A042741 Denominators of continued fraction convergents to sqrt(901).

1, 60, 3601, 216120, 12970801, 778464180, 46720821601, 2804027760240, 168288386436001, 10100107213920300, 606174721221654001, 36380583380513160360, 2183441177552011275601, 131042851236501189696420, 7864754515367623393060801, 472016313773293904773344480

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• frac
• easy

Comments

From Michael A. Allen, Jan 22 2024: (Start)

Also called the 60-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 60 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 (60,1).

Crossrefs

Cf. A042740, A040870.

Row n=60 of A073133, A172236 and A352361 and column k=60 of A157103.

Programs

Magma

I:=[1,60]; [n le 2 select I[n] else 60*Self(n-1)+Self(n-2): n in [1..30]]; // Vincenzo Librandi, Jan 28 2014

Mathematica

Denominator[Convergents[Sqrt[901],30]] (* or *) LinearRecurrence[{60,1},{1,60},30] (* Harvey P. Dale, Sep 09 2012 *)

Formula

a(n) = F(n, 60), the n-th Fibonacci polynomial evaluated at x=60. - T. D. Noe, Jan 19 2006

From Philippe Deléham, Nov 23 2008: (Start)

a(n) = 60*a(n-1) + a(n-2) for n>1; a(0)=1, a(1)=60.

G.f.: 1/(1 - 60*x - x^2). (End)

E.g.f.: exp(30*x)*cosh(sqrt(901)*x) + 30*exp(30*x)*sinh(sqrt(901)*x)/sqrt(901). - Stefano Spezia, May 14 2023

Extensions

Additional term from Colin Barker, Dec 22 2013