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A353965 Number of tilings of a 3 X n rectangle using 2 X 2 and 1 X 1 tiles and right trominoes.

%I A353965 #12 Jun 07 2025 09:37:43
%S A353965 1,1,13,47,259,1189,5877,28167,136723,660173,3194613,15445007,
%T A353965 74699811,361230229,1746933205,8448061879,40854753875,197572345789,
%U A353965 955455626773,4620559362303,22344915889827,108059470995013,522573007884725,2527150465444071,12221238828079379
%N A353965 Number of tilings of a 3 X n rectangle using 2 X 2 and 1 X 1 tiles and right trominoes.
%C A353965 For tiling algorithm see A351322.
%H A353965 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (3,9,-1,2).
%F A353965 G.f.: (1 - 2*x + x^2) / (1 - 3*x - 9*x^2 + x^3 - 2*x^4).
%F A353965 a(n) = 3*a(n-1) + 9*a(n-2) - a(n-3) + 2*a(n-4).
%F A353965 31*a(n) = 18*(-2)^n +13*A200739(n+3) +2*A200739(n+2) +9*A200739(n+1). - _R. J. Mathar_, Jun 07 2025
%e A353965 a(2) = 13:
%e A353965     v    h,v   h=v   h,v
%e A353965    ___   ___   ___   ___   ___
%e A353965   |   | | |_| |  _| |  _| |_|_|    mirroring included
%e A353965   |___| |___| |_| | |_|_| |_|_|    h: horizontal, v: vertical
%e A353965   |_|_| |_|_| |___| |_|_| |_|_|
%e A353965     2  +  4  +  2  +  4  +  1 = 13
%o A353965 (Maxima) See A352589.
%Y A353965 Cf. A351322, A352589, A353963, A353964.
%K A353965 nonn,easy
%O A353965 0,3
%A A353965 _Gerhard Kirchner_, May 13 2022