A041105 Denominators of continued fraction convergents to sqrt(60).

1, 1, 3, 4, 59, 63, 185, 248, 3657, 3905, 11467, 15372, 226675, 242047, 710769, 952816, 14050193, 15003009, 44056211, 59059220, 870885291, 929944511, 2730774313, 3660718824, 53980837849, 57641556673, 169263951195, 226905507868, 3345941061347, 3572846569215

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• cofr
• easy
• frac

Comments

Interspersion of 4 linear recurrences with constant coefficients. - Gerry Martens, Jun 10 2015

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• Index entries for linear recurrences with constant coefficients, signature (0,0,0,62,0,0,0,-1).

Crossrefs

Cf. A041104, A040052, A020817, A010513.

Cf. A258684.

Programs

Magma

I:=[1, 1, 3, 4, 59, 63, 185, 248]; [n le 8 select I[n] else 62*Self(n-4)-Self(n-8): n in [1..40]]; // Vincenzo Librandi, Dec 11 2013

Maple

numtheory:-cfrac(sqrt(60),100,'con','den'):
den[1..-2]; # Robert Israel, Jun 09 2015

Mathematica

Denominator[Convergents[Sqrt[60], 30]] (* Vincenzo Librandi, Dec 11 2013 *)
d0 := LinearRecurrence[{62, -1}, {1, 59}, 20]
d1 := LinearRecurrence[{62, -1}, {1, 63}, 20] (* A258684  *)
d2 := LinearRecurrence[{62, -1}, {3, 185}, 20]
d3 := LinearRecurrence[{62, -1}, {4, 248}, 20]
Flatten[MapIndexed[{d0[[#]] , d1[[#]], d2[[#]] , d3[[#]]} &,
  Range[10]]] (* Gerry Martens, Jun 09 2015 *)
LinearRecurrence[{0, 0, 0, 62, 0, 0, 0, -1},{1, 1, 3, 4, 59, 63, 185, 248},30] (* Ray Chandler, Aug 03 2015 *)

Formula

G.f.: -(x^2-x-1)*(x^4+4*x^2+1) / ((x^4-8*x^2+1)*(x^4+8*x^2+1)). - Colin Barker, Nov 12 2013

a(n) = 62*a(n-4) - a(n-8). - Vincenzo Librandi, Dec 11 2013

Extensions

More terms from Colin Barker, Nov 12 2013