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A359886 Number of 3-dimensional tilings of a 2 X 2 X n box using 2 X 2 X 1 plates and trominos (three 1 X 1 X 1 cubes).

%I A359886 #21 Jun 24 2024 18:05:39
%S A359886 1,1,3,49,231,789,4771,27225,122799,607469,3255979,16253649,80098519,
%T A359886 409480005,2079921395,10411734921,52523676351,266059774429,
%U A359886 1341128940795,6758479842689,34138205819239,172324729379509,869131661400259,4386075013348025,22138673661637327
%N A359886 Number of 3-dimensional tilings of a 2 X 2 X n box using 2 X 2 X 1 plates and trominos (three 1 X 1 X 1 cubes).
%C A359886 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 6.
%H A359886 Paolo Xausa, <a href="/A359886/b359886.txt">Table of n, a(n) for n = 0..1000</a>
%H A359886 Gerhard Kirchner, <a href="/A359884/a359884.txt">Maxima code</a>
%H A359886 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (2,1,58,72,32,-128).
%F A359886 G.f.: (1 - x - 16*x^3) / (1 - 2*x - x^2 - 58*x^3 - 72*x^4 - 32*x^5 + 128*x^6).
%F A359886 Recurrence 1:
%F A359886 a(n) = a(n-1) + 3*c(n-2) + 2*a(n-2) + 4*c(n-3) + 8*a(n-3),
%F A359886 c(n) = 12*a(n-1) + c(n-1) + 16*a(n-2) + 16*c(n-3),
%F A359886 with a(n),c(n) <= 0 for n <= 0 except for a(0)=1.
%F A359886 Recurrence 2:
%F A359886 a(n) = 2*a(n-1) + a(n-2) + 58*a(n-3) + 72*a(n-4) + 32*a(n-5) - 128*a(n-6) for n >= 6.
%F A359886 For n < 6, recurrence 1 can be used.
%e A359886 a(3) = 49.
%e A359886 The number of tilings only using plates is A001045(3) = 5.
%e A359886 The number of tilings only using trominos is A359885(1) = 44.
%e A359886 These terms are to be added as, for n=3, there is no tiling using both tiles.
%t A359886 LinearRecurrence[{2, 1, 58, 72, 32, -128}, {1, 1, 3, 49, 231, 789}, 30] (* _Paolo Xausa_, Jun 24 2024 *)
%o A359886 (Maxima) /* See link "Maxima code". */
%Y A359886 Cf. A006253, A001045, A335559, A359884, A359885.
%K A359886 nonn,easy
%O A359886 0,3
%A A359886 _Gerhard Kirchner_, Jan 20 2023