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