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A226348 Number of ways to tile a fixed 3 x n square grid with 1 x 1, 2 x 2, and axially-oriented 1 x 2 tiles.

%I A226348 #22 Jun 13 2015 00:54:40
%S A226348 1,12,39,233,1018,5191,24471,119998,576801,2800623,13529080,65524257,
%T A226348 316934081,1533990732,7422176263,35918097881,173803358634,
%U A226348 841050066135,4069827389207,19694046692606,95299690379681
%N A226348 Number of ways to tile a fixed 3 x n square grid with 1 x 1, 2 x 2, and axially-oriented 1 x 2 tiles.
%H A226348 Andrew Woods, <a href="/A226348/b226348.txt">Table of n, a(n) for n = 1..100</a>
%H A226348 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (2,15,-1,-26,5,6).
%F A226348 Recurrence: a(n) = 2*a(n-1)+15*a(n-2)-a(n-3)-26*a(n-4)+5*a(n-5)+6*a(n-6) for n>6, a(1)=1, a(2)=12, a(3)=39, a(4)=233, a(5)=1018, a(6)=5191.
%F A226348 G.f.: x*(1+10*x-24*x^3+5*x^4+6*x^5)/((1-x)*(1-x-16*x^2-15*x^3+11*x^4+6*x^5)). [_Bruno Berselli_, Jun 04 2013]
%e A226348 For n=2 the a(2)=12 solutions are:
%e A226348 '_ _ _    _ _ _    _ _ _    _ _ _    _ ___    _ _ _
%e A226348 |_|_|_|  |_|_| |  |_| |_|  |_| | |  |_|   |  | |_|_|
%e A226348 |_|_|_|  |_|_|_|  |_|_|_|  |_|_|_|  |_|___|  |_|_|_|
%e A226348 '_ _ _    _ _ _    ___ _    _ _ _    _ ___    ___ _
%e A226348 | |_| |  | | |_|  |   |_|  | | | |  | |   |  |   | |
%e A226348 |_|_|_|  |_|_|_|  |___|_|  |_|_|_|  |_|___|  |___|_|
%e A226348 '
%e A226348 Note that the 1 x 2 tiles must point in the direction of increasing n.
%t A226348 CoefficientList[Series[(1 + 10 x - 24 x^3 + 5 x^4 + 6 x^5)/((1 - x) (1 - x - 16 x^2 - 15 x^3 + 11 x^4 + 6 x^5)), {x, 0, 20}], x] (* _Bruno Berselli_, Jun 04 2013 *)
%t A226348 LinearRecurrence[{2, 15, -1, -26, 5, 6}, {1, 12, 39, 233, 1018, 5191}, 21] (* _T. D. Noe_, Jun 04 2013 *)
%K A226348 nonn,easy
%O A226348 1,2
%A A226348 _Andrew Woods_, Jun 04 2013