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

Author

N. J. A. Sloane

Keywords

• nonn
• frac
• easy

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)

Take any recurrence (t) of the form (12,1). Then a(n) = (t(i-n)*(-1)^n + t(i+n+2))/(t(i) + t(i+2)) always applies for integer i >= n >= 1. - Klaus Purath, Aug 02 2025

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).

Crossrefs

Cf. A010491, A041060.

Cf. A243399.

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

Programs

Mathematica

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

Sage

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

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

a(2*n) = a(n-1)^2 + a(n)^2 = A097730(n), a(2*n+1) = 12*A097728(n). - Klaus Purath, Aug 02 2025

E.g.f.: exp(6*x)*(cosh(sqrt(37)*x) + 6*sinh(sqrt(37)*x)/sqrt(37)). - Stefano Spezia, Aug 09 2025