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

0,1

Comments

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

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.

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

Example

0.340537329550999142826273184432902896710608217124302097763236105377791969...

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 *)

Prog

(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
(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

Crossrefs

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

Sequence in context: A021750 A246770 A197809 * A197485 A158677 A337164

Adjacent sequences: A086227 A086228 A086229 * A086231 A086232 A086233

Keyword

nonn,cons

Author

Eric W. Weisstein, Jul 12 2003

Status

approved