A042405 Denominators of continued fraction convergents to sqrt(730).

1, 54, 2917, 157572, 8511805, 459795042, 24837444073, 1341681774984, 72475653293209, 3915026959608270, 211483931472139789, 11424047326455156876, 617110039560050611093, 33335366183569188155898, 1800726883952296211029585, 97272587099607564583753488

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• frac
• easy

Comments

From Michael A. Allen, Dec 17 2023: (Start)

Also called the 54-metallonacci sequence; the g.f. 1/(1-k*x-x^2) gives the k-metallonacci sequence.

a(n) is the number of tilings of an n-board (a board with dimensions n X 1) using unit squares and dominoes (with dimensions 2 X 1) if there are 54 kinds of squares available. (End)

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• Michael A. Allen and Kenneth Edwards, Fence tiling derived identities involving the metallonacci numbers squared or cubed, Fib. Q. 60:5 (2022) 5-17.
• Tanya Khovanova, Recursive Sequences
• Index entries for linear recurrences with constant coefficients, signature (54,1).

Crossrefs

Cf. A042404.

Row n=54 of A073133, A172236 and A352361 and column k=54 of A157103.

Programs

Mathematica

Denominator[Convergents[Sqrt[730], 30]] (* Vincenzo Librandi, Jan 21 2014 *)

Formula

a(n) = F(n, 54), the n-th Fibonacci polynomial evaluated at x=54. - T. D. Noe, Jan 19 2006

From Philippe Deléham, Nov 23 2008: (Start)

a(n) = 54*a(n-1) + a(n-2) for n > 1; a(0)=1, a(1)=54.

G.f.: 1/(1 - 54*x - x^2). (End)

Extensions

a(15) from Colin Barker, Dec 11 2013