A010127 Continued fraction for sqrt(23).

4, 1, 3, 1, 8, 1, 3, 1, 8, 1, 3, 1, 8, 1, 3, 1, 8, 1, 3, 1, 8, 1, 3, 1, 8, 1, 3, 1, 8, 1, 3, 1, 8, 1, 3, 1, 8, 1, 3, 1, 8, 1, 3, 1, 8, 1, 3, 1, 8, 1, 3, 1, 8, 1, 3, 1, 8, 1, 3, 1, 8, 1, 3, 1, 8, 1, 3, 1, 8, 1, 3, 1, 8, 1, 3, 1, 8, 1, 3, 1, 8, 1, 3, 1, 8, 1, 3, 1, 8

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• cofr
• easy
• mult

Examples

			4.795831523312719541597438064... = 4 + 1/(1 + 1/(3 + 1/(1 + 1/(8 + ...)))). - _Harry J. Smith_, Jun 03 2009
		

References

• Albert H. Beiler, Recreations in the theory of numbers, New York, Dover, (2nd ed.) 1966. See Table 95 at p. 262.
• James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, pages 275-276.

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 (0,0,0,1).

Crossrefs

Cf. A010479 (decimal expansion).

Programs

Mathematica

ContinuedFraction[Sqrt[23],300] (* Vladimir Joseph Stephan Orlovsky, Mar 05 2011 *)
PadRight[{4},120,{8,1,3,1}] (* Harvey P. Dale, Oct 23 2024 *)

PARI

{ allocatemem(932245000); default(realprecision, 17000); x=contfrac(sqrt(23)); for (n=0, 20000, write("b010127.txt", n, " ", x[n+1])); } \\ Harry J. Smith, Jun 03 2009

Formula

From Amiram Eldar, Nov 12 2023: (Start)

Multiplicative with a(2) = 3, a(2^e) = 8 for e >= 2, and a(p^e) = 1 for an odd prime p.

Dirichlet g.f.: zeta(s) * (1 + 1/2^(s-1) + 5/4^s). (End)

From Stefano Spezia, Aug 17 2024: (Start)

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

E.g.f.: (5*cos(x) + 11*cosh(x) + 2*sinh(x) - 8)/2. (End)