%I #17 Jul 26 2023 19:28:20
%S 1,1,6,30,145,733,3540,17300,84479,411963,2011408,9816506,47911847,
%T 233851991,1141365064,5570761346,27189615925,132706261547,
%U 647709321582,3161321546320,15429691961077,75308819284819,367565220881250,1794002281279416,8756117243124305
%N Number of tilings of a 3 X n rectangle using dominoes and trominoes (of any shape).
%H Alois P. Heinz, <a href="/A364460/b364460.txt">Table of n, a(n) for n = 0..1453</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Domino_(mathematics)">Domino (mathematics)</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Tromino">Tromino</a>
%H <a href="/index/Rec#order_18">Index entries for linear recurrences with constant coefficients</a>, signature (3,6,13,0,53,18,69,48,35,-125,-76,-24,38,8,18,1,-3,-1).
%F G.f.: -(x^15 +2*x^14 +4*x^13 -5*x^12 -9*x^11 -18*x^10 +16*x^9 +5*x^8 +8*x^7 -10*x^6 +13*x^5 -6*x^4 +7*x^3 +3*x^2 +2*x -1) / (x^18 +3*x^17 -x^16 -18*x^15 -8*x^14 -38*x^13 +24*x^12 +76*x^11 +125*x^10 -35*x^9 -48*x^8 -69*x^7 -18*x^6 -53*x^5 -13*x^3 -6*x^2 -3*x +1).
%F a(n) mod 2 = A133872(n).
%e a(2) = 6:
%e .___. .___. .___. .___. .___. .___.
%e | | | |___| | | | |___| | ._| |_. |
%e | | | |___| |_|_| | | | |_| | | |_|
%e |_|_| |___| |___| |_|_| |___| |___| .
%Y Column k=3 of A364457.
%Y Cf. A133872.
%K nonn,easy
%O 0,3
%A _Alois P. Heinz_, Jul 25 2023