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

0,1

References

Harold Davenport, The Higher Arithmetic, Cambridge University Press, 8th ed., 2008, p. 97.

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).

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)

Example

5.65685424949238019520675489... = 5 + 1/(1 + 1/(1 + 1/(1 + 1/(10 + ...)))). - Harry J. Smith, Jun 04 2009

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 *)

Prog

(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

Crossrefs

Cf. A010487 (decimal expansion).

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

Sequence in context: A140210 A380400 A360720 * A361063 A363972 A206773

Adjacent sequences: A010127 A010128 A010129 * A010131 A010132 A010133

Keyword

nonn,cofr,easy,mult

Author

N. J. A. Sloane

Status

approved