

A074903


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


3



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
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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 ("BCFsign").
Still another description: Decimal expansion of expected number of iterations of Gaussian reduction of a 2dimensional 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 averagecase 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 MathWorld, 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.  JeanFrançois Alcover, Apr 23 2015


EXAMPLE

1.351131574491659001793868005256521068360651508742701687345147211...
(Only the first 31 digits are the same as those given by Flajolet & Vallée.  JeanFrançois 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 (* JeanFrançois 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 A305470
Adjacent sequences: A074900 A074901 A074902 * A074904 A074905 A074906


KEYWORD

nonn,cons


AUTHOR

N. J. A. Sloane, Sep 15 2002


EXTENSIONS

Corrected and extended by JeanFrançois 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 JeanFrançois Alcover, Apr 23 2015) that formerly described this same constant.


STATUS

approved



