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A365967 a(2*n) = A030186(n), a(2*n+1) = A033505(n).

%I A365967 #11 Sep 24 2023 05:55:34
%S A365967 1,1,2,3,7,10,22,32,71,103,228,331,733,1064,2356,3420,7573,10993,
%T A365967 24342,35335,78243,113578,251498,365076,808395,1173471,2598440,
%U A365967 3771911,8352217,12124128,26846696,38970824,86293865,125264689,277376074,402640763,891575391
%N A365967 a(2*n) = A030186(n), a(2*n+1) = A033505(n).
%C A365967 a(n) is the number of ways to tile a double-height board of n cells with squares and dominos. For example, here is the board for n=9:
%C A365967  _______
%C A365967 |_|_|_|_|_
%C A365967 |_|_|_|_|_|
%C A365967 and here is one of the a(9)=103 possible tilings of this board:
%C A365967  _______
%C A365967 | |_|_|_|_
%C A365967 |_|_|___|_|.
%H A365967 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (0,3,0,1,0,-1).
%F A365967 a(n) = 3*a(n-2) + a(n-4) - a(n-6).
%F A365967 a(2*n) = a(2*n-1) + a(2*n-2) + a(2*n-3) + a(2*n-4).
%F A365967 a(2*n+1) = a(2*n) + a(2*n-1).
%F A365967 G.f.: (1+x-x^2)/(1-3*x^2-x^4+x^6).
%t A365967 a[0] = 1; a[1] = 1; a[2] = 2; a[3] = 3;
%t A365967 a[n_] := a[n] = If[EvenQ[n], a[n-1] + a[n-2] + a[n-3] + a[n-4], a[n-1] + a[n-2]];
%t A365967 Table[a[n],{n,0,30}]
%Y A365967 Cf. A030186, A033505.
%K A365967 nonn,easy
%O A365967 0,3
%A A365967 _Greg Dresden_, Sep 23 2023