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A041061 Denominators of continued fraction convergents to sqrt(37). 12
1, 12, 145, 1752, 21169, 255780, 3090529, 37342128, 451196065, 5451694908, 65871534961, 795910114440, 9616792908241, 116197425013332, 1403985893068225, 16964028141832032, 204972323595052609, 2476631911282463340, 29924555258984612689, 361571295019097815608 (list; graph; refs; listen; history; text; internal format)
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. [From 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

LINKS

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

Tanya Khovanova, Recursive Sequences

Pablo Lam-Estrada, Myriam Rosalía Maldonado-Ramírez, José Luis López-Bonilla, 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

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). [Philippe Deléham, Nov 21 2008]

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

MATHEMATICA

a=0; lst={}; s=0; Do[a=s-(a-1); AppendTo[lst, a]; s+=a*12, {n, 3*4!}]; lst (* Vladimir Joseph Stephan Orlovsky, Oct 27 2009 *)

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.

Sequence in context: A075619 A055332 A288792 * A174227 A041266 A015501

Adjacent sequences:  A041058 A041059 A041060 * A041062 A041063 A041064

KEYWORD

nonn,frac,easy,changed

AUTHOR

N. J. A. Sloane.

STATUS

approved

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Last modified December 7 14:24 EST 2019. Contains 329845 sequences. (Running on oeis4.)