%I #50 Dec 03 2024 09:17:53

%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 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

%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,changed

%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.