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

Author

N. J. A. Sloane

Keywords

• nonn
• cofr
• frac
• easy

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).

Crossrefs

Cf. A041046.

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

Programs

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

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 *)

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