromino">Tetromino</a>
%H A242636 <a href="/index/Rec#order_12">Index entries for linear recurrences with constant coefficients</a>, signature (0,6,13,3,-18,-13,-3,1,-2,-4,0,-2).
%F A242636 G.f.: (x^6-x^5-2*x^4+x^3+3*x^2-1) / (-2*x^12 -4*x^10 -2*x^9 +x^8 -3*x^7 -13*x^6 -18*x^5 +3*x^4 +13*x^3 +6*x^2 -1).
%e A242636 a(3) = 12:
%e A242636 ._____.  ._____.  .___._.  ._.___.  ._____.  ._____.
%e A242636 | .___|  |___. |  |   | |  | |   |  |___. |  | .___|
%e A242636 |_|_. |  | ._|_|  |___| |  | |___|  |   |_|  |_|   |
%e A242636 |   | |  | |   |  | |___|  |___| |  |___| |  | |___|
%e A242636 |___|_|  |_|___|  |_____|  |_____|  |_____|  |_____|
%e A242636 ._____.  ._____.  ._.___.  .___._.  ._____.  ._____.
%e A242636 | .___|  |___. |  | |_. |  | ._| |  | .___|  |___. |
%e A242636 |_| ._|  |_. |_|  |_. | |  | | ._|  |_| | |  | | |_|
%e A242636 |___| |  | |___|  | |_|_|  |_|_| |  | ._| |  | |_. |
%e A242636 |_____|  |_____|  |_____|  |_____|  |_|___|  |___|_|.
%p A242636 gf:= (x^6-x^5-2*x^4+x^3+3*x^2-1) / (-2*x^12 -4*x^10 -2*x^9 +x^8 -3*x^7 -13*x^6 -18*x^5 +3*x^4 +13*x^3 +6*x^2 -1):
%p A242636 a:= n-> coeff(series(gf, x, n+1), x, n):
%p A242636 seq(a(n), n=0..40);
%Y A242636 Cf. A084480, A174248, A226322, A230031, A232497, A233139, A233191, A233266.
%K A242636 nonn,easy
%O A242636 0,3
%A A242636 _Alois P. Heinz_, May 19 2014