A040019 Continued fraction for sqrt(24).

4, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• cofr
• easy
• mult
• changed

Comments

Decimal expansion of 23/55. - R. J. Mathar, Aug 25 2025

Examples

			4.898979485566356196394568149... = 4 + 1/(1 + 1/(8 + 1/(1 + 1/(8 + ...)))). - _Harry J. Smith_, Jun 03 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. A010480 (decimal expansion), A010689.

Programs

Maple

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

Mathematica

ContinuedFraction[Sqrt[24],300] (* Vladimir Joseph Stephan Orlovsky, Mar 05 2011 *)
PadRight[{4},120,{8,1}] (* Harvey P. Dale, Oct 24 2022 *)

PARI

{ allocatemem(932245000); default(realprecision, 21000); x=contfrac(sqrt(24)); for (n=0, 20000, write("b040019.txt", n, " ", x[n+1])); } \\ Harry J. Smith, Jun 03 2009

Formula

From Amiram Eldar, Nov 12 2023: (Start)

Multiplicative with a(2^e) = 8, and a(p^e) = 1 for an odd prime p.

Dirichlet g.f.: zeta(s) * (1 + 7/2^s). (End)

G.f.: (4 + x + 4*x^2)/(1 - x^2). - Stefano Spezia, Jul 26 2025