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%I #26 Sep 06 2022 13:28:37
%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.: -(x-1)^2*(x^2+x+1)^2 / (x^9+x^7-x^6-2*x^4+3*x^3+x-1).
%F a(n) = 1 + Sum_{i=0..n-3} a(i)*(1 + B*(B-1)) where B=floor((n-i)/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:= -(x-1)^2*(x^2+x+1)^2 / (x^9+x^7-x^6-2*x^4+3*x^3+x-1):
%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
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