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

0,2

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).

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

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 *)

Prog

(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

Crossrefs

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

Sequence in context: A222543 A042221 A341252 * A279919 A041467 A180436

Adjacent sequences: A041148 A041149 A041150 * A041152 A041153 A041154

Keyword

nonn,frac,easy

Author

N. J. A. Sloane.

Status

approved