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A360576 Number of 3-dimensional tilings of a 2 X 2 X n box using 1 X 1 X 1 cubes, 2 X 2 X 1 plates and trominos (L-shaped connection of 3 cubes).

%I A360576 #7 Feb 13 2023 03:11:15
%S A360576 1,6,122,1768,28844,457592,7318760,116806896,1865305376,29782666544,
%T A360576 475549098160,7593154541264,121241257906000,1935879286697296,
%U A360576 30910512661708432,493553365105565264,7880649886335326608,125831666350680625104
%N A360576 Number of 3-dimensional tilings of a 2 X 2 X n box using 1 X 1 X 1 cubes, 2 X 2 X 1 plates and trominos (L-shaped connection of 3 cubes).
%C A360576 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 12.
%H A360576 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (15,28,-214,192,384,-128).
%F A360576 G.f.: (1-9*x+4*x^2-16*x^3) / (1-15*x-28*x^2+214*x^3-192*x^4-384*x^5+128*x^6).
%F A360576 Recurrence 1:
%F A360576 a(n) = 8*a(n-1) + 3*b(n-1) + 2*c(n-1) + d(n-1) + e(n-1) + 7*a(n-2)
%F A360576 b(n) = 12*a(n-1) + 5*b(n-1) + 2*c(n-1) + 2*d(n-1) + e(n-1)
%F A360576 c(n) = 16*a(n-1) + 4*b(n-1) + 2*c(n-1)
%F A360576 d(n) = 2*a(n-1) + b(n-1) + d(n-1)
%F A360576 e(n) = 12*a(n-1) + 3*b(n-1)
%F A360576 with a(n),b(n),c(n),d(n),e(n)= 0 for n<=0 except for a(0)=1.
%F A360576 Recurrence 2:
%F A360576 a(n)=15*a(n-1) + 28*a(n-2) - 214*a(n-3) + 192*a(n-4) + 384*a(n-5) - 128*a(n-6)
%F A360576 for n>=6. For n<6, recurrence 1 can be used.
%Y A360576 Cf. A006253, A001045, A033516, A335559, A359884, A359885, A360064, A360065, A360575, A360577.
%K A360576 nonn
%O A360576 0,2
%A A360576 _Gerhard Kirchner_, Feb 12 2023