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

Author

N. J. A. Sloane

Keywords

• nonn
• frac
• easy

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).

Crossrefs

Cf. A042512, A040756.

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

Programs

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 *)

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)

Extensions

Additional term from Colin Barker, Dec 17 2013