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A366143 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].

%I A366143 #9 Sep 30 2023 21:47:35
%S A366143 1,1,1,1,1,2,3,5,6,9,11,18,22,35,43,69,84,134,164,263,321,513,627,
%T A366143 1004,1226,1961,2396,3835,4684,7494,9155,14651,17896,28635,34980,
%U A366143 55976,68376,109411,133652,213869,261249,418040,510657,817143,998175,1597247,1951113
%N A366143 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 A366143 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 A366143        ___     ___
%C A366143    ___|   |___|   |___
%C A366143   |   |___|   |___|   |___
%C A366143   |___|   |___|   |___|   |
%C A366143   |   |___|   |___|   |___|
%C A366143   |___|   |___|   |___|,
%C A366143 and here is the strip of 3 cells (which can be reflected)
%C A366143            ___
%C A366143        ___|   |
%C A366143    ___|    ___|
%C A366143   |    ___|
%C A366143   |___|
%C A366143 As an example, here is one of the a(11) = 18 ways to tile the zig-zag strip of 11 cells:
%C A366143        ___     ___
%C A366143    ___|   |___|   |___
%C A366143   |   |___|   |___    |___
%C A366143   |___|    ___|   |___    |
%C A366143   |    ___|   |___|   |___|
%C A366143   |___|   |___|   |___|
%H A366143 <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 A366143 a(n) = a(n-2) + 2*a(n-4) - a(n-10).
%F A366143 a(2*n) = a(2*n-1) + a(2*n-4) - a(2*n-5) + a(2*n-6).
%F A366143 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 A366143 G.f.: (x^8+x^7-x^5-2*x^4+x+1)/(x^10-2*x^4-x^2+1).
%t A366143 LinearRecurrence[{0, 1, 0, 2, 0, 0, 0, 0, 0, -1}, {1, 1, 1, 1, 1, 2,
%t A366143   3, 5, 6, 9}, 40]
%Y A366143 Cf. A135318.
%K A366143 nonn,easy
%O A366143 0,6
%A A366143 _Greg Dresden_ and Ziyi Xie, Sep 30 2023