A086230 Decimal expansion of probability that a random walk on a 3-D lattice returns to the origin.

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, 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, 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

Offset: 0

Author

Eric W. Weisstein, Jul 12 2003

Keywords

• nonn
• cons

Comments

Pólya (1921) proved that this constant is < 1. McCrea and Whipple (1940) evaluated it by 0.34. - Amiram Eldar, Aug 28 2020

Examples

			0.340537329550999142826273184432902896710608217124302097763236105377791969...
		

References

• Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, p. 322-331.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..10000
• W. H. McCrea and F. J. W. Whipple, Random Paths in Two and Three Dimensions, Proceedings of the Royal Society of Edinburgh, Vol. 60, No. 3 (1940), pp. 281-298. See p. 297.
• Georg Pólya, Über eine Aufgabe der Wahrscheinlichkeitsrechnung betreffend die Irrfahrt im Straßennetz, Mathematische Annalen, Vol. 84, No. 1-2 (1921), pp. 149-160.
• Eric Weisstein's World of Mathematics, Pólya's Random Walk Constants.

Crossrefs

Cf. A086231, A086232, A086233, A086234, A086235, A086236.

Programs

Magma

C := 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

Mathematica

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 *)

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

Formula

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

Equals 1 - 1/A086231. - Amiram Eldar, Aug 28 2020