A010122 Continued fraction for sqrt(13).

3, 1, 1, 1, 1, 6, 1, 1, 1, 1, 6, 1, 1, 1, 1, 6, 1, 1, 1, 1, 6, 1, 1, 1, 1, 6, 1, 1, 1, 1, 6, 1, 1, 1, 1, 6, 1, 1, 1, 1, 6, 1, 1, 1, 1, 6, 1, 1, 1, 1, 6, 1, 1, 1, 1, 6, 1, 1, 1, 1, 6, 1, 1, 1, 1, 6, 1, 1, 1, 1, 6, 1, 1, 1, 1, 6, 1, 1, 1, 1, 6, 1, 1, 1, 1, 6, 1, 1, 1, 1, 6

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• cofr
• easy
• mult

Comments

Eventual period is (1, 1, 1, 1, 6). - Zak Seidov, Mar 05 2011

Examples

			3.605551275463989293119221267... = 3 + 1/(1 + 1/(1 + 1/(1 + 1/(1 + ...)))). - _Harry J. Smith_, Jun 02 2009
		

References

• Albert H. Beiler, Recreations in the theory of numbers, New York, Dover, (2nd ed.) 1966. See Table 96 at p. 264.
• Florian Cajori, A History of Mathematical Notations, Dover edition (2012), par. 428.

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 (0,0,0,0,1).

Crossrefs

Cf. A010470 (decimal expansion).

Programs

Mathematica

ContinuedFraction[Sqrt[13],300] (* Vladimir Joseph Stephan Orlovsky, Mar 05 2011 *)

PARI

{ allocatemem(932245000); default(realprecision, 13000); x=contfrac(sqrt(13)); for (n=0, 20000, write("b010122.txt", n, " ", x[n+1])); } \\ Harry J. Smith, Jun 02 2009

Formula

From Amiram Eldar, Nov 12 2023: (Start)

Multiplicative with a(5^e) = 6, and a(p^e) = 1 for p != 5.

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

G.f.: (3 + x + x^2 + x^3 + x^4 + 3*x^5)/(1 - x^5). - Stefano Spezia, Aug 17 2024