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A359885 Number of 3-dimensional tilings of a 2 X 2 X 3n box using trominos (three 1 X 1 X 1 cubes).

%I A359885 #19 Jun 25 2024 02:07:10
%S A359885 1,44,2512,145088,8383744,484453376,27994083328,1617634967552,
%T A359885 93474855387136,5401434047381504,312121261353336832,
%U A359885 18035892123135377408,1042202005934895529984,60223526164332403490816,3480009713100277581611008,201091971436982107249836032
%N A359885 Number of 3-dimensional tilings of a 2 X 2 X 3n box using trominos (three 1 X 1 X 1 cubes).
%C A359885 The first recurrence 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 5.
%C A359885 The example uses two cross section profiles with two overstanding cubes: C (with a common square) and D (with one common edge).
%H A359885 Paolo Xausa, <a href="/A359885/b359885.txt">Table of n, a(n) for n = 0..500</a>
%H A359885 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (60,-128).
%F A359885 G.f.: (1 - 16*x) / (1 - 60*x + 128*x^2).
%F A359885 a(n)  = 44*a(n-1) + 6*e(n-1) where e(n) = 96*a(n-1) + 16*e(n-1) with a(n),e(n) <= 0 for n < =0 except for a(0)=1.
%F A359885 a(n) = 60*a(n-1) - 128*a(n-2) for n >= 2.
%F A359885 E.g.f.: exp(30*x)*cosh(2*sqrt(193)*x) + 7*exp(30*x)*sinh(2*sqrt(193)*x)/sqrt(193). - _Stefano Spezia_, Jan 21 2023
%e A359885 a(1)=44.
%e A359885 t1,t2,t3 is a tromino standing on 1,2,3 cubes respectively.
%e A359885 1) Two t2-tiles generate a C-profile or a D-profile in 4 ways each.
%e A359885    C,D-profile: 4,2 rotation images, D-profile: 2 ways for each image.
%e A359885     C-profile                      D-profiles
%e A359885 .     ___                      ___                   ___
%e A359885 .   /__ /|               ___ /__ /|            ___ /__ /|
%e A359885 . /__ /| |___          /__ /|   | |          /__ /|   | |
%e A359885 .|   | |/__ /|        |   | |___| |         |   | |___| |
%e A359885 .|   |/__ /| |        |   |/__ /| |         |   |/__ /  |
%e A359885 .|       | |/         |       | |/          |   |   |  /
%e A359885 .|_______|/           |_______|/            |___|___|/
%e A359885 2) t1+t3 generates a C-profile in 4*2=8 ways.
%e A359885 .     ___
%e A359885 .   /   /|                       ______
%e A359885 . /__ /  |    _______          /_____ /|    _______
%e A359885 .|   |  /   /__     /|        |      | |  /__     /|
%e A359885 .|   | |   |  /__ /  |   or   |    __|/  |  /__ /  |
%e A359885 .|   | |   |_|   |  /         |   | |    |_|   |  /
%e A359885 .|___|/      |___|/           |___|/       |___|/
%e A359885 1,2) There are 12 ways to generate a C-profile. The connection of two C-profiles is a 2 X 2 X 3 cuboid. Starting with a C-profile, there are 4*3*3=36 ways to generate this cuboid.
%e A359885 3) There are 4*2=8 ways to generate the cuboid by starting with a D-profile. Therefore, a(1)=36+8=44.
%e A359885 .     ___
%e A359885 .   /   /|          ___       ___
%e A359885 . /__ /  |    ___ /__ /|    /   /|
%e A359885 .|   |   |  /__ /|   | |  /__ /  |
%e A359885 .|___|/| | |   | |___| | |   |  /
%e A359885 .  |___|/  |   |/__ /| | |   | |    or
%e A359885 .          |       | |/  |   | |
%e A359885 .          |_______|/    |___|/
%e A359885 .   _______
%e A359885 . /______ /|          ___
%e A359885 .|       | |    ___ /__ /|    _______
%e A359885 .|    ___|/   /__ /|   | |  /______ /|
%e A359885 .|   | |     |   | |___| | |       | |
%e A359885 .|___|/      |   |/__ /| | |___    | |
%e A359885 .            |       | |/      |   | |
%e A359885 .            |_______|/        |___|/
%t A359885 LinearRecurrence[{60, -128}, {1, 44}, 20] (* _Paolo Xausa_, Jun 24 2024 *)
%o A359885 (Maxima) /* See A359884. */
%Y A359885 Cf. A006253, A001045, A335559, A359884, A359886.
%K A359885 nonn,easy
%O A359885 0,2
%A A359885 _Gerhard Kirchner_, Jan 20 2023