%I
%S 1,1,1,2,3,4,8,13,19,35,58,89,154,256,405,681,1131,1822,3025,5012,
%T 8156,13465,22257,36415,59976,98961,162370,267184,440335,723521,
%U 1190237,1960146,3223045,5301876,8727650,14355677,23615683,38865307,63937660,105184761
%N Number of tilings of a 3 X n rectangle using straight (3 X 1) trominoes and 2 X 2 tiles.
%H Alois P. Heinz, <a href="/A219968/b219968.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,3,2,0,1,1,0,1).
%F G.f.: (x1)^2*(x^2+x+1)^2 / (x^9+x^7x^62*x^4+3*x^3+x1).
%F a(n) = 1 + Sum_{i=0..n3} a(i)*(1 + B*(B1)) where B=floor((ni)/3). E.g. a(7) = 1 + a(0)*3 + a(1)*3 + a(2)*1 + a(3)*1 + a(4)*1 = 13.  _Greg Dresden_ and Andrew Chang, Aug 23 2022
%e a(6) = 8, because there are 8 tilings of a 3 X 6 rectangle using straight (3 X 1) trominoes and 2 X 2 tiles:
%e ._._._._._._. ._____._._._. ._._____._._. ._._._____._.
%e        _____     _____     _____ 
%e        _____     _____     _____ 
%e ______ ________ ________ ________
%e ._._._._____. ._____._____. .___.___.___. ._____._____.
%e    _____ __________     __________
%e    _____ __________ ____.____    
%e ________ __________ __________ _________
%p gf:= (x1)^2*(x^2+x+1)^2 / (x^9+x^7x^62*x^4+3*x^3+x1):
%p a:= n> coeff(series(gf, x, n+1), x, n):
%p seq(a(n), n=0..50);
%Y Column k=3 of A219967.
%K nonn,easy
%O 0,4
%A _Alois P. Heinz_, Dec 02 2012
