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A363958 Expansion of (1 + x + x^3)/(1 - x^2 - 2*x^4 - 2*x^6 + x^8).

%I A363958 #18 Jul 04 2023 11:52:41
%S A363958 1,1,1,2,3,4,7,10,16,23,37,53,86,123,199,285,461,660,1068,1529,2474,
%T A363958 3542,5731,8205,13276,19007,30754,44030,71242,101996,165033,236275,
%U A363958 382301,547334,885605,1267906,2051515,2937120
%N A363958 Expansion of (1 + x + x^3)/(1 - x^2 - 2*x^4 - 2*x^6 + x^8).
%C A363958 a(n) is the number of ways to tile a zig-zag strip of n cells using squares (of length 1), strips (of length 3), and triangles (using 3 cells), where the zig-zag strip begins below the center line. Here is the zig-zag strip corresponding to n=12, with 12 cells:
%C A363958        ___     ___     ___
%C A363958       |   |   |   |   |   |
%C A363958      _|_ _|_ _|_ _|_ _|_ _|_
%C A363958     |   |   |   |   |   |   |
%C A363958    _|___|___|___|___|_ _|___|
%C A363958   |   |   |   |   |   |
%C A363958   |___|   |___|   |___|,
%C A363958 and here are the three possible triangles and strips (which can also be rotated or reflected):
%C A363958        ___
%C A363958       |   |
%C A363958      _|  _|                  ___
%C A363958     |   |                   |   |
%C A363958    _| __|   ___ ___ ___    _|   |_
%C A363958   |   |    |           |  |       |
%C A363958   |___|,   |___ ___ ___|, |___ ___|.
%C A363958 As an example, here is one of the a(12) = 86 ways to tile the skew double-strip of 12 cells:
%C A363958        ___     ___     ___
%C A363958       |   |   |   |   |   |
%C A363958      _|  _|___|_ _|___|___|_
%C A363958     |   |       |           |
%C A363958    _| __|_     _|___________|
%C A363958   |   |   |   |   |   |
%C A363958   |___|   |___|   |___|.
%H A363958 <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (0,1,0,2,0,2,0,1).
%F A363958 a(n) = a(n-2) + 2*a(n-4) + 2*a(n-6) + a(n-8).
%F A363958 a(2*n) = a(2*n-1) + a(2*n-3) + a(2*n-5).
%F A363958 a(2*n) = A123392(n).
%F A363958 a(2*n+1) = a(2*n) + a(2*n-2).
%F A363958 a(2*n+1) = A210460(n+1).
%t A363958 LinearRecurrence[{0, 1, 0, 2, 0, 2, 0, 1}, {1, 1, 1, 2, 3, 4, 7, 10}, 50]
%Y A363958 Cf. A077998. Alternate terms are A123392 and A210460.
%K A363958 nonn,easy
%O A363958 0,4
%A A363958 _Greg Dresden_ and Yunxin Li, Jun 29 2023