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%I #11 Jul 19 2024 14:39:54
%S 0,1,2,4,7,12,21,40,76,139,254,466,855,1576,2905,5340,9816,18053,
%T 33202,61076,112351,206636,380045,699012,1285684,2364759,4349502,
%U 7999954,14714159,27063568,49777681,91555464,168396816,309729961,569682082,1047808756
%N Number of tilings using squares, dominos, and flexible trominos of a strip of length n-1 and with an n-th cell placed on top of the middle of the strip.
%C As an illustration, here are the figures for n=8 and n=9, respectively.
%C _ _
%C _____|_|_____ _______|_|_____
%C |_|_|_|_|_|_|_|, |_|_|_|_|_|_|_|_|.
%H <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,2,0,2,2,0,-1,-1).
%F a(n) = a(n-1) + 2*a(n-3) + 2*a(n-5) + 2*a(n-6) - a(n-8) - a(n-9).
%F a(2*n) = a(2*n-1) + a(2*n-3) + a(2*n-4) + 3*a(2*n-5) + 2*a(2*n-6) + a(2*n-7).
%F a(2*n) = A000073(2*n+1) + A000073(n+1)*(A000073(n+1) + 2*A000073(n)).
%F a(2*n+1) = a(2*n) + a(2*n-1) + a(2*n-3) + a(2*n-4) + a(2*n-5).
%F a(2*n+1) = A000073(2*n+2) + A000073(n+1)^2 + A000073(n+2)*(A000073(n+1) + A000073(n)).
%F G.f.: x*(1 + x + 2*x^2 + x^3 + x^4 - x^5 - x^6)/(1 - x - 2*x^3 - 2*x^5 -
%F 2*x^6 + x^8 + x^9).
%e For n=8, here is one of a(8)=76 possible tilings with squares, dominos, and flexible trominos.
%e _
%e _____| |_____
%e |___|_|___|___|.
%t LinearRecurrence[{1, 0, 2, 0, 2, 2, 0, -1, -1}, {0, 1, 2, 4, 7, 12, 21, 40, 76}, 40]
%Y Cf. A000073.
%K nonn,easy
%O 0,3
%A _Greg Dresden_ and Yinuo Zhu, Jul 17 2024