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A360065 Number of 3-dimensional tilings of a 2 X 2 X n box using 2 X 1 X 1 dominos and trominos (L-shaped connection of 3 cubes).

%I A360065 #16 Oct 02 2024 10:52:56
%S A360065 1,2,45,412,4705,50374,549109,5955544,64683649,702259786,7625147293,
%T A360065 82791470836,898931464993,9760376329678,105975828745957,
%U A360065 1150659965697328,12493588746237697,135652375422278290,1472880803124594061,15992184812239930060,173639288800074705121
%N A360065 Number of 3-dimensional tilings of a 2 X 2 X n box using 2 X 1 X 1 dominos and trominos (L-shaped connection of 3 cubes).
%C A360065 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 9.
%H A360065 Paolo Xausa, <a href="/A360065/b360065.txt">Table of n, a(n) for n = 0..950</a>
%H A360065 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (7,42,6,-81,27).
%F A360065 G.f.: (1 - 5*x - 11*x^2 + 7*x^3) / (1 - 7*x - 42*x^2 - 6*x^3 + 81*x^4 - 27*x^5).
%F A360065 Recurrence 1:
%F A360065 a(n) = 2*a(n-1) + b(n-1) + c(n-1) + 13*a(n-2) + 2*b(n-2) + c(n-2) + 2*d(n-2),
%F A360065 b(n) = 12*a(n-1) + 2*b(n-1) + 2*c(n-1) + e(n-1),
%F A360065 c(n) = 16*a(n-1) + 6*b(n-1) + c(n-1) + 2*e(n-1),
%F A360065 d(n) = 4*a(n-1) + 2*b(n-1) + d(n-1),
%F A360065 e(n) = 16*a(n-1) + 5*b(n-1) + 2*c(n-1) + 2*d(n-1),
%F A360065 with a(n), b(n), c(n), d(n), e(n) = 0 for n <= 0 except for a(0)=1.
%F A360065 Recurrence 2:
%F A360065 a(n) = 7*a(n-1) + 42*a(n-2) + 6*a(n-3) - 81*a(n-4) + 27*a(n-5) for n >= 5.
%F A360065   For n < 5, recurrence 1 can be used.
%e A360065 a(2)=45
%e A360065 1) Two parallel trominos and one domino: There are 3 middle axes of the 2 X 2 cube with 4 rotation images each: 12 images.
%e A360065        ___             ___         ___ ___
%e A360065      /__ /|          /   /|      /__ /   /|
%e A360065    /__ /| |___     /__ /  |    /__ /__ /  |
%e A360065   |   | |/__ /|   |   |  /    |   |   |  /|
%e A360065   |   |/__ /| | + |___|/   =  |   |___|/| |
%e A360065   |       | |/                |       | |/
%e A360065   |_______|/                  |_______|/
%e A360065 2) Two "linked" trominos and one domino: 12 rotation images and, as there is no symmetry plane, 12 mirror images: 24 images.
%e A360065        ___                       ___         ___ ___
%e A360065      /   /|                    /   /|      /   /   /|
%e A360065    /__ /  |      _______     /__ /  |    /__ /__ /  |
%e A360065   |   |  /     /__     /|   |   |  /    |   |   |  /|
%e A360065   |   | |  +  |  /__ /  | + |___|/   =  |   |___|/  |
%e A360065   |   | |     |_|   |  /                |   |   |  /
%e A360065   |___|/        |___|/                  |___|___|/
%e A360065 3) Using only dominos: A006253(2)=9 ways, Sum: a(2) = 12 + 24 + 9 = 45.
%t A360065 LinearRecurrence[{7, 42, 6, -81, 27}, {1, 2, 45, 412, 4705}, 25] (* _Paolo Xausa_, Oct 02 2024 *)
%Y A360065 Cf. A006253, A001045, A033516, A335559, A359884, A359885, A360064, A360066.
%K A360065 nonn,easy
%O A360065 0,2
%A A360065 _Gerhard Kirchner_, Jan 30 2023