A040005 Continued fraction for sqrt(8).

2, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• cofr
• easy
• mult

Examples

			2.828427124746190097603377448... = 2 + 1/(1 + 1/(4 + 1/(1 + 1/(4 + ...)))). - _Harry J. Smith_, Jun 02 2009
		

References

• 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. A010466 (decimal expansion).

Essentially the same as A010685.

Programs

Maple

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

Mathematica

ContinuedFraction[Sqrt[8],300] (* Vladimir Joseph Stephan Orlovsky, Mar 04 2011 *)

PARI

{ allocatemem(932245000); default(realprecision, 16000); x=contfrac(sqrt(8)); for (n=0, 20000, write("b040005.txt", n, " ", x[n+1])); } \\ Harry J. Smith, Jun 02 2009

Formula

From Amiram Eldar, Nov 12 2023: (Start)

Multiplicative with a(2^e) = 4, and a(p^e) = 1 for an odd prime p.

Dirichlet g.f.: zeta(s) * (1 + 3/2^s). (End)

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