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%I #27 Sep 08 2022 08:45:11
%S 3,4,0,5,3,7,3,2,9,5,5,0,9,9,9,1,4,2,8,2,6,2,7,3,1,8,4,4,3,2,9,0,2,8,
%T 9,6,7,1,0,6,0,8,2,1,7,1,2,4,3,0,2,0,9,7,7,6,3,2,3,6,1,0,5,3,7,7,7,9,
%U 1,9,6,9,4,5,8,9,6,2,3,8,4,6,4,2,5,2,8,0,8,1,8,8,9,0,5,7,1,3,0,9,9,4
%N Decimal expansion of probability that a random walk on a 3-D lattice returns to the origin.
%C Pólya (1921) proved that this constant is < 1. McCrea and Whipple (1940) evaluated it by 0.34. - _Amiram Eldar_, Aug 28 2020
%D Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, p. 322-331.
%H G. C. Greubel, <a href="/A086230/b086230.txt">Table of n, a(n) for n = 0..10000</a>
%H W. H. McCrea and F. J. W. Whipple, <a href="https://doi.org/10.1017/S0370164600020265">Random Paths in Two and Three Dimensions</a>, Proceedings of the Royal Society of Edinburgh, Vol. 60, No. 3 (1940), pp. 281-298. See p. 297.
%H Georg Pólya, <a href="https://doi.org/10.1007/BF01458701">Über eine Aufgabe der Wahrscheinlichkeitsrechnung betreffend die Irrfahrt im Straßennetz</a>, Mathematische Annalen, Vol. 84, No. 1-2 (1921), pp. 149-160.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PolyasRandomWalkConstants.html">Pólya's Random Walk Constants</a>.
%F Equals 1 - (16*Sqrt(2/3)*Pi^3)/(Gamma(1/24)* Gamma(5/24)*Gamma(7/24)* Gamma(11/24)). - _G. C. Greubel_, Jan 25 2018
%F Equals 1 - 1/A086231. - _Amiram Eldar_, Aug 28 2020
%e 0.340537329550999142826273184432902896710608217124302097763236105377791969...
%t RealDigits[1 - (16*Sqrt[2/3]*Pi^3) / (Gamma[1/24]*Gamma[5/24]*Gamma[7/24]*Gamma[11/24]), 10, 102] // First (* _Jean-François Alcover_, Feb 08 2013, after _Eric W. Weisstein_ *)
%o (PARI) 1-32*Pi^3/sqrt(6)/gamma(1/24)/gamma(5/24)/gamma(7/24)/gamma(11/24) \\ _Charles R Greathouse IV_, Jul 22 2013
%o (Magma) C<i> := ComplexField(); 1 - (16*Sqrt(2/3)*Pi(C)^3)/(Gamma(1/24)* Gamma(5/24)*Gamma(7/24)*Gamma(11/24)); // _G. C. Greubel_, Jan 25 2018
%Y Cf. A086231, A086232, A086233, A086234, A086235, A086236.
%K nonn,cons
%O 0,1
%A _Eric W. Weisstein_, Jul 12 2003
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