A041014 Numerators of continued fraction convergents to sqrt(11).

3, 10, 63, 199, 1257, 3970, 25077, 79201, 500283, 1580050, 9980583, 31521799, 199111377, 628855930, 3972246957, 12545596801, 79245827763, 250283080090, 1580944308303, 4993116004999, 31539640338297

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• cofr
• frac
• easy

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• Index entries for linear recurrences with constant coefficients, signature (0,20,0,-1).

Crossrefs

Cf. A010468, A041015 (denominators).

Analog for other sqrt(m): A001333 (m=2), A002531 (m=3), A001077 (m=5), A041006 (m=6), A041008 (m=7), A041010 (m=8), A005667 (m=10), A041016 (m=12), ..., A042936 (m=1000).

Programs

Mathematica

Table[Numerator[FromContinuedFraction[ContinuedFraction[Sqrt[11],n]]],{n,1,50}] (* Vladimir Joseph Stephan Orlovsky, Mar 16 2011 *)
Numerator[Convergents[Sqrt[11], 30]] (* Vincenzo Librandi, Oct 28 2013 *)

PARI

A041014=contfracpnqn(c=contfrac(sqrt(11)), #c)[1,][^-1] \\ Discard last element which may be incorrect. Use e.g. \p999 to get more terms, or extend as follows:
{A041014_upto(N,A=Vec(A041014,N))=for(n=#A041014+1,N, A[n]=20*A[n-2]-A[n-4]); A041014=A} \\ M. F. Hasler, Nov 01 2019

Formula

G.f.: (3 + 10*x + 3*x^2 - x^3)/(1 - 20*x^2 + x^4).