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A086230 Decimal expansion of probability that a random walk on a 3-D lattice returns to the origin. 14
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 (list; constant; graph; refs; listen; history; text; internal format)
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
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
Sequence in context: A021750 A246770 A197809 * A197485 A158677 A337164
KEYWORD
nonn,cons
AUTHOR
Eric W. Weisstein, Jul 12 2003
STATUS
approved

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Last modified June 14 21:35 EDT 2024. Contains 373401 sequences. (Running on oeis4.)