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A360577 Number of 3-dimensional tilings of a 2 X 2 X n box using 2 X 2 X 1 plates, 2 X 1 X 1 dominos and trominos (L-shaped connection of 3 cubes).

%I A360577 #6 Feb 13 2023 03:18:00
%S A360577 1,3,60,657,8311,101284,1246049,15292819,187803572,2305968393,
%T A360577 28315208039,347681742812,4269186204201,52421329940803,
%U A360577 643681521419708,7903765218510353,97050331862075975,1191681006432895780,14632650860374551265,179674317212728197891,2206220907971874345652
%N A360577 Number of 3-dimensional tilings of a 2 X 2 X n box using 2 X 2 X 1 plates, 2 X 1 X 1 dominos and trominos (L-shaped connection of 3 cubes).
%C A360577 Recurrence 1 is derived in A359884, "3d-tilings of a 2 X 2 X n box" as a special case of a more general tiling problem: III, example 13.
%H A360577 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (8,51,27,-96,-43,66).
%F A360577 G.f.: (1-5*x-15*x^2-3*x^3+10*x^4) / (1-8*x-51*x^2-27*x^3+96*x^4+43*x^5-66*x^6).
%F A360577 Recurrence 1:
%F A360577 a(n) = 3*a(n-1) + b(n-1) + c(n-1) + 19*a(n-2) + 4*b(n-2) + c(n-2) + 2*d(n-2)
%F A360577 b(n) = 12*a(n-1) + 2*b(n-1) + 2*c(n-1) + e(n-1)
%F A360577 c(n) = 20*a(n-1) + 6*b(n-1) + 2*c(n-1) + 2*e(n-1)
%F A360577 d(n) = 4*a(n-1) + 2*b(n-1) + d(n-1)
%F A360577 e(n) = 24*a(n-1) + 7*b(n-1) + 2*c(n-1) + 2*d(n-1)
%F A360577 with a(n),b(n),c(n),d(n),e(n)= 0 for n<=0 except for a(0)=1.
%F A360577 Recurrence 2:
%F A360577 a(n)=8*a(n-1) + 51*a(n-2) + 27*a(n-3) - 96*a(n-4) - 43*a(n-5) + 66*a(n-6)
%F A360577 for n>=6. For n<6, recurrence 1 can be used.
%Y A360577 Cf. A006253, A001045, A033516, A335559, A359884, A359885, A360064, A360065, A360575, A360576.
%K A360577 nonn
%O A360577 0,2
%A A360577 _Gerhard Kirchner_, Feb 12 2023