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

0,2

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).

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)

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 *)

Crossrefs

Cf. A041760, A040380.

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

Sequence in context: A207740 A208082 A009984 * A229635 A278431 A229584

Adjacent sequences: A041758 A041759 A041760 * A041762 A041763 A041764

Keyword

nonn,frac,easy

Author

N. J. A. Sloane

Status

approved