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A190759 Number of tilings of a 5 X n rectangle using right trominoes and 2 X 2 tiles.

%I A190759 #16 Aug 20 2024 09:39:20
%S A190759 1,0,4,0,16,0,136,0,1128,384,8120,6912,60904,75136,491960,720640,
%T A190759 4023592,6828928,32819320,63472640,270471784,574543744,2256221368,
%U A190759 5119155712,18940876712,45266369152,159625747960,397949457408,1350573713256
%N A190759 Number of tilings of a 5 X n rectangle using right trominoes and 2 X 2 tiles.
%H A190759 Alois P. Heinz, <a href="/A190759/b190759.txt">Table of n, a(n) for n = 0..650</a>
%H A190759 <a href="/index/Rec#order_14">Index entries for linear recurrences with constant coefficients</a>, signature (0, 13, 2, -57, -12, 190, -10, -453, 396, -2, -88, 308, -160, -80).
%F A190759 G.f.: (20*x^12+40*x^11 +18*x^10+52*x^9 +35*x^8-26*x^7 +34*x^6-4*x^5 -21*x^4 +2*x^3 +9*x^2-1) / (-80*x^14-160*x^13 +308*x^12-88*x^11 -2*x^10+396*x^9 -453*x^8-10*x^7 +190*x^6-12*x^5 -57*x^4+2*x^3 +13*x^2-1).
%e A190759 a(2) = 4, because there are 4 tilings of a 5 X 2 rectangle using right trominoes and 2 X 2 tiles:
%e A190759 .___. .___. .___. .___.
%e A190759 | . | | . | | ._| |_. |
%e A190759 |___| |___| |_| | | |_|
%e A190759 | ._| |_. | |___| |___|
%e A190759 |_| | | |_| | . | | . |
%e A190759 |___| |___| |___| |___|
%p A190759 a:= n-> (Matrix(14, (i, j)-> `if`(i=j-1, 1, `if`(i=14, [-80, -160, 308, -88, -2, 396, -453, -10, 190, -12, -57, 2, 13, 0][j], 0)))^n. <<0, 1/4, 0, 1, 0, 4, 0, 16, 0, 136, 0, 1128, 384, 8120>>)[4,1]: seq(a(n), n=0..30);
%t A190759 a[n_] := (MatrixPower[ Table[ If[i == j-1, 1, If[i == 14, {-80, -160, 308, -88, -2, 396, -453, -10, 190, -12, -57, 2, 13, 0}[[j]], 0]], {i, 1, 14}, {j, 1, 14}], n] . {0, 1/4, 0, 1, 0, 4, 0, 16, 0, 136, 0, 1128, 384, 8120})[[4]]; Table[a[n], {n, 0, 30}] (* _Jean-François Alcover_, Dec 05 2013, translated from _Alois P. Heinz_'s Maple program *)
%Y A190759 Cf. A165799, A165791, A165716, A054854, A054856.
%Y A190759 Column k=5 of A219946.
%K A190759 easy,nice,nonn
%O A190759 0,3
%A A190759 _Alois P. Heinz_, May 18 2011