A041151 Denominators of continued fraction convergents to sqrt(85).

1, 4, 5, 9, 41, 747, 3029, 3776, 6805, 30996, 564733, 2289928, 2854661, 5144589, 23433017, 426938895, 1731188597, 2158127492, 3889316089, 17715391848, 322766369353, 1308780869260, 1631547238613, 2940328107873, 13392859670105, 244011802169763, 989440068349157

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• frac
• easy

Comments

From Johannes W. Meijer, Jun 12 2010: (Start)

The a(n) terms of this sequence can be constructed with the terms of sequence A099371.

For the terms of the periodic sequence of the continued fraction for sqrt(85) see A010158. We observe that its period is five. The decimal expansion of sqrt(85) is A010536. (End)

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,756,0,0,0,0,1).

Crossrefs

Cf. A041150, A041019, A041047, A041091, A041151, A041227, A041319, A041427, A041551.

Programs

Magma

I:=[1, 4, 5, 9, 41, 747, 3029, 3776, 6805, 30996]; [n le 10 select I[n] else 756*Self(n-5)+Self(n-10): n in [1..30]]; // Vincenzo Librandi, Dec 12 2013

Mathematica

Table[Denominator[FromContinuedFraction[ContinuedFraction[Sqrt[85], n]]], {n, 1, 50}] (* Vladimir Joseph Stephan Orlovsky, Jun 23 2011 *)
Denominator[Convergents[Sqrt[85], 30]] (* Vincenzo Librandi, Dec 12 2013 *)

Formula

From Johannes W. Meijer, Jun 12 2010: (Start)

a(5*n) = A099371(3*n+1), a(5*n+1) = (A099371(3*n+2)-A099371(3*n+1))/2, a(5*n+2) = (A099371(3*n+2)+A099371(3*n+1))/2, a(5*n+3):= A099371(3*n+2) and a(5*n+4) = A099371(3*n+3)/2. (End)

G.f.: -(x^8-4*x^7+5*x^6-9*x^5+41*x^4+9*x^3+5*x^2+4*x+1) / (x^10+756*x^5-1). - Colin Barker, Nov 11 2013

a(n) = 756*a(n-5) + a(n-10). - Vincenzo Librandi, Dec 12 2013