A041265 Denominators of continued fraction convergents to sqrt(145).

1, 24, 577, 13872, 333505, 8017992, 192765313, 4634385504, 111418017409, 2678666803320, 64399421297089, 1548264777933456, 37222754091700033, 894894362978734248, 21514687465581321985, 517247393536930461888, 12435452132351912407297

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• frac
• easy
• less

Comments

From Michael A. Allen, May 04 2023: (Start)

Also called the 24-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 24 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 (24,1).

Crossrefs

Cf. A041264, A176910, A040132.

Row n=24 of A073133, A172236 and A352361 and column k=24 of A157103.

Programs

Mathematica

Denominator[Convergents[Sqrt[145], 30]] (* Vincenzo Librandi, Dec 14 2013 *)

Formula

a(n) = F(n, 24), the n-th Fibonacci polynomial evaluated at x=24. - T. D. Noe, Jan 19 2006

From Philippe Deléham, Nov 21 2008: (Start)

a(n) = 24*a(n-1) + a(n-2) for n > 1, a(0)=1, a(1)=24.

G.f.: 1/(1-24*x-x^2). (End)

Extensions

More terms from Colin Barker, Nov 14 2013