A040030 Continued fraction for sqrt(37).

6, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• cofr
• easy
• changed

Examples

			6.08276253029821968899968... = 6 + 1/(12 + 1/(12 + 1/(12 + 1/(12 + ...)))). - _Harry J. Smith_, Jun 04 2009
		

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. A010491 (decimal expansion), A041060/A041061 (convergents), A248263 (Egyptian fraction).

Cf. A040000, A040002, A040006, A010851

Programs

Maple

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

Mathematica

ContinuedFraction[Sqrt[37],300] (* Vladimir Joseph Stephan Orlovsky, Mar 06 2011 *)
PadRight[{6},120,{12}] (* Harvey P. Dale, Jan 02 2017 *)

PARI

{ allocatemem(932245000); default(realprecision, 44000); x=contfrac(sqrt(37)); for (n=0, 20000, write("b040030.txt", n, " ", x[n+1])); } \\ Harry J. Smith, Jun 04 2009

Formula

From Elmo R. Oliveira, Feb 06 2024: (Start)

a(n) = 12 for n >= 1.

G.f.: 6*(1+x)/(1-x).

E.g.f.: 12*exp(x) - 6.

a(n) = 6*A040000(n) = 3*A040002(n) = 2*A040006(n). (End)