A074903 Decimal expansion of the mean number of iterations in comparing two numbers via their continued fractions.

1, 3, 5, 1, 1, 3, 1, 5, 7, 4, 4, 9, 1, 6, 5, 9, 0, 0, 1, 7, 9, 3, 8, 6, 8, 0, 0, 5, 2, 5, 6, 5, 2, 1, 0, 6, 8, 3, 6, 0, 6, 5, 1, 5, 0, 8, 7, 4, 2, 7, 0, 1, 6, 8, 7, 3, 4, 5, 1, 4, 7, 2, 1, 1, 0, 1, 3, 7, 4, 2, 2, 7, 7, 1, 1, 9, 5, 5, 0, 1, 7, 1, 2, 8, 6, 9, 1, 3, 0, 7, 5, 1, 5, 9, 7, 8, 0, 2, 3, 9

Offset

1,2

Comments

Another description: Decimal expansion of the mean number of comparisons (moment sum of index 2) in the basic continued fraction sign algorithm ("BCF-sign").

Still another description: Decimal expansion of expected number of iterations of Gaussian reduction of a 2-dimensional lattice.

References

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, page 161.

Philippe Flajolet and Brigitte Vallée, Continued fraction algorithms and constants, in "Constructive, Experimental, and Nonlinear Analysis", Michel Théra Editor, CMS Conference Proceedings, Canadian Mathematical Society Volume 27 (2000), p. 67.

Links

Table of n, a(n) for n=1..100.

H. Daude, P. Flajolet and B. Vallee, An average-case analysis of the Gaussian algorithm for lattice reduction, INRIA, 1996. [alternative link]

Philippe Flajolet, Continued Fractions, Comparison Algorithms and Fine Structure Constants.

Eric Weisstein's World of Mathematics, Polylogarithm.

Eric Weisstein's World of Mathematics, Vallée Constant.

Formula

(-60/Pi^4)*(24*Li_4(1/2) - Pi^2*log(2)^2 + 21*zeta(3)*log(2) + log(2)^4) + 17, with Li_4 the tetralogarithm function. - Jean-François Alcover, Apr 23 2015

Example

1.351131574491659001793868005256521068360651508742701687345147211...
(Only the first 31 digits are the same as those given by Flajolet & Vallée. - Jean-François Alcover, Apr 23 2015)

Mathematica

17 - 60/Pi^4 (24*PolyLog[4, 1/2] - Pi^2*Log[2]^2 + 21*Zeta[3]*Log[2] + Log[2]^4) // RealDigits[#, 10, 100]& // First (* Jean-François Alcover, Mar 19 2013, after Steven Finch *)

Prog

(PARI) 17 - 60*(24*polylog(4, 1/2) - Pi^2*log(2)^2 + 21*zeta(3)*log(2) + log(2)^4)/Pi^4 \\ Charles R Greathouse IV, Aug 27 2014

Crossrefs

Cf. A099218.

Sequence in context: A307860 A123701 A143303 * A091084 A016610 A356551

Adjacent sequences: A074900 A074901 A074902 * A074904 A074905 A074906

Keyword

nonn,cons

Author

N. J. A. Sloane, Sep 15 2002

Extensions

Corrected and extended by Jean-François Alcover, Mar 19 2013

Entry revised by N. J. A. Sloane, Apr 24 2015 to include information from two other entries (submitted respectively by Eric W. Weisstein, Aug 05 2008 and Jean-François Alcover, Apr 23 2015) that formerly described this same constant.

Status

approved