A042513 Denominators of continued fraction convergents to sqrt(785).

1, 56, 3137, 175728, 9843905, 551434408, 30890170753, 1730400996576, 96933345979009, 5429997775821080, 304176808791959489, 17039331290125552464, 954506729055822897473, 53469416158416207810952, 2995241811600363460310785, 167787010865778769985214912

Offset

0,2

Comments

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

Also called the 56-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 56 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 (56,1).

Formula

a(n) = F(n, 56), the n-th Fibonacci polynomial evaluated at x=56. - T. D. Noe, Jan 19 2006

From Philippe Deléham, Nov 23 2008: (Start)

a(n) = 56*a(n-1) + a(n-2) for n > 1, a(0)=1, a(1)=56.

G.f.: 1/(1 - 56*x - x^2). (End)

Mathematica

a=0; lst={}; s=0; Do[a = s-(a-1); AppendTo[lst, a]; s+=a*56, {n, 3*4!}]; lst (* Vladimir Joseph Stephan Orlovsky, Nov 03 2009 *)
Denominator[Convergents[Sqrt[785], 30]] (* Harvey P. Dale, Jun 26 2012 *)
CoefficientList[Series[1/(1 - 56 x - x^2), {x, 0, 25}], x] (* Vincenzo Librandi, Jan 23 2014 *)

Crossrefs

Cf. A042512, A040756.

Row n=56 of A073133, A172236 and A352361 and column k=56 of A157103.

Sequence in context: A223990 A224388 A224021 * A264941 A358115 A216704

Adjacent sequences: A042510 A042511 A042512 * A042514 A042515 A042516

Keyword

nonn,frac,easy

Author

N. J. A. Sloane

Extensions

Additional term from Colin Barker, Dec 17 2013

Status

approved