

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
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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 LamEstrada, Myriam Rosalía MaldonadoRamírez, José Luis LópezBonilla, Fausto JarquínZárate, 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 nth Fibonacci polynomial evaluated at x=12.  T. D. Noe, Jan 19 2006
a(n) = 12*a(n1)+a(n2), n>1 ; a(0)=1, a(1)=12. G.f.: 1/(112*xx^2). [Philippe Deléham, Nov 21 2008]
a(n) = ((6+sqrt(37))^(n+1)(6sqrt(37))^(n+1))/(2*sqrt(37)). [Rolf Pleisch, May 14 2011]


MATHEMATICA

a=0; lst={}; s=0; Do[a=s(a1); 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



