A040012 Continued fraction for sqrt(17).

4, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• cofr
• easy

Comments

Decimal expansion of 22/45. - Elmo R. Oliveira, Feb 06 2024

Examples

			4.123105625617660549821409855... = 4 + 1/(8 + 1/(8 + 1/(8 + 1/(8 + ...)))). - _Harry J. Smith_, Jun 03 2009
		

References

• Jan Gullberg, Mathematics from the Birth of Numbers, W. W. Norton & Co., NY & London, 1997, §4.4 Powers and Roots, p. 144.
• James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, Pages 275-276.

Links

• Harry J. Smith, Table of n, a(n) for n = 0..20000
• G. Xiao, Contfrac.
• Index entries for continued fractions for constants.
• Index to divisibility sequences.
• Index entries for linear recurrences with constant coefficients, signature (1).

Crossrefs

Cf. A041024/A041025 (convergents), A010473 (decimal expansion), A248245 (Egyptian fraction).

Cf. A040000.

Programs

Maple

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

Mathematica

ContinuedFraction[Sqrt[17],300] (* Vladimir Joseph Stephan Orlovsky, Mar 05 2011 *)
PadRight[{4},100,8] (* Harvey P. Dale, Jun 22 2015 *)

PARI

{ allocatemem(932245000); default(realprecision, 37000); x=contfrac(sqrt(17)); for (n=0, 20000, write("b040012.txt", n, " ", x[n+1])); } \\ Harry J. Smith, Jun 03 2009

Formula

a(n) = 4*A040000(n). - Stefano Spezia, May 14 2023

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

a(n) = 8 for n >= 1.

G.f.: 4*(1+x)/(1-x).

E.g.f.: 8*exp(x) - 4. (End)