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

Author

N. J. A. Sloane

Keywords

• nonn
• cofr
• easy

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

Examples

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

References

• 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 (1).

Crossrefs

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

Cf. A040000.

Programs

Magma

[6-3*(Binomial(2*n,n) mod 2): n in [0..100]]; // Vincenzo Librandi, Jan 03 2016

Maple

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

Mathematica

ContinuedFraction[Sqrt[10],300] (* Vladimir Joseph Stephan Orlovsky, Mar 04 2011 *)
PadRight[{3},120,{6}] (* Harvey P. Dale, Aug 25 2024 *)

PARI

contfrac(sqrt(10)) \\ For illustration.

PARI

A040006(n)=if(n,6,3) \\ M. F. Hasler, Nov 02 2019

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)