A041047 Denominators of continued fraction convergents to sqrt(29).

1, 2, 3, 5, 13, 135, 283, 418, 701, 1820, 18901, 39622, 58523, 98145, 254813, 2646275, 5547363, 8193638, 13741001, 35675640, 370497401, 776670442, 1147167843, 1923838285, 4994844413, 51872282415

Offset

0,2

Comments

The terms of this sequence can be constructed with the terms of sequence A052918.

For the terms of the periodical sequence of the continued fraction for sqrt(29) see A010128. We observe that its period is five. The decimal expansion of sqrt(29) is A010484. - Johannes W. Meijer, Jun 12 2010

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,140,0,0,0,0,1).

Formula

a(5*n) = A052918(3*n), a(5*n+1) = (A052918(3*n+1) - A052918(3*n))/2, a(5*n+2) = (A052918(3*n+1) + A052918(3*n))/2, a(5*n+3) = A052918(3*n+1) and a(5*n+4) = A052918(3*n+2)/2. - Johannes W. Meijer, Jun 12 2010

G.f.: (1 + 2*x + 3*x^2 + 5*x^3 + 13*x^4 - 5*x^5 + 3*x^6 - 2*x^7 + x^8)/(1 - 140*x^5 - x^10). - Peter J. C. Moses, Jul 29 2013

a(n) = 140*a(n-5) + a(n-10). - Vincenzo Librandi, Dec 10 2013

Mathematica

Table[Denominator[FromContinuedFraction[ContinuedFraction[Sqrt[29], n]]], {n, 1, 50}] (* Vladimir Joseph Stephan Orlovsky, Mar 18 2011 *)
Denominator[Convergents[Sqrt[29], 30]] (* Vincenzo Librandi, Dec 10 2013 *)

Prog

(Magma) I:=[1, 2, 3, 5, 13, 135, 283, 418, 701, 1820]; [n le 10 select I[n] else 140*Self(n-5)+Self(n-10): n in [1..50]]; // Vincenzo Librandi, Dec 10 2013

Crossrefs

Cf. A041046.

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

Sequence in context: A139095 A005478 A117740 * A120494 A038601 A372228

Adjacent sequences: A041044 A041045 A041046 * A041048 A041049 A041050

Keyword

nonn,cofr,frac,easy

Author

N. J. A. Sloane.

Status

approved