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

0,2

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).

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

Mathematica

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

Prog

(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

Crossrefs

Cf. A042740, A040870.

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

Sequence in context: A267930 A267977 A159991 * A166772 A293091 A074076

Adjacent sequences: A042738 A042739 A042740 * A042742 A042743 A042744

Keyword

nonn,frac,easy

Author

N. J. A. Sloane

Extensions

Additional term from Colin Barker, Dec 22 2013

Status

approved