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%I #25 Sep 22 2023 11:59:28
%S 1,5,7,4,5,9,7,2,3,7,5,5,1,8,9,3,6,5,7,4,9,4,6,9,2,1,8,3,0,7,6,5,1,9,
%T 6,9,0,2,2,1,6,6,6,1,8,0,7,5,8,5,1,9,1,7,0,1,9,3,6,9,3,0,9,8,3,0,1,8,
%U 3,1,1,8,0,5,9,4,4,5,4,3,8,2,1,3,1,0,8,5,3,1,3,3,6,2,2,4,1,9,5,3
%N Decimal expansion of the average distance traveled in three steps of length 1 for a random walk in the plane starting at the origin.
%H J. M. Borwein, A. Straub, J. Wan, and W. Zudilin, <a href="http://arxiv.org/abs/1103.2995">Densities of short uniform random walks</a>, arXiv:1103.2995 [math.CA], (11-August-2011)
%H Steven R. Finch, <a href="https://doi.org/10.1017/9781316997741">Mathematical Constants II</a>, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 464.
%F Integral_(0..3) x*p(x) dx, where p(x) = 2*sqrt(3)/Pi*x/(3+x^2) * 2F1(1/3, 2/3; 1; x^2*(9-x^2)^2/(3+x^2)^3), 2F1 being the hypergeometric function.
%F Re(3F2(-1/2, -1/2, 1/2; 1, 1; 4)).
%F (3*2^(1/3))/(16*Pi^4)*Gamma(1/3)^6 + (27*2^(2/3))/(4*Pi^4)*Gamma(2/3)^6.
%e 1.5745972375518936574946921830765...
%t (3*2^(1/3))/(16*Pi^4)*Gamma[1/3]^6 + (27*2^(2/3))/(4*Pi^4)*Gamma[2/3]^6 //
%t RealDigits[#, 10, 100]& // First (* updated May 20 2015 *)
%Y Cf. A088538 (two steps).
%K nonn,cons,walk
%O 1,2
%A _Jean-François Alcover_, Aug 04 2014
%E More digits from _Jean-François Alcover_, May 20 2015
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