A042301 Denominators of continued fraction convergents to sqrt(677).

1, 52, 2705, 140712, 7319729, 380766620, 19807183969, 1030354333008, 53598232500385, 2788138444353028, 145036797338857841, 7544701600064960760, 392469520000716817361, 20415959741637339463532, 1062022376085142368921025, 55245579516169040523356832

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• frac
• easy

Comments

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

Also called the 52-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 52 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 (52,1).

Crossrefs

Cf. A042300, A040650.

Row n=52 of A073133, A172236 and A352361 and column k=52 of A157103.

Programs

Mathematica

a = 0; lst = {}; s = 0; Do[a = s - (a - 1); AppendTo[lst, a]; s += a*52, {n, 3*4!}]; lst (* Vladimir Joseph Stephan Orlovsky, Nov 03 2009 *)
Denominator[Convergents[Sqrt[677], 30]] (* Vincenzo Librandi, Jan 19 2014 *)
LinearRecurrence[{52,1},{1,52},20] (* Harvey P. Dale, Mar 24 2023 *)

Formula

a(n) = F(n, 52), the n-th Fibonacci polynomial evaluated at x=52. - T. D. Noe, Jan 19 2006

From Philippe Deléham, Nov 23 2008: (Start)

a(n) = 52*a(n-1) + a(n-2) for n > 1, a(0)=1, a(1)=52.

G.f.: 1/(1 - 52*x - x^2). (End)

Extensions

Additional term from Colin Barker, Dec 07 2013