Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I #22 Jan 05 2025 19:51:42
%S 1,15,50,254,1202,5774,27650,132494,634802,3041534,14572850,69822734,
%T 334540802,1602881294,7679865650,36796446974,176302369202,
%U 844715399054,4047274626050,19391657731214,92911014030002,445163412418814,2132906048064050,10219366827901454,48963928091443202
%N Number of 4 X n Fibonacci minimal checkerboards.
%C 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
%H Andrew Howroyd, <a href="/A349817/b349817.txt">Table of n, a(n) for n = 1..1000</a>
%H Yifan Zhang and George Grossman, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Papers1/55-3/ZhangGrossman03112017.pdf">A Combinatorial Proof for the Generating Function of Powers of the Fibonacci Sequence</a>, Fib. Q., 55:3 (2017), 235-242.
%F Lemma 2.23 of Zhang-Grossman gives a formula.
%F G.f.: 1 - 1/B(x) where x*B(x) is the g.f. of A056571. - _Andrew Howroyd_, Feb 28 2023
%e 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
%o (PARI) \\ x*F(n,4) gives g.f. of A056571.
%o F(n,k)=sum(i=0, n, fibonacci(i+1)^k*x^i, O(x*x^n))
%o Vec(1 - 1/F(25,4)) \\ _Andrew Howroyd_, Feb 28 2023
%Y Cf. A056571.
%K nonn,changed
%O 1,2
%A _N. J. A. Sloane_, Dec 24 2021
%E Terms a(11) and beyond from _Andrew Howroyd_, Feb 28 2023