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A086702 Decimal expansion of Lévy's constant. 14
3, 2, 7, 5, 8, 2, 2, 9, 1, 8, 7, 2, 1, 8, 1, 1, 1, 5, 9, 7, 8, 7, 6, 8, 1, 8, 8, 2, 4, 5, 3, 8, 4, 3, 8, 6, 3, 6, 0, 8, 4, 7, 5, 5, 2, 5, 9, 8, 2, 3, 7, 4, 1, 4, 9, 4, 0, 5, 1, 9, 8, 9, 2, 4, 1, 9, 0, 7, 2, 3, 2, 1, 5, 6, 4, 4, 9, 6, 0, 3, 5, 5, 1, 8, 1, 2, 7, 7, 5, 4, 0, 4, 7, 9, 1, 7, 4, 5, 2, 9, 4, 9, 2, 6, 9 (list; constant; graph; refs; listen; history; text; internal format)
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
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
Cf. A002210.
Sequence in context: A358139 A358654 A200714 * A156140 A324556 A069888
KEYWORD
cons,nonn
AUTHOR
Benoit Cloitre, Jul 28 2003
EXTENSIONS
Offset corrected by R. J. Mathar, Feb 05 2009
STATUS
approved

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Last modified April 13 18:35 EDT 2024. Contains 371644 sequences. (Running on oeis4.)