A042105 Denominators of continued fraction convergents to sqrt(577).

1, 48, 2305, 110688, 5315329, 255246480, 12257146369, 588598272192, 28264974211585, 1357307360428272, 65179018274768641, 3129950184549323040, 150302787876642274561, 7217663768263378501968, 346598163664518810369025, 16643929519665166276215168

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• frac
• easy

Comments

From Michael A. Allen, Dec 02 2023: (Start)

Also called the 48-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 48 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 (48,1).

Crossrefs

Cf. A042104, A040552.

Row n=48 of A073133, A172236 and A352361 and column k=48 of A157103.

Programs

Mathematica

a=0;lst={};s=0;Do[a=s-(a-1);AppendTo[lst,a];s+=a*48,{n,3*4!}];lst (* Vladimir Joseph Stephan Orlovsky, Nov 03 2009 *)
Denominator[Convergents[Sqrt[577], 30]] (* Vincenzo Librandi, Jan 14 2014 *)
LinearRecurrence[{48,1},{1,48},20] (* Harvey P. Dale, Aug 21 2019 *)

Formula

a(n) = F(n, 48), the n-th Fibonacci polynomial evaluated at x=48. - T. D. Noe, Jan 19 2006

From Philippe Deléham, Nov 23 2008: (Start)

a(n) = 48*a(n-1) + a(n-2), n>1; a(0)=1, a(1)=48.

G.f.: 1/(1 - 48*x - x^2). (End)

Extensions

Additional term from Colin Barker, Dec 01 2013