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A232497 Number of tilings of a 4 X n rectangle using L and Z tetrominoes.

%I A232497 #27 Jan 27 2025 10:36:39
%S A232497 1,0,2,6,14,32,102,238,652,1696,4480,11658,30870,80644,212292,556858,
%T A232497 1463390,3840686,10090218,26490280,69575414,182693434,479789138,
%U A232497 1259906496,3308668718,8688615148,22817011182,59918425698,157349755400,413208421354,1085110433096
%N A232497 Number of tilings of a 4 X n rectangle using L and Z tetrominoes.
%H A232497 Alois P. Heinz, <a href="/A232497/b232497.txt">Table of n, a(n) for n = 0..1000</a>
%H A232497 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 A232497 Wikipedia, <a href="https://en.wikipedia.org/wiki/Tetromino">Tetromino</a>
%H A232497 <a href="/index/Rec#order_12">Index entries for linear recurrences with constant coefficients</a>, signature (0,5,7,2,-13,-13,-6,-6,0,-4,0,-2).
%F A232497 G.f.: -(x^6-x^5-2*x^4+x^3+3*x^2-1) / (2*x^12 +4*x^10 +6*x^8 +6*x^7 +13*x^6 +13*x^5 -2*x^4 -7*x^3 -5*x^2+1).
%e A232497 a(3) = 6:
%e A232497 ._._._.  ._._._.  ._._._.  ._._._.  ._._._.  ._._._.
%e A232497 | .___|  |___. |  | |_. |  | ._| |  | .___|  |___. |
%e A232497 |_| ._|  |_. |_|  |_. | |  | | ._|  |_| | |  | | |_|
%e A232497 |___| |  | |___|  | |_|_|  |_|_| |  | ._| |  | |_. |
%e A232497 |_____|  |_____|  |_____|  |_____|  |_|___|  |___|_|.
%p A232497 a:= n-> coeff(series(-(x^6-x^5-2*x^4+x^3+3*x^2-1)/
%p A232497     (2*x^12+4*x^10+6*x^8+6*x^7+13*x^6+13*x^5-2*x^4-7*x^3-5*x^2+1),
%p A232497     x, n+1), x, n);
%p A232497 seq(a(n), n=0..40);
%Y A232497 Cf. A084480, A174248, A226322, A233139, A233191, A233266.
%K A232497 nonn,easy
%O A232497 0,3
%A A232497 _Alois P. Heinz_, Nov 24 2013