OFFSET
1,1
COMMENTS
Let P(k)/Q(k) denote the k-th convergent of x. Then for almost all irrational values of x, lim_{k->inf} Q(k)^(1/k) = L. [edited by Jared Kish, Oct 17 2014; edited by A.H.M. Smeets, Jun 26 2018]
The conditions for x, such that lim_{k->inf} Q(k)^(1/k) = L, are that the terms occurring in the continued fraction for the value of x must satisfy the Gauss-Kuzmin distribution and the terms must occur in random order in the continued fraction sequence. - A.H.M. Smeets, Jun 26 2018
Named after the French mathematician Paul Lévy (1886 - 1971). - Amiram Eldar, Sep 25 2022
REFERENCES
Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, pp. 59-65.
Paul Lévy, Théorie de l'addition des variables aléatoires, 2nd. ed., Editions Jacques Gabay, 1954, chap. IX, pp. 316-320.
LINKS
G. C. Greubel, Table of n, a(n) for n = 1..10000
Paul Lévy, Sur le développement en fraction continue d'un nombre choisi au hasard, Compositio Mathematica, Vol. 3 (1936), pp. 286-303.
Steven R. Finch, Khintchine's Constant. [Broken link]
Steven R. Finch, Khintchine's Constant. [From the Wayback machine]
Simon Plouffe, The Levy constant. [broken link]
Eric Weisstein's World of Mathematics, Levy Constant.
Eric Weisstein's World of Mathematics, Khinchin Constant.
Eric Weisstein's World of Mathematics, Continued Fraction.
Wikipedia, Lévy's constant.
FORMULA
L = exp(Pi^2/(12*log(2))).
EXAMPLE
3.27582291872181115978768...
MATHEMATICA
RealDigits[E^(Pi^2/Log[4096]), 10, 111][[1]] (* Robert G. Wilson v, May 19 2004 *)
PROG
(PARI) exp(Pi^2/12/log(2)) \\ Michel Marcus, Apr 18 2015
(Magma) C<i> := ComplexField(); [Exp((Pi(C))^2/(12*Log(2)))]; // G. C. Greubel, Nov 06 2017
CROSSREFS
KEYWORD
cons,nonn
AUTHOR
Benoit Cloitre, Jul 28 2003
EXTENSIONS
Offset corrected by R. J. Mathar, Feb 05 2009
STATUS
approved