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%I #9 Sep 30 2023 21:47:35
%S 1,1,1,1,1,2,3,5,6,9,11,18,22,35,43,69,84,134,164,263,321,513,627,
%T 1004,1226,1961,2396,3835,4684,7494,9155,14651,17896,28635,34980,
%U 55976,68376,109411,133652,213869,261249,418040,510657,817143,998175,1597247,1951113
%N a(n) = a(n-2) + 2*a(n-4) - a(n-10), with a[0..9] = [1, 1, 1, 1, 1, 2, 3, 5, 6, 9].
%C a(n) is the number of ways to tile a zig-zag strip of n cells using squares (of 1 cell) and strips (of 3 cells). Here is the zig-zag strip corresponding to n=11, with 11 cells:
%C ___ ___
%C ___| |___| |___
%C | |___| |___| |___
%C |___| |___| |___| |
%C | |___| |___| |___|
%C |___| |___| |___|,
%C and here is the strip of 3 cells (which can be reflected)
%C ___
%C ___| |
%C ___| ___|
%C | ___|
%C |___|
%C As an example, here is one of the a(11) = 18 ways to tile the zig-zag strip of 11 cells:
%C ___ ___
%C ___| |___| |___
%C | |___| |___ |___
%C |___| ___| |___ |
%C | ___| |___| |___|
%C |___| |___| |___|
%H <a href="/index/Rec#order_10">Index entries for linear recurrences with constant coefficients</a>, signature (0,1,0,2,0,0,0,0,0,-1).
%F a(n) = a(n-2) + 2*a(n-4) - a(n-10).
%F a(2*n) = a(2*n-1) + a(2*n-4) - a(2*n-5) + a(2*n-6).
%F a(2*n+1) = a(2*n) + 2*a(2*n-3) - a(2*n-4) + a(2*n-6) - a(2*n-7).
%F G.f.: (x^8+x^7-x^5-2*x^4+x+1)/(x^10-2*x^4-x^2+1).
%t LinearRecurrence[{0, 1, 0, 2, 0, 0, 0, 0, 0, -1}, {1, 1, 1, 1, 1, 2,
%t 3, 5, 6, 9}, 40]
%Y Cf. A135318.
%K nonn,easy
%O 0,6
%A _Greg Dresden_ and Ziyi Xie, Sep 30 2023
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