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A233191 Number of tilings of a 4 X n rectangle using L and T tetrominoes.

%I A233191 #20 Jan 27 2025 10:32:25
%S A233191 1,0,2,4,12,16,76,128,386,832,2368,5024,13946,31680,82632,193696,
%T A233191 498174,1182464,2993384,7213648,18061074,43832960,109163384,266217472,
%U A233191 660116398,1615451648,3995295112,9796774896,24189684402,59396496000,146494223160,360026507808
%N A233191 Number of tilings of a 4 X n rectangle using L and T tetrominoes.
%H A233191 Alois P. Heinz, <a href="/A233191/b233191.txt">Table of n, a(n) for n = 0..1000</a>
%H A233191 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 A233191 Wikipedia, <a href="https://en.wikipedia.org/wiki/Tetromino">Tetromino</a>
%H A233191 <a href="/index/Rec#order_12">Index entries for linear recurrences with constant coefficients</a>, signature (0, 4, 4, 5, -8, 4, 12, -18, -8, 0, 0, -8).
%F A233191 G.f.: (2*x^6+x^4+2*x^2-1) / (-8*x^12 -8*x^9 -18*x^8 +12*x^7 +4*x^6 -8*x^5 +5*x^4 +4*x^3 +4*x^2 -1).
%e A233191 a(3) = 4:
%e A233191 ._____.  ._____.  ._____.  ._____.
%e A233191 |_. ._|  |_. ._|  | |_. |  | ._| |
%e A233191 | |_| |  | |_| |  | ._| |  | |_. |
%e A233191 | ._| |  | |_. |  |_| |_|  |_| |_|
%e A233191 |_|___|  |___|_|  |_____|  |_____|.
%p A233191 gf:= (2*x^6+x^4+2*x^2-1) / (-8*x^12 -8*x^9 -18*x^8
%p A233191      +12*x^7 +4*x^6 -8*x^5 +5*x^4 +4*x^3 +4*x^2 -1):
%p A233191 a:= n-> coeff(series(gf, x, n+1), x, n):
%p A233191 seq(a(n), n=0..40);
%Y A233191 Cf. A084480, A174248, A226322, A230031, A232497, A233139, A233266.
%K A233191 nonn,easy
%O A233191 0,3
%A A233191 _Alois P. Heinz_, Dec 05 2013