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A335242 a(n) = 2*a(n-1) + a(n-3) for n >= 4, with initial values a(0) = 1, a(1) = 0, a(2) = 2, and a(3) = 3.

%I A335242 #40 Jun 20 2025 08:12:28
%S A335242 1,0,2,3,6,14,31,68,150,331,730,1610,3551,7832,17274,38099,84030,
%T A335242 185334,408767,901564,1988462,4385691,9672946,21334354,47054399,
%U A335242 103781744,228897842,504850083,1113481910,2455861662,5416573407,11946628724,26349119110,58114811627
%N A335242 a(n) = 2*a(n-1) + a(n-3) for n >= 4, with initial values a(0) = 1, a(1) = 0, a(2) = 2, and a(3) = 3.
%C A335242 a(n) is the number of ways to tile this 2 X n strip (with one extra square added at the top left) with dominoes and L-shaped trominoes (also called polyominoes):
%C A335242 ._
%C A335242 |_|_ _ _ _
%C A335242 |_|_|_|_|_| . . .
%C A335242 |_|_|_|_|_| . . .
%H A335242 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (2,0,1).
%F A335242 a(n) = 2*a(n-1) + a(n-3) for n >= 4.
%F A335242 a(n) = A008998(n-2) + A052980(n-2) for n >= 2.
%F A335242 G.f.: (2*x^3-2*x^2+2*x-1)/(x^3+2*x-1).
%e A335242 a(2) = 2 thanks to the following two tilings (where the L-shaped trominoes are tiled with X's and the dominoes are left blank):
%e A335242 ._            _
%e A335242 |X|_         | |_
%e A335242 |X|X|  and   |_|X|
%e A335242 |_ _|        |X X|
%t A335242 LinearRecurrence[{2, 0, 1}, {1, 0, 2, 3}, 40]
%Y A335242 Cf. A052980, A008998.
%K A335242 nonn,easy
%O A335242 0,3
%A A335242 _Greg Dresden_, May 28 2020