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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

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, chap IX, pp. 316-320.

LINKS

G. C. Greubel, Table of n, a(n) for n = 1..10000

Steven R. Finch, Khintchine's Constant [Broken link]

Steven R. Finch, Khintchine's Constant [From the Wayback machine]

Simon Plouffe, The Levy constant

Eric Weisstein's World of Mathematics, Levy Constant

Eric Weisstein's World of Mathematics, Khinchin Constant

Eric Weisstein's World of Mathematics, Continued Fraction

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: A307544 A126314 A200714 * A156140 A324556 A069888

Adjacent sequences:  A086699 A086700 A086701 * A086703 A086704 A086705

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 October 15 10:15 EDT 2019. Contains 328026 sequences. (Running on oeis4.)