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

0,2

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).

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)

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 *)

Crossrefs

Cf. A042104, A040552.

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

Sequence in context: A158783 A227139 A009992 * A206046 A079240 A360904

Adjacent sequences: A042102 A042103 A042104 * A042106 A042107 A042108

Keyword

nonn,frac,easy

Author

N. J. A. Sloane

Extensions

Additional term from Colin Barker, Dec 01 2013

Status

approved