A293238 Decimal expansion of the escape probability for a random walk on the 3D bcc lattice.

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

Offset

0,1

Comments

The return probability equals unity minus this constant. The expected number of visits to the origin is the inverse of this constant, A091670.

Links

G. C. Greubel, Table of n, a(n) for n = 0..10000

Shunya Ishioka and Masahiro Koiwa, Random walks on diamond and hexagonal close packed lattices, Phil. Mag. A, 37 (1978), 517-533.

G. L. Montet, Integral methods in the calculation of correlation factors in diffusion, Phys. Rev. B 7 (1973), 650-662.

Index entries for sequences related to b.c.c. lattice.

Index entries for sequences related to walks.

Formula

Equals Pi^2/(4*K(1/sqrt(2))^2), where K is the complete elliptic integral of the first kind.

Equals (4*Pi^3)/Gamma(1/4)^4. - G. C. Greubel, Oct 26 2018

Equals Product_{n>=1} exp(beta(2n)/n), where beta(n) is the Dirichlet beta function. - Antonio Graciá Llorente, Apr 03 2025

Equals Gamma(3/4)^4/Pi. - Stefano Spezia, Apr 05 2025

Conjecture: Equals Sum_{k>=0} A004405(k) / exp(k*Pi). - Simon Plouffe, Sep 09 2025

Example

0.7177700110461299978211932236657794...

Mathematica

RealDigits[(4*Pi^3)/Gamma[1/4]^4, 10, 100][[1]] (* G. C. Greubel, Oct 26 2018 *)

Prog

(PARI) default(realprecision, 100); (4*Pi^3)/gamma(1/4)^4 \\ G. C. Greubel, Oct 26 2018
(Magma) SetDefaultRealField(RealField(100)); R:= RealField(); (4*Pi(R)^3)/Gamma(1/4)^4; // G. C. Greubel, Oct 26 2018

Crossrefs

Cf. A004405, A091670, A242761, A293237.

Sequence in context: A010772 A383824 A199732 * A210708 A210529 A151785

Adjacent sequences: A293235 A293236 A293237 * A293239 A293240 A293241

Keyword

nonn,cons

Author

Andrey Zabolotskiy, Oct 03 2017

Status

approved