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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.
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%I #37 Jun 15 2022 01:51:40

%S 1,0,2,3,6,14,31,68,150,331,730,1610,3551,7832,17274,38099,84030,

%T 185334,408767,901564,1988462,4385691,9672946,21334354,47054399,

%U 103781744,228897842,504850083,1113481910,2455861662,5416573407,11946628724,26349119110,58114811627

%N 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 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 ._

%C |_|_ _ _ _

%C |_|_|_|_|_| . . .

%C |_|_|_|_|_| . . .

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (2,0,1).

%F a(n) = 2*a(n-1) + a(n-3) for n >= 4.

%F a(n) = A008998(n-2) + A052980(n-2) for n >= 2.

%F G.f.: (2*x^3-2*x^2+2*x-1)/(x^3+2*x-1).

%e 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 ._ _

%e |X|_ | |_

%e |X|X| and |_|X|

%e |_ _| |X X|

%t Join[{1}, LinearRecurrence[{2, 0, 1}, {0, 2, 3}, 40]

%Y Cf. A052980, A008998.

%K nonn,easy

%O 0,3

%A _Greg Dresden_, May 28 2020