A365967 - OEIS

%I #11 Sep 24 2023 05:55:34

%S 1,1,2,3,7,10,22,32,71,103,228,331,733,1064,2356,3420,7573,10993,

%T 24342,35335,78243,113578,251498,365076,808395,1173471,2598440,

%U 3771911,8352217,12124128,26846696,38970824,86293865,125264689,277376074,402640763,891575391

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

%C 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  _______

%C |_|_|_|_|_

%C |_|_|_|_|_|

%C and here is one of the a(9)=103 possible tilings of this board:

%C  _______

%C | |_|_|_|_

%C |_|_|___|_|.

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (0,3,0,1,0,-1).

%F a(n) = 3*a(n-2) + a(n-4) - a(n-6).

%F a(2*n) = a(2*n-1) + a(2*n-2) + a(2*n-3) + a(2*n-4).

%F a(2*n+1) = a(2*n) + a(2*n-1).

%F G.f.: (1+x-x^2)/(1-3*x^2-x^4+x^6).

%t a[0] = 1; a[1] = 1; a[2] = 2; a[3] = 3;

%t 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 Table[a[n],{n,0,30}]

%Y Cf. A030186, A033505.

%K nonn,easy

%O 0,3

%A _Greg Dresden_, Sep 23 2023