A010131 Continued fraction for sqrt(33).

5, 1, 2, 1, 10, 1, 2, 1, 10, 1, 2, 1, 10, 1, 2, 1, 10, 1, 2, 1, 10, 1, 2, 1, 10, 1, 2, 1, 10, 1, 2, 1, 10, 1, 2, 1, 10, 1, 2, 1, 10, 1, 2, 1, 10, 1, 2, 1, 10, 1, 2, 1, 10, 1, 2, 1, 10, 1, 2, 1, 10, 1, 2, 1, 10, 1, 2, 1, 10, 1, 2, 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) = 2, a(2^e) = 10 for e >= 2, and a(p^e) = 1 for an odd prime p.

Dirichlet g.f.: zeta(s) * (1 + 1/2^(2*s-3) + 1/2^s). (End)

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

Example

5.74456264653802865985061146... = 5 + 1/(1 + 1/(2 + 1/(1 + 1/(10 + ...)))). - Harry J. Smith, Jun 04 2009

Mathematica

ContinuedFraction[Sqrt[33], 300] (* Vladimir Joseph Stephan Orlovsky, Mar 06 2011 *)
LinearRecurrence[{0, 0, 0, 1}, {5, 1, 2, 1, 10}, 100] (* or *) PadRight[{5}, 100, {10, 1, 2, 1}] (* Harvey P. Dale, Dec 31 2017 *)

Prog

(PARI) { allocatemem(932245000); default(realprecision, 17000); x=contfrac(sqrt(33)); for (n=0, 20000, write("b010131.txt", n, " ", x[n+1])); } \\ Harry J. Smith, Jun 04 2009

Crossrefs

Cf. A010488 (decimal expansion).

Sequence in context: A115038 A231990 A369915 * A176323 A260821 A068115

Adjacent sequences: A010128 A010129 A010130 * A010132 A010133 A010134

Keyword

nonn,cofr,easy,mult

Author

N. J. A. Sloane

Status

approved