A041761 Denominators of continued fraction convergents to sqrt(401).

1, 40, 1601, 64080, 2564801, 102656120, 4108809601, 164455040160, 6582310416001, 263456871680200, 10544857177624001, 422057743976640240, 16892854616243233601, 676136242393705984280, 27062342550364482604801, 1083169838256973010176320

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• frac
• easy

Comments

From Michael A. Allen, Aug 08 2023: (Start)

Also called the 40-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 40 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 (40,1).

Crossrefs

Cf. A041760, A040380.

Row n=40 of A073133, A172236 and A352361 and column k=40 of A157103.

Programs

Mathematica

a=0;lst={};s=0;Do[a=s-(a-1);AppendTo[lst,a];s+=a*40,{n,3*4!}];lst (* Vladimir Joseph Stephan Orlovsky, Nov 03 2009 *)
Denominator[Convergents[Sqrt[401],20]] (* Harvey P. Dale, Aug 18 2012 *)
CoefficientList[Series[1/(1 - 40 x - x^2), {x, 0, 30}], x] (* Vincenzo Librandi, Dec 24 2013 *)

Formula

a(n) = F(n, 40), the n-th Fibonacci polynomial evaluated at x=40. - T. D. Noe, Jan 19 2006

From Philippe Deléham, Nov 23 2008: (Start)

a(n) = 40*a(n-1) + a(n-2), n>1; a(0)=1, a(1)=40.

G.f.: 1/(1 - 40*x - x^2). (End)