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 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 (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. - 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

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