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.
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).
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