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

1, 44, 2512, 145088, 8383744, 484453376, 27994083328, 1617634967552, 93474855387136, 5401434047381504, 312121261353336832, 18035892123135377408, 1042202005934895529984, 60223526164332403490816, 3480009713100277581611008, 201091971436982107249836032

Offset: 0

Author

Gerhard Kirchner, Jan 20 2023

Keywords

• nonn
• easy

Comments

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.

The example uses two cross section profiles with two overstanding cubes: C (with a common square) and D (with one common edge).

Examples

			a(1)=44.
t1,t2,t3 is a tromino standing on 1,2,3 cubes respectively.
1) Two t2-tiles generate a C-profile or a D-profile in 4 ways each.
   C,D-profile: 4,2 rotation images, D-profile: 2 ways for each image.
    C-profile                      D-profiles
.     ___                      ___                   ___
.   /__ /|               ___ /__ /|            ___ /__ /|
. /__ /| |___          /__ /|   | |          /__ /|   | |
.|   | |/__ /|        |   | |___| |         |   | |___| |
.|   |/__ /| |        |   |/__ /| |         |   |/__ /  |
.|       | |/         |       | |/          |   |   |  /
.|_______|/           |_______|/            |___|___|/
2) t1+t3 generates a C-profile in 4*2=8 ways.
.     ___
.   /   /|                       ______
. /__ /  |    _______          /_____ /|    _______
.|   |  /   /__     /|        |      | |  /__     /|
.|   | |   |  /__ /  |   or   |    __|/  |  /__ /  |
.|   | |   |_|   |  /         |   | |    |_|   |  /
.|___|/      |___|/           |___|/       |___|/
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.
3) There are 4*2=8 ways to generate the cuboid by starting with a D-profile. Therefore, a(1)=36+8=44.
.     ___
.   /   /|          ___       ___
. /__ /  |    ___ /__ /|    /   /|
.|   |   |  /__ /|   | |  /__ /  |
.|___|/| | |   | |___| | |   |  /
.  |___|/  |   |/__ /| | |   | |    or
.          |       | |/  |   | |
.          |_______|/    |___|/
.   _______
. /______ /|          ___
.|       | |    ___ /__ /|    _______
.|    ___|/   /__ /|   | |  /______ /|
.|   | |     |   | |___| | |       | |
.|___|/      |   |/__ /| | |___    | |
.            |       | |/      |   | |
.            |_______|/        |___|/
		

Links

• Paolo Xausa, Table of n, a(n) for n = 0..500
• Index entries for linear recurrences with constant coefficients, signature (60,-128).

Crossrefs

Cf. A006253, A001045, A335559, A359884, A359886.

Programs

Mathematica

LinearRecurrence[{60, -128}, {1, 44}, 20] (* Paolo Xausa, Jun 24 2024 *)

Maxima

/* See A359884. */

Formula

G.f.: (1 - 16*x) / (1 - 60*x + 128*x^2).

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.

a(n) = 60*a(n-1) - 128*a(n-2) for n >= 2.

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