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

Author

Andrey Zabolotskiy, Oct 03 2017

Keywords

• nonn
• cons

Comments

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

Examples

			0.7177700110461299978211932236657794...
		

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.

Crossrefs

Cf. A091670, A242761, A293237.

Programs

Magma

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

Mathematica

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

PARI

default(realprecision, 100); (4*Pi^3)/gamma(1/4)^4 \\ G. C. Greubel, Oct 26 2018

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