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A365274 a(n) = a(n-2) + 4*a(n-4) - 2*a(n-8) - a(n-10), with a[0..9] = [1, 1, 1, 2, 3, 5, 7, 13, 18, 31].

%I A365274 #20 Oct 09 2023 20:29:32
%S A365274 1,1,1,2,3,5,7,13,18,31,43,78,108,190,263,471,652,1156,1600,2853,3949,
%T A365274 7019,9715,17299,23944,42592,58952,104926,145230,258403,357659,636490,
%U A365274 880976,1567619,2169764,3861135,5344256,9509879,13162764,23423036,32420177
%N A365274 a(n) = a(n-2) + 4*a(n-4) - 2*a(n-8) - a(n-10), with a[0..9] = [1, 1, 1, 2, 3, 5, 7, 13, 18, 31].
%C A365274 a(n) is the number of ways to tile a zig-zag strip of n cells using squares (of 1 cell) and strips and triangles (of 3 cells). Here is the zig-zag strip corresponding to n=11, with 11 cells:
%C A365274        ___     ___
%C A365274    ___|   |___|   |___
%C A365274   |   |___|   |___|   |___
%C A365274   |___|   |___|   |___|   |
%C A365274   |   |___|   |___|   |___|
%C A365274   |___|   |___|   |___|,
%C A365274 and here are the strip and triangle of 3 cells (which can be reflected):
%C A365274           ___     ___
%C A365274       ___|   |   |   |___
%C A365274   ___|    ___|   |       |
%C A365274  |    ___|       |    ___|
%C A365274  |___|           |___|.
%C A365274 As an example, here is one of the a(11) = 78 ways to tile the zig-zag strip of 11 cells:
%C A365274       ___     ___
%C A365274   ___|   |___|   |___
%C A365274  |   |___|   |       |___
%C A365274  |___|    ___|    ___|   |
%C A365274  |    ___|   |___|   |___|
%C A365274  |___|   |___|   |___|.
%H A365274 <a href="/index/Rec#order_10">Index entries for linear recurrences with constant coefficients</a>, signature (0,1,0,4,0,0,0,-2,0,-1).
%F A365274 a(n) = a(n-2) + 4*a(n-4) - 2*a(n-8) - a(n-10).
%F A365274 a(2*n) = a(2*n-1) + a(2*n-2) - a(2*n-3) + a(2*n-4).
%F A365274 a(2*n+1) = a(2*n) + a(2*n-2) +2*a(2*n-3) - a(2*n-4) - a(2*n-7).
%F A365274 G.f.: (x^8-x^5-2*x^4+x^3+x+1)/(x^10+2*x^8-4*x^4-x^2+1).
%t A365274 LinearRecurrence[{0, 1, 0, 4, 0, 0, 0, -2, 0, -1}, {1, 1, 1, 2, 3, 5, 7, 13, 18, 31}, 40]
%Y A365274 Cf. A135318, A366143.
%K A365274 nonn,easy
%O A365274 0,4
%A A365274 _Greg Dresden_ and Ziyi Xie, Oct 01 2023