A040007 Continued fraction for sqrt(11).

3, 3, 6, 3, 6, 3, 6, 3, 6, 3, 6, 3, 6, 3, 6, 3, 6, 3, 6, 3, 6, 3, 6, 3, 6, 3, 6, 3, 6, 3, 6, 3, 6, 3, 6, 3, 6, 3, 6, 3, 6, 3, 6, 3, 6, 3, 6, 3, 6, 3, 6, 3, 6, 3, 6, 3, 6, 3, 6, 3, 6, 3, 6, 3, 6, 3, 6, 3, 6, 3, 6, 3, 6, 3, 6, 3, 6, 3, 6, 3

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• cofr
• easy
• changed

Comments

Eventual period is (3,6). - Zak Seidov, Mar 05 2011

Decimal expansion of 37/110. - R. J. Mathar, Aug 22 2025

Examples

			3.316624790355399849114932736... = 3 + 1/(3 + 1/(6 + 1/(3 + 1/(6 + ...)))). - _Harry J. Smith_, Jun 02 2009
		

References

• Jan Gullberg, Mathematics from the Birth of Numbers, W. W. Norton & Co., NY & London, 1997, §4.4 Powers and Roots, p. 144.
• 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. A010468, A010704.

Programs

Maple

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

Mathematica

ContinuedFraction[Sqrt[11],300] (* Vladimir Joseph Stephan Orlovsky, Mar 05 2011 *)
PadRight[{3},120,{6,3}] (* Harvey P. Dale, Jan 18 2025 *)

PARI

{ allocatemem(932245000); default(realprecision, 27000); x=contfrac(sqrt(11)); for (n=0, 20000, write("b040007.txt", n, " ", x[n+1])); } \\ Harry J. Smith, Jun 02 2009

Formula

From Stefano Spezia, Jan 18 2025: (Start)

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

E.g.f.: 3*(2*cosh(x) + sinh(x) - 1). (End)