%I #45 Feb 28 2023 04:09:13
%S 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,
%T 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,
%U 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
%N Decimal expansion of the mean number of iterations in comparing two numbers via their continued fractions.
%C Another description: Decimal expansion of the mean number of comparisons (moment sum of index 2) in the basic continued fraction sign algorithm ("BCF-sign").
%C Still another description: Decimal expansion of expected number of iterations of Gaussian reduction of a 2-dimensional lattice.
%D Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, page 161.
%D 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.
%H H. Daude, P. Flajolet and B. Vallee, <a href="http://algo.inria.fr/flajolet/Publications/RR2798.ps.gz">An average-case analysis of the Gaussian algorithm for lattice reduction</a>, INRIA, 1996. [<a href="https://www.semanticscholar.org/paper/An-Average-Case-Analysis-of-the-Gaussian-Algorithm-Daud%C3%A9-Flajolet/5783d4bfc398819b106c216bfea14347dd58550b">alternative link</a>]
%H Philippe Flajolet, <a href="http://algo.inria.fr/seminars/sem99-00/flajolet.pdf">Continued Fractions, Comparison Algorithms and Fine Structure Constants</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Polylogarithm.html">Polylogarithm</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/ValleeConstant.html">Vallée Constant</a>.
%F (-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
%e 1.351131574491659001793868005256521068360651508742701687345147211...
%e (Only the first 31 digits are the same as those given by Flajolet & Vallée. - _Jean-François Alcover_, Apr 23 2015)
%t 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_ *)
%o (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
%Y Cf. A099218.
%K nonn,cons
%O 1,2
%A _N. J. A. Sloane_, Sep 15 2002
%E Corrected and extended by _Jean-François Alcover_, Mar 19 2013
%E 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.