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

0,1

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).

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

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 *)

Crossrefs

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

Sequence in context: A035108 A022136 A042385 * A041117 A041491 A058025

Adjacent sequences: A041043 A041044 A041045 * A041047 A041048 A041049

Keyword

nonn,frac,easy

Author

N. J. A. Sloane.

Status

approved