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A086702 Decimal expansion of Lévy's constant.

%I A086702 #79 Feb 16 2025 08:32:50
%S A086702 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,
%T A086702 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,
%U A086702 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
%N A086702 Decimal expansion of Lévy's constant.
%C A086702 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]
%C A086702 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
%C A086702 Named after the French mathematician Paul Lévy (1886 - 1971). - _Amiram Eldar_, Sep 25 2022
%D A086702 Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, pp. 59-65.
%D A086702 Paul Lévy, Théorie de l'addition des variables aléatoires, 2nd. ed., Editions Jacques Gabay, 1954, chap. IX, pp. 316-320.
%H A086702 G. C. Greubel, <a href="/A086702/b086702.txt">Table of n, a(n) for n = 1..10000</a>
%H A086702 Paul Lévy, <a href="http://www.numdam.org/item/?id=CM_1936__3__286_0">Sur le développement en fraction continue d'un nombre choisi au hasard</a>, Compositio Mathematica, Vol. 3 (1936), pp. 286-303.
%H A086702 Steven R. Finch, <a href="http://www.people.fas.harvard.edu/~sfinch/constant/khntchn/khntchn.html">Khintchine's Constant</a>. [Broken link]
%H A086702 Steven R. Finch, <a href="http://web.archive.org/web/20011108021950/http://www.mathsoft.com/asolve/constant/khntchn/khntchn.html">Khintchine's Constant</a>. [From the Wayback machine]
%H A086702 Simon Plouffe, <a href="http://www.worldwideschool.org/library/books/sci/math/MiscellaneousMathematicalConstants/chap55.html">The Levy constant</a>. [broken link]
%H A086702 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LevyConstant.html">Levy Constant</a>.
%H A086702 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/KhinchinConstant.html">Khinchin Constant</a>.
%H A086702 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ContinuedFraction.html">Continued Fraction</a>.
%H A086702 Wikipedia, <a href="https://en.wikipedia.org/wiki/L%C3%A9vy%27s_constant">Lévy's constant</a>.
%F A086702 L = exp(Pi^2/(12*log(2))).
%e A086702 3.27582291872181115978768...
%t A086702 RealDigits[E^(Pi^2/Log[4096]), 10, 111][[1]] (* _Robert G. Wilson v_, May 19 2004 *)
%o A086702 (PARI) exp(Pi^2/12/log(2)) \\ _Michel Marcus_, Apr 18 2015
%o A086702 (Magma) C<i> := ComplexField(); [Exp((Pi(C))^2/(12*Log(2)))]; // _G. C. Greubel_, Nov 06 2017
%Y A086702 Cf. A002210, A100199.
%K A086702 cons,nonn
%O A086702 1,1
%A A086702 _Benoit Cloitre_, Jul 28 2003
%E A086702 Offset corrected by _R. J. Mathar_, Feb 05 2009