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A360064 Number of 3-dimensional tilings of a 2 X 2 X n box using 1 X 1 X 1 cubes and trominos (L-shaped connection of 3 cubes).

%I A360064 #18 Oct 02 2024 07:30:40
%S A360064 1,5,89,1177,16873,237977,3366793,47599097,673035625,9516252633,
%T A360064 134553882441,1902506043833,26900227288361,380352114739609,
%U A360064 5377937177440009,76040613721296249,1075165950495479017,15202163218500810073,214948926180739194569
%N A360064 Number of 3-dimensional tilings of a 2 X 2 X n box using 1 X 1 X 1 cubes and trominos (L-shaped connection of 3 cubes).
%C A360064 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 8.
%H A360064 Paolo Xausa, <a href="/A360064/b360064.txt">Table of n, a(n) for n = 0..850</a>
%H A360064 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (13,20,-64,112,224,-128).
%F A360064 G.f.: (1 - 8*x + 4*x^2 - 16*x^3) / (1 - 13*x - 20*x^2 + 64*x^3 - 112*x^4 - 224*x^5 + 128*x^6).
%F A360064 Recurrence 1:
%F A360064 a(n) = 5*a(n-1) + 2*b(n-1) + c(n-1) + d(n-1) + e(n-1) + 8*a(n-2) + 4*b(n-2) + c(n-2) + 2*d(n-2),
%F A360064 b(n) = 8*a(n-1) + 4*b(n-1) + 2*c(n-1),
%F A360064 c(n) = 20*a(n-1) + 6*b(n-1) + 4*c(n-1) + 4*d(n-1) + 2*e(n-1),
%F A360064 d(n) = 4*a(n-1), e(n) = 16*a(n-1) + 4*b(n-1),
%F A360064 with a(n), b(n), c(n), d(n), e(n) = 0 for n <= 0 except for a(0)=1.
%F A360064 Recurrence 2:
%F A360064 a(n) = 13*a(n-1) + 20*a(n-2) - 64*a(n-3) + 112*a(n-4) + 224*a(n-5) - 128*a(n-6) for n >= 6. For n < 6, recurrence 1 can be used.
%e A360064 4 rotations:
%e A360064    ___ ___     ___ ___
%e A360064   |   |   |   |   |   | (cross sections)
%e A360064   |   |___|   |___|___|
%e A360064   |       |   |   |   |
%e A360064   |_______|   |___|___| a(1) = 4 + 1 = 5.
%t A360064 LinearRecurrence[{13, 20, -64, 112, 224, -128}, {1, 5, 89, 1177, 16873, 237977}, 25] (* _Paolo Xausa_, Oct 02 2024 *)
%Y A360064 Cf. A006253, A001045, A033516, A335559, A359885, A359886, A360065, A360066.
%K A360064 nonn,easy
%O A360064 0,2
%A A360064 _Gerhard Kirchner_, Jan 30 2023