A041060 -id:A041060 - cp's OEIS frontend

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

N. J. A. Sloane

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

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).

Cf. A010491, A041060.
Cf. A243399.
Row n=12 of A073133, A172236 and A352361 and column k=12 of A157103.

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

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

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

A040030 Continued fraction for sqrt(37).

Original entry on oeis.org

6, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12

Offset: 0

N. J. A. Sloane

			6.08276253029821968899968... = 6 + 1/(12 + 1/(12 + 1/(12 + 1/(12 + ...)))). - _Harry J. Smith_, Jun 04 2009

James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 276.

Harry J. Smith, Table of n, a(n) for n = 0..20000
G. Xiao, Contfrac
Index entries for continued fractions for constants
Index entries for linear recurrences with constant coefficients, signature (1).

Cf. A010491 (decimal expansion), A041060/A041061 (convergents), A248263 (Egyptian fraction).

Cf. A040000, A040002, A040006, A010851

Digits := 100: convert(evalf(sqrt(N)),confrac,90,'cvgts'):

ContinuedFraction[Sqrt[37],300] (* Vladimir Joseph Stephan Orlovsky, Mar 06 2011 *)
PadRight[{6},120,{12}] (* Harvey P. Dale, Jan 02 2017 *)

{ allocatemem(932245000); default(realprecision, 44000); x=contfrac(sqrt(37)); for (n=0, 20000, write("b040030.txt", n, " ", x[n+1])); } \\ Harry J. Smith, Jun 04 2009

From Elmo R. Oliveira, Feb 06 2024: (Start)
a(n) = 12 for n >= 1.
G.f.: 6*(1+x)/(1-x).
E.g.f.: 12*exp(x) - 6.
a(n) = 6*A040000(n) = 3*A040002(n) = 2*A040006(n). (End)

A089926 a(n) = 12*a(n-1) + a(n-2), a(0)=1, a(1)=6.

Original entry on oeis.org

1, 6, 73, 882, 10657, 128766, 1555849, 18798954, 227143297, 2744518518, 33161365513, 400680904674, 4841332221601, 58496667563886, 706801342988233, 8540112783422682, 103188154744060417, 1246797969712147686

Offset: 0

Paul Barry, Nov 15 2003

The family of recurrences a(n) = 2*k*a(n-1) + a(n-2), a(0)=1, a(1)=k has solution a(n) = ((k+sqrt(k^2+1))^n + (k-sqrt(k^2+1))^n)/2; a(n) = Sum_{j=0..floor(n/2)} C(n,2k)*(k^2+1)^jk^(n-2j); a(n) = T(n,ki)*(-i)^n; e.g.f. exp(kx)*cosh(sqrt(k^2+1)*x).

Tanya Khovanova, Recursive Sequences
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (12,1).

Cf. A088320, A088317, A005667, A001077.

Essentially the same as A041060.

E.g.f.: exp(6x)*cosh(sqrt(37)x);
a(n) = ((6+sqrt(37))^n + (6-sqrt(37))^n)/2;
a(n) = Sum_{k=0..floor(n/2)} C(n, 2k)*37^k*6^(n-2k).
a(n) = T(n, 6i)*(-i)^n with T(n, x) Chebyshev's polynomials of the first kind (see A053120) and i^2 = -1.
G.f.: (1-6x)/(1-12*x-x^2). - Philippe Deléham, Nov 21 2008

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A041061 Denominators of continued fraction convergents to sqrt(37).

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A040030 Continued fraction for sqrt(37).

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A089926 a(n) = 12*a(n-1) + a(n-2), a(0)=1, a(1)=6.

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