A349817 Number of 4 X n Fibonacci minimal checkerboards.

1, 15, 50, 254, 1202, 5774, 27650, 132494, 634802, 3041534, 14572850, 69822734, 334540802, 1602881294, 7679865650, 36796446974, 176302369202, 844715399054, 4047274626050, 19391657731214, 92911014030002, 445163412418814, 2132906048064050, 10219366827901454, 48963928091443202

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

Cf. A056571.

Sequence in context: A318084 A191746 A029941 * A278909 A194851 A075928

Adjacent sequences: A349814 A349815 A349816 * A349818 A349819 A349820

Keyword

nonn

Author

N. J. A. Sloane, Dec 24 2021

Extensions

Terms a(11) and beyond from Andrew Howroyd, Feb 28 2023

Status

approved