A040006 Continued fraction for sqrt(10).

3, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6

Offset

0,1

Comments

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

The convergents are given in A005667(n+1)/A005668(n+1), n >= 0. - Wolfdieter Lang, Nov 23 2017

Decimal expansion of 11/30. - Elmo R. Oliveira, Feb 16 2024

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 (1).

Formula

a(n) = 3 + 3*sign(n). a(n) = 6, n > 0. - Wesley Ivan Hurt, Nov 01 2013

From Elmo R. Oliveira, Feb 16 2024: (Start)

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

E.g.f.: 6*exp(x) - 3.

a(n) = 3*A040000(n). (End)

Example

3.162277660168379331998893544... = 3 + 1/(6 + 1/(6 + 1/(6 + 1/(6 + ...)))).

Maple

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

Mathematica

ContinuedFraction[Sqrt[10], 300] (* Vladimir Joseph Stephan Orlovsky, Mar 04 2011 *)

Prog

(PARI) contfrac(sqrt(10)) \\ For illustration.
(PARI) A040006(n)=if(n, 6, 3) \\ M. F. Hasler, Nov 02 2019
(Magma) [6-3*(Binomial(2*n, n) mod 2): n in [0..100]]; // Vincenzo Librandi, Jan 03 2016

Crossrefs

Cf. A010467 (decimal expansion), A005667(n+1)/A005668(n+1) (convergents), A248239 (Egyptian fraction).

Cf. A040000.

Sequence in context: A292165 A327576 A331058 * A358548 A155067 A094011

Adjacent sequences: A040003 A040004 A040005 * A040007 A040008 A040009

Keyword

nonn,cofr,easy

Author

N. J. A. Sloane

Status

approved