A041061 Denominators of continued fraction convergents to sqrt(37).

1, 12, 145, 1752, 21169, 255780, 3090529, 37342128, 451196065, 5451694908, 65871534961, 795910114440, 9616792908241, 116197425013332, 1403985893068225, 16964028141832032, 204972323595052609, 2476631911282463340, 29924555258984612689, 361571295019097815608

Offset

0,2

Comments

Sqrt(37) = 6.08276253... = 12/2 + 12/145 + 12/(145*21169) + 12/(21169*3090529) + ... - Gary W. Adamson, Jun 13 2008

For positive n, a(n) equals the permanent of the n X n tridiagonal matrix with 12's along the main diagonal and 1's along the superdiagonal and the subdiagonal. - John M. Campbell, Jul 08 2011

a(n) equals the number of words of length n on alphabet {0,1,...,12} avoiding runs of zeros of odd lengths. - Milan Janjic, Jan 28 2015

From Michael A. Allen, Apr 02 2023: (Start)

Also called the 12-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 12 kinds of squares available. (End)

Links

Nathaniel Johnston, Table of n, a(n) for n = 0..500

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

Pablo Lam-Estrada, Myriam Rosalía Maldonado-Ramírez, José Luis López-Bonilla and Fausto Jarquín-Zárate, The sequences of Fibonacci and Lucas for each real quadratic fields Q(Sqrt(d)), arXiv:1904.13002 [math.NT], 2019.

Index entries for linear recurrences with constant coefficients, signature (12,1).

Formula

a(n) = F(n, 12), the n-th Fibonacci polynomial evaluated at x=12. - T. D. Noe, Jan 19 2006

From Philippe Deléham, Nov 21 2008: (Start)

a(n) = 12*a(n-1) + a(n-2), n>1; a(0)=1, a(1)=12.

G.f.: 1/(1 - 12*x - x^2). (End)

a(n) = ((6+sqrt(37))^(n+1) - (6-sqrt(37))^(n+1))/(2*sqrt(37)). - Rolf Pleisch, May 14 2011

Mathematica

Denominator[Convergents[Sqrt[37], 30]] (* or *) LinearRecurrence[{12, 1}, {1, 12}, 30] (* Harvey P. Dale, May 26 2014 *)

Prog

(Sage) [lucas_number1(n, 12, -1) for n in range(1, 18)] # Zerinvary Lajos, Apr 28 2009

Crossrefs

Cf. A010491, A041060.

Cf. A243399.

Row n=12 of A073133, A172236 and A352361 and column k=12 of A157103.

Sequence in context: A075619 A055332 A288792 * A174227 A041266 A015501

Adjacent sequences: A041058 A041059 A041060 * A041062 A041063 A041064

Keyword

nonn,frac,easy

Author

N. J. A. Sloane

Status

approved