A010130 Continued fraction for sqrt(32).

5, 1, 1, 1, 10, 1, 1, 1, 10, 1, 1, 1, 10, 1, 1, 1, 10, 1, 1, 1, 10, 1, 1, 1, 10, 1, 1, 1, 10, 1, 1, 1, 10, 1, 1, 1, 10, 1, 1, 1, 10, 1, 1, 1, 10, 1, 1, 1, 10, 1, 1, 1, 10, 1, 1, 1, 10, 1, 1, 1, 10, 1, 1, 1, 10, 1, 1, 1, 10, 1, 1, 1, 10

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• cofr
• easy
• mult

Examples

			5.65685424949238019520675489... = 5 + 1/(1 + 1/(1 + 1/(1 + 1/(10 + ...)))). - _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 (0,0,0,1).

Crossrefs

Cf. A010487 (decimal expansion).

Cf. A041052/A041053 (convergents), A248259 (Egyptian fraction).

Programs

Mathematica

ContinuedFraction[Sqrt[32],300] (* Vladimir Joseph Stephan Orlovsky, Mar 06 2011 *)
PadRight[{5},100,{10,1,1,1}] (* Harvey P. Dale, Aug 20 2014 *)

PARI

{ allocatemem(932245000); default(realprecision, 16000); x=contfrac(sqrt(32)); for (n=0, 20000, write("b010130.txt", n, " ", x[n+1])); } \\ Harry J. Smith, Jun 04 2009

Formula

From Amiram Eldar, Nov 12 2023: (Start)

Multiplicative with a(2) = 1, a(2^e) = 10 for e >= 2, and a(p^e) = 1 for an odd prime p.

Dirichlet g.f.: zeta(s) * (1 + 9/4^s). (End)

From Elmo R. Oliveira, Aug 05 2024: (Start)

G.f.: (5 + x + x^2 + x^3 + 5*x^4)/((1 - x)*(1 + x + x^2 + x^3)).

a(n) = a(n-4), n > 4. (End)