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

Author

N. J. A. Sloane, Dec 24 2021

Keywords

• nonn

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

Examples

			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
		

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.

Crossrefs

Cf. A056571.

Programs

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

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

Extensions

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