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).

1, 2, 45, 412, 4705, 50374, 549109, 5955544, 64683649, 702259786, 7625147293, 82791470836, 898931464993, 9760376329678, 105975828745957, 1150659965697328, 12493588746237697, 135652375422278290, 1472880803124594061, 15992184812239930060, 173639288800074705121

Offset: 0

Author

Gerhard Kirchner, Jan 30 2023

Keywords

• nonn
• easy

Comments

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.

Examples

			a(2)=45
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.
       ___             ___         ___ ___
     /__ /|          /   /|      /__ /   /|
   /__ /| |___     /__ /  |    /__ /__ /  |
  |   | |/__ /|   |   |  /    |   |   |  /|
  |   |/__ /| | + |___|/   =  |   |___|/| |
  |       | |/                |       | |/
  |_______|/                  |_______|/
2) Two "linked" trominos and one domino: 12 rotation images and, as there is no symmetry plane, 12 mirror images: 24 images.
       ___                       ___         ___ ___
     /   /|                    /   /|      /   /   /|
   /__ /  |      _______     /__ /  |    /__ /__ /  |
  |   |  /     /__     /|   |   |  /    |   |   |  /|
  |   | |  +  |  /__ /  | + |___|/   =  |   |___|/  |
  |   | |     |_|   |  /                |   |   |  /
  |___|/        |___|/                  |___|___|/
3) Using only dominos: A006253(2)=9 ways, Sum: a(2) = 12 + 24 + 9 = 45.
		

Links

• Paolo Xausa, Table of n, a(n) for n = 0..950
• Index entries for linear recurrences with constant coefficients, signature (7,42,6,-81,27).

Crossrefs

Cf. A006253, A001045, A033516, A335559, A359884, A359885, A360064, A360066.

Programs

Mathematica

LinearRecurrence[{7, 42, 6, -81, 27}, {1, 2, 45, 412, 4705}, 25] (* Paolo Xausa, Oct 02 2024 *)

Formula

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).

Recurrence 1:

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),

b(n) = 12*a(n-1) + 2*b(n-1) + 2*c(n-1) + e(n-1),

c(n) = 16*a(n-1) + 6*b(n-1) + c(n-1) + 2*e(n-1),

d(n) = 4*a(n-1) + 2*b(n-1) + d(n-1),

e(n) = 16*a(n-1) + 5*b(n-1) + 2*c(n-1) + 2*d(n-1),

with a(n), b(n), c(n), d(n), e(n) = 0 for n <= 0 except for a(0)=1.

Recurrence 2:

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.

For n < 5, recurrence 1 can be used.