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

0,1

Comments

Decimal expansion of 31213/99990. - Elmo R. Oliveira, Oct 04 2025

References

Harold Davenport, The Higher Arithmetic, Cambridge University Press, 8th ed., 2008, p. 97.

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).

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

E.g.f.: 2*cos(x) + 4*cosh(x) + sinh(x) - 3. - Elmo R. Oliveira, Oct 04 2025

Example

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

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 *)

Prog

(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

Crossrefs

Cf. A010471 (decimal expansion), A041020/A041021 (convergents), A248243 (Egyptian fraction).

Sequence in context: A339913 A102774 A131918 * A039620 A008296 A351397

Adjacent sequences: A010120 A010121 A010122 * A010124 A010125 A010126

Keyword

nonn,cofr,easy,mult

Author

N. J. A. Sloane

Status

approved