A359884 Number of 3-dimensional tilings of a 2 X 2 X n box using 2 X 2 X 1 plates and 1 X 2 X 1 dominos.

1, 3, 24, 133, 839, 5056, 30969, 188603, 1150952, 7018621, 42811231, 261110416, 1592592465, 9713598835, 59245780536, 361354997685, 2203996629559, 13442737199456, 81990685695721, 500082110459883, 3050128402768520, 18603511408241453, 113467563119685583

Offset: 0

Author

Gerhard Kirchner, Jan 20 2023

Keywords

• nonn
• easy

Comments

The first recurrence is derived in "3d-tilings of a 2 X 2 X n box" as a special case of a more general tiling problem: III, example 4.

Examples

			a(1) = 3
      _______         _______          _______
    /       /|      /   /   /|       /______ /|
   /______ / |     /__ /__ / |      /______ /||
   |       | /     |   |   | /      |       ||/
   |_______|/      |___|___|/       |_______|/
		

Links

• Paolo Xausa, Table of n, a(n) for n = 0..1000
• Gerhard Kirchner, Maxima code
• Gerhard Kirchner, 3d-tilings of a 2X2Xn box
• Index entries for linear recurrences with constant coefficients, signature (5,9,-14).

Crossrefs

Cf. A006253, A001045, A335559, A359885, A359886.

Programs

Mathematica

LinearRecurrence[{5, 9, -14}, {1, 3, 24}, 25] (* Paolo Xausa, Jun 24 2024 *)

Maxima

/* See link "Maxima code". */

Formula

G.f.: (1 - 2*x) / (1 - 5*x - 9*x^2 + 14*x^3).

a(n) = 3*a(n-1) + c(n-1) + 7*a(n-2) where c(n) = 8*a(n-1) + 2*c(n-1) with a(n),c(n) <= 0 for n <= 0 except for a(0)=1.

a(n) = 5*a(n-1) + 9*a(n-2) - 14*a(n-3) for n >= 3.