OFFSET
1,2
COMMENTS
a(n) is the number of tilings of a 4 X n board by monominoes and dominoes with all dominoes placed horizontally which cannot be decomposed into two or more such tilings placed end to end. - Andrew Howroyd, Feb 28 2023
LINKS
Andrew Howroyd, Table of n, a(n) for n = 1..1000
Yifan Zhang and George Grossman, A Combinatorial Proof for the Generating Function of Powers of the Fibonacci Sequence, Fib. Q., 55:3 (2017), 235-242.
FORMULA
Lemma 2.23 of Zhang-Grossman gives a formula.
G.f.: 1 - 1/B(x) where x*B(x) is the g.f. of A056571. - Andrew Howroyd, Feb 28 2023
EXAMPLE
a(2) = 15. Each row of a 4 X 2 board can be tiled with either a domino or two monominoes giving a total of 2^4 = 16 tilings. The tiling consisting of all monominoes is not minimal so a(2) = 16 - 1 = 15. - Andrew Howroyd, Feb 28 2023
PROG
(PARI) \\ x*F(n, 4) gives g.f. of A056571.
F(n, k)=sum(i=0, n, fibonacci(i+1)^k*x^i, O(x*x^n))
Vec(1 - 1/F(25, 4)) \\ Andrew Howroyd, Feb 28 2023
CROSSREFS
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, Dec 24 2021
EXTENSIONS
Terms a(11) and beyond from Andrew Howroyd, Feb 28 2023
STATUS
approved