A240946 Decimal expansion of the average distance traveled in three steps of length 1 for a random walk in the plane starting at the origin.

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, 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, 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

Offset

1,2

Links

Table of n, a(n) for n=1..100.

J. M. Borwein, A. Straub, J. Wan, and W. Zudilin, Densities of short uniform random walks, arXiv:1103.2995 [math.CA], (11-August-2011)

Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 464.

Formula

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.

Re(3F2(-1/2, -1/2, 1/2; 1, 1; 4)).

(3*2^(1/3))/(16*Pi^4)*Gamma(1/3)^6 + (27*2^(2/3))/(4*Pi^4)*Gamma(2/3)^6.

Example

1.5745972375518936574946921830765...

Mathematica

(3*2^(1/3))/(16*Pi^4)*Gamma[1/3]^6 + (27*2^(2/3))/(4*Pi^4)*Gamma[2/3]^6 //
  RealDigits[#, 10, 100]& // First (* updated May 20 2015 *)

Crossrefs

Cf. A088538 (two steps).

Sequence in context: A010488 A300081 A293843 * A343039 A021178 A201506

Adjacent sequences: A240943 A240944 A240945 * A240947 A240948 A240949

Keyword

nonn,cons,walk

Author

Jean-François Alcover, Aug 04 2014

Extensions

More digits from Jean-François Alcover, May 20 2015

Status

approved