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
Examples
1.351131574491659001793868005256521068360651508742701687345147211... (Only the first 31 digits are the same as those given by Flajolet & Vallée. - _Jean-François Alcover_, Apr 23 2015)
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
- 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.
Crossrefs
Cf. A099218.
Programs
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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 *)
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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
Formula
Equals (-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
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.
Comments