A010125 Continued fraction for sqrt(21).

4, 1, 1, 2, 1, 1, 8, 1, 1, 2, 1, 1, 8, 1, 1, 2, 1, 1, 8, 1, 1, 2, 1, 1, 8, 1, 1, 2, 1, 1, 8, 1, 1, 2, 1, 1, 8, 1, 1, 2, 1, 1, 8, 1, 1, 2, 1, 1, 8, 1, 1, 2, 1, 1, 8, 1, 1, 2, 1, 1, 8, 1, 1, 2, 1, 1, 8, 1, 1, 2, 1, 1, 8, 1, 1, 2, 1, 1, 8, 1, 1

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• cofr
• easy

Examples

			4.582575694955840006588047193... = 4 + 1/(1 + 1/(1 + 1/(2 + 1/(1 + ...)))). - _Harry J. Smith_, Jun 03 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 (0,0,0,0,0,1).

Crossrefs

Cf. A041032/A041033 (convergents), A010477 (decimal expansion).

Programs

Mathematica

ContinuedFraction[Sqrt[21],300] (* Vladimir Joseph Stephan Orlovsky, Mar 05 2011 *)
PadRight[{4},120,{8,1,1,2,1,1}] (* Harvey P. Dale, May 16 2020 *)

PARI

{ allocatemem(932245000); default(realprecision, 14000); x=contfrac(sqrt(21)); for (n=0, 20000, write("b010125.txt", n, " ", x[n+1])); } \\ Harry J. Smith, Jun 03 2009

Formula

a(n) = gcd(Fibonacci(n), Fibonacci(n-6)), n > 0. - Gary Detlefs, Dec 29 2010

G.f.: (4 + x + x^2 + 2*x^3 + x^4 + x^5 + 4*x^6)/(1 - x^6). - Stefano Spezia, Jul 26 2025