A040003 Continued fraction for sqrt(6).

2, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• cofr
• easy

Examples

			2.449489742783178098197284074... = 2 + 1/(2 + 1/(4 + 1/(2 + 1/(4 + ...)))). - _Harry J. Smith_, Jun 01 2009
		

References

• Jan Gullberg, Mathematics from the Birth of Numbers, W. W. Norton & Co., NY & London, 1997, §4.4 Powers and Roots, p. 143.
• James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 276.

Links

• Harry J. Smith, Table of n, a(n) for n = 0..20000
• G. Xiao, Contfrac.
• Index entries for continued fractions for constants.
• Index entries for linear recurrences with constant coefficients, signature (0,1).

Crossrefs

Cf. A010464 (decimal expansion).

Equals twice A040001.

Essentially the same as A010694.

Programs

Maple

Digits := 100: convert(evalf(sqrt(N)),confrac,90,'cvgts'):

Mathematica

ContinuedFraction[Sqrt[6], 300] (* Vladimir Joseph Stephan Orlovsky, Mar 04 2011 *)

PARI

{ allocatemem(932245000); default(realprecision, 21000); x=contfrac(sqrt(6)); for (n=0, 20000, write("b040003.txt", n, " ", x[n+1])); } \\ Harry J. Smith, Jun 01 2009

Formula

a(n-1) = gcd(2^n, 3^n+1) (empirical). - Michel Marcus, Sep 03 2020

G.f.: 2*(1 + x + x^2)/(1 - x^2). - Stefano Spezia, Jul 26 2025