A086702 Decimal expansion of Lévy's constant.

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

Offset: 1

Author

Benoit Cloitre, Jul 28 2003

Keywords

• cons
• nonn

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

Examples

			3.27582291872181115978768...
		

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.

Crossrefs

Cf. A002210, A100199.

Programs

Magma

C := ComplexField(); [Exp((Pi(C))^2/(12*Log(2)))]; // G. C. Greubel, Nov 06 2017

Mathematica

RealDigits[E^(Pi^2/Log[4096]), 10, 111][[1]] (* Robert G. Wilson v, May 19 2004 *)

PARI

exp(Pi^2/12/log(2)) \\ Michel Marcus, Apr 18 2015

Formula

L = exp(Pi^2/(12*log(2))).

Extensions

Offset corrected by R. J. Mathar, Feb 05 2009