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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.
11
1, 3, 24, 133, 839, 5056, 30969, 188603, 1150952, 7018621, 42811231, 261110416, 1592592465, 9713598835, 59245780536, 361354997685, 2203996629559, 13442737199456, 81990685695721, 500082110459883, 3050128402768520, 18603511408241453, 113467563119685583
OFFSET
0,2
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.
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.
EXAMPLE
a(1) = 3
_______ _______ _______
/ /| / / /| /______ /|
/______ / | /__ /__ / | /______ /||
| | / | | | / | ||/
|_______|/ |___|___|/ |_______|/
MATHEMATICA
LinearRecurrence[{5, 9, -14}, {1, 3, 24}, 25] (* Paolo Xausa, Jun 24 2024 *)
PROG
(Maxima) /* See link "Maxima code". */
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Gerhard Kirchner, Jan 20 2023
STATUS
approved