A041046 Numerators of continued fraction convergents to sqrt(29).

5, 11, 16, 27, 70, 727, 1524, 2251, 3775, 9801, 101785, 213371, 315156, 528527, 1372210, 14250627, 29873464, 44124091, 73997555, 192119201, 1995189565, 4182498331, 6177687896, 10360186227, 26898060350

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• frac
• easy

Comments

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

The terms of this sequence can be constructed with the terms of sequence A087130.

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. (End)

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,140,0,0,0,0,1).

Crossrefs

Cf. A041047, A041018, A041046, A041090, A041150, A041226, A041318, A041426, A041550, A010484.

Programs

Mathematica

Table[Numerator[FromContinuedFraction[ContinuedFraction[Sqrt[29],n]]],{n,1,50}] (* Vladimir Joseph Stephan Orlovsky, Mar 18 2011 *)
Numerator[Convergents[Sqrt[29], 30]] (* Vincenzo Librandi, Oct 28 2013 *)
LinearRecurrence[ {0,0,0,0,140,0,0,0,0,1},{5,11,16,27,70,727,1524,2251,3775,9801},30] (* Harvey P. Dale, Jun 10 2021 *)

Formula

a(5*n) = A087130(3*n+1), a(5*n+1) = (A087130(3*n+2) - A087130(3*n+1))/2, a(5*n+2) = ( A087130(3*n+2) + A087130(3*n+1))/2, a(5*n+3) = A087130(3*n+2) and a(5*n+4) = A087130(3*n+3)/2. - Johannes W. Meijer, Jun 12 2010

G.f.: (5 + 11*x + 16*x^2 + 27*x^3 + 70*x^4 + 27*x^5 - 16*x^6 + 11*x^7 - 5*x^8 + x^9)/(1 - 140*x^5 - x^10) - Peter J. C. Moses, Jul 29 2013