A010123 Continued fraction for sqrt(14).

3, 1, 2, 1, 6, 1, 2, 1, 6, 1, 2, 1, 6, 1, 2, 1, 6, 1, 2, 1, 6, 1, 2, 1, 6, 1, 2, 1, 6, 1, 2, 1, 6, 1, 2, 1, 6, 1, 2, 1, 6, 1, 2, 1, 6, 1, 2, 1, 6, 1, 2, 1, 6, 1, 2, 1, 6, 1, 2, 1, 6, 1, 2, 1, 6, 1, 2, 1, 6, 1, 2, 1, 6, 1, 2, 1, 6, 1, 2, 1, 6

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• cofr
• easy
• mult

Examples

			3.741657386773941385583748732... = 3 + 1/(1 + 1/(2 + 1/(1 + 1/(6 + ...)))). - _Harry J. Smith_, Jun 02 2009
		

References

• Roger Penrose, "The Road to Reality, A complete guide to the Laws of the Universe", Jonathan Cape, London, 2004, page 56. [From Olivier Gérard, May 22 2009]
• 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 (0,0,0,1).

Crossrefs

Cf. A010471 (decimal expansion).

Programs

Mathematica

ContinuedFraction[Sqrt[14],300] (* Vladimir Joseph Stephan Orlovsky, Mar 05 2011 *)
PadRight[{3},120,{6,1,2,1}] (* Harvey P. Dale, Jan 16 2017 *)

PARI

{ allocatemem(932245000); default(realprecision, 15000); x=contfrac(sqrt(14)); for (n=0, 20000, write("b010123.txt", n, " ", x[n+1])); } \\ Harry J. Smith, Jun 02 2009

Formula

a(n) = 1 + floor((n+2)/4) - floor((n+1)/4) + 5*(floor((n+4)/4) - floor((n+3)/4)) for n > 0. - Wesley Ivan Hurt, Apr 10 2017

From Amiram Eldar, Nov 12 2023: (Start)

Multiplicative with a(2) = 2, a(2^e) = 6 for e >= 2, and a(p^e) = 1 for an odd prime p.

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

G.f.: (3 + x + 2*x^2 + x^3 + 3*x^4)/(1 - x^4). - Stefano Spezia, Jul 26 2025