

A041061


Denominators of continued fraction convergents to sqrt(37).


14



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.  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
Also called the 12metallonacci sequence; the g.f. 1/(1k*xx^2) gives the kmetallonacci sequence.
a(n) is the number of tilings of an nboard (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



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/(1  12*x  x^2). (End)
a(n) = ((6+sqrt(37))^(n+1)  (6sqrt(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



KEYWORD

nonn,frac,easy


AUTHOR



STATUS

approved



