A086231 Decimal expansion of value of Watson's integral.

1, 5, 1, 6, 3, 8, 6, 0, 5, 9, 1, 5, 1, 9, 7, 8, 0, 1, 8, 1, 5, 6, 0, 1, 2, 1, 5, 9, 6, 8, 1, 4, 2, 0, 7, 7, 9, 9, 5, 5, 3, 8, 7, 0, 4, 4, 4, 5, 2, 2, 6, 2, 6, 7, 6, 5, 6, 6, 9, 8, 0, 4, 6, 3, 6, 5, 8, 0, 8, 6, 3, 2, 0, 3, 5, 3, 5, 2, 1, 4, 5, 0, 4, 0, 1, 6, 1, 1, 7, 4, 1, 2, 0, 9, 6, 8, 8, 1, 1, 3, 9, 2

Offset

1,2

Links

G. C. Greubel, Table of n, a(n) for n = 1..10000

Steven R. Finch, Errata and Addenda to Mathematical Constants, p. 40.

M. Lawrence Glasser and I. John Zucker, Extended Watson integrals for the cubic lattices, Proceedings of the National Academy of Sciences, Vol. 74, No. 5 (1977), pp. 1800-1801, alternative link.

Anthony J. Guttmann, Lattice Green's functions in all dimensions, J. Phys. A.: Math. Theor., Vol. 43, No. 30 (2010) 305205.

George N. Watson, Three triple integrals, The Quarterly Journal of Mathematics, Vol. os-10, No. 1 (1939), pp. 266-276.

Eric Weisstein's World of Mathematics, Pólya's Random Walk Constants.

Formula

Equals (sqrt(3)-1)*(gamma(1/24)*gamma(11/24))^2/(32*Pi^3). - G. C. Greubel, Jan 07 2018

Equals 1/(1 - A086230). - Amiram Eldar, Aug 28 2020

Equals Sum_{k>=0} A002896(k)/36^k. - Vaclav Kotesovec, Apr 23 2023

Example

1.51638605915197801815601215968142077995538704445226267656698...

Maple

evalf((sqrt(3)-1)*(GAMMA(1/24)*GAMMA(11/24))^2 / (32*Pi^3), 120); # Vaclav Kotesovec, Sep 16 2014

Mathematica

RealDigits[ N[ Sqrt[6]/32/Pi^3*Gamma[1/24]*Gamma[5/24]*Gamma[7/24]*Gamma[11/24], 102]][[1]] (* Jean-François Alcover, Nov 12 2012, after Eric W. Weisstein *)

Prog

(PARI) (sqrt(3)-1)*(gamma(1/24)*gamma(11/24))^2 / (32*Pi^3) \\ Altug Alkan, Apr 13 2016
(Magma) C<i> := ComplexField(); (Sqrt(3)-1)*(Gamma(1/24)*Gamma(11/24))^2/(32*Pi(C)^3) // G. C. Greubel, Jan 07 2018

Crossrefs

Cf. A002896, A086230, A242812, A242813, A242814, A242815, A242816, A273086.

Sequence in context: A180595 A088401 A077491 * A201419 A163336 A173898

Adjacent sequences: A086228 A086229 A086230 * A086232 A086233 A086234

Keyword

nonn,cons

Author

Eric W. Weisstein, Jul 12 2003

Status

approved