cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-2 of 2 results.

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

Original entry on oeis.org

1, 12, 145, 1752, 21169, 255780, 3090529, 37342128, 451196065, 5451694908, 65871534961, 795910114440, 9616792908241, 116197425013332, 1403985893068225, 16964028141832032, 204972323595052609, 2476631911282463340, 29924555258984612689, 361571295019097815608
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

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)
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

Links

Crossrefs

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

Programs

  • Mathematica
    Denominator[Convergents[Sqrt[37],30]] (* or *) LinearRecurrence[{12,1},{1,12},30] (* Harvey P. Dale, May 26 2014 *)
  • Sage
    [lucas_number1(n,12,-1) for n in range(1, 18)] # Zerinvary Lajos, Apr 28 2009

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
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

A097729 Pell equation solutions (6*a(n))^2 - 37*b(n)^2 = -1 with b(n):=A097730(n), n >= 0.

Original entry on oeis.org

1, 147, 21461, 3133159, 457419753, 66780150779, 9749444593981, 1423352130570447, 207799661618691281, 30337327244198356579, 4429041977991341369253, 646609791459491641554359, 94400600511107788325567161, 13781841064830277603891251147, 2012054394864709422379797100301
Offset: 0

Views

Author

Wolfdieter Lang, Aug 31 2004

Keywords

Examples

			(x,y) = (6,1), (882,145), (128766,21169), ... give the positive integer solutions to x^2 - 37*y^2 =-1.

Links

Crossrefs

Cf. A097728 for S(n, 2*73).
Cf. similar sequences of the type (1/k)*sinh((2*n+1)*arcsinh(k)) listed in A097775.

Programs

  • Mathematica
    LinearRecurrence[{146, -1}, {1, 147}, 20] (* Harvey P. Dale, Sep 24 2012 *)
  • PARI
    x='x+O('x^99); Vec((1+x)/(1-2*73*x+x^2)) \\ Altug Alkan, Apr 05 2018

Formula

G.f.: (1 + x)/(1 - 2*73*x + x^2).
a(n) = S(n, 2*73) + S(n-1, 2*73) = S(2*n, 2*sqrt(37)), with Chebyshev polynomials of the second kind. See A049310 for the triangle of S(n, x) = U(n, x/2) coefficients. S(-1, x) := 0 =: U(-1, x).
a(n) = ((-1)^n)*T(2*n+1, 6*i)/(6*i) with the imaginary unit i and Chebyshev polynomials of the first kind. See the T-triangle A053120.
a(n) = 146*a(n-1) - a(n-2), n > 1; a(0)=1, a(1)=147. - Philippe Deléham, Nov 18 2008
a(n) = (1/6)*sinh((2*n + 1)*arcsinh(6)). - Bruno Berselli, Apr 03 2018

Extensions

a(12)-a(13) from Harvey P. Dale, Sep 24 2012
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Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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