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A346054 Number of ways to tile a 3 X n strip with dominoes and L-shaped 5-minoes.

%I A346054 #33 Dec 01 2022 08:57:47
%S A346054 1,0,3,8,13,52,119,308,873,2184,5867,15552,40581,107836,283871,748076,
%T A346054 1976545,5208784,13743315,36260088,95627773,252289476,665499975,
%U A346054 1755466916,4630903129,12215645848,32223689915,85003275440,224228961909,591494654412,1560303157679
%N A346054 Number of ways to tile a 3 X n strip with dominoes and L-shaped 5-minoes.
%H A346054 G. C. Greubel, <a href="/A346054/b346054.txt">Table of n, a(n) for n = 0..1000</a>
%H A346054 Greg Dresden and Michael Tulskikh, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL24/Dresden2/dresden9.html">Tilings of 2 X n boards with dominos and L-shaped trominos</a>, Journal of Integer Sequences 24 (2021), article 21.4.5.
%H A346054 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (1,3,5,-4).
%F A346054 a(n) = a(n-1) + 3*a(n-2) + 5*a(n-3) - 4*a(n-4).
%F A346054 G.f.: (1 - x)/(1 - x - 3*x^2 - 5*x^3 + 4*x^4).
%e A346054 Here are two such tilings for a 3 X 3 strip; each has four rotations thus demonstrating that a(3)=8.
%e A346054   ._____.  ._____.
%e A346054   | | | |  | |___|
%e A346054   | |_|_|  | |___|
%e A346054   |_____|  |_____|
%e A346054 For a 3 X 4 strip, here are three of the possible a(4)=13 tilings.
%e A346054   ._______.  ._______.  ._______.
%e A346054   | |___  |  |  ___| |  |___|___|
%e A346054   | |___| |  | |___| |  | |___| |
%e A346054   |_____|_|  |_|_____|  |_|___|_|
%e A346054 For a 3 X 5 strip, here are three of the possible a(5)=52 tilings.
%e A346054   ._________.  ._________.  ._________.
%e A346054   | | |___| |  |  ___|___|  | |___|___|
%e A346054   | |_|___|_|  | | |___| |  | |___|___|
%e A346054   |_____|___|  |_|_|___|_|  |_____|___|
%t A346054 LinearRecurrence[{1, 3, 5, -4}, {1, 0, 3, 8}, 50];
%o A346054 (Magma) I:=[1,0,3,8]; [n le 4 select I[n] else Self(n-1) +3*Self(n-2) +5*Self(n-3) -4*Self(n-4): n in [1..50]]; // _G. C. Greubel_, Dec 01 2022
%o A346054 (SageMath)
%o A346054 @CachedFunction
%o A346054 def a(n): # a = A346054
%o A346054     if (n<4): return (1,0,3,8)[n]
%o A346054     else: return a(n-1) + 3*a(n-2) + 5*a(n-3) - 4*a(n-4)
%o A346054 [a(n) for n in range(51)] # _G. C. Greubel_, Dec 01 2022
%Y A346054 Cf. A052980.
%K A346054 nonn,easy
%O A346054 0,3
%A A346054 _Greg Dresden_ and _Ziyao Geng_, Jul 02 2021
%E A346054 Corrected by _Greg Dresden_, Sep 04 2021