A242816 Decimal expansion of the expected number of returns to the origin of a random walk on an 8-d lattice.

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

Offset: 1

Author

Jean-François Alcover, May 23 2014

Keywords

• nonn
• cons

Examples

			1.0786470120...
		

References

• Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.9 Polya's random walk constants, p. 323.

Links

• Marc Mezzarobba, Table of n, a(n) for n = 1..10000
• Eric Weisstein's World of Mathematics, Pólya's Random Walk Constants.

Crossrefs

Cf. A086230, A086231, A086232, A086233, A086234, A086235, A086236, A242812, A242813, A242814, A242815.

Programs

Maple

m8:= int(exp(-t)*BesselI(0, t/8)^8, t=0..infinity):
s:= convert(evalf(m8, 120), string):
map(parse, subs("."=NULL, [seq(i, i=s)]))[]; # Alois P. Heinz, May 23 2014

Mathematica

d = 8; d/Pi^d*NIntegrate[(d - Sum[Cos[t[k]], {k, 1, d}])^-1, Sequence @@ Table[{t[k], 0, Pi}, {k, 1, d}] // Evaluate] // RealDigits[#, 10, 7]& // First

Formula

m(d) = d/(2*Pi)^d*multipleIntegral(-Pi..Pi) (d-sum_(k=1..d) cos(t_k))^(-1) dt_1 dt_2 ... dt_d, where d is the lattice dimension.

m(d) = Integral_{t>0} exp(-t)*BesselI(0,t/d)^d dt where BesselI(0,x) is the zeroth modified Bessel function.

Equals 1/(1 - A086236). - Amiram Eldar, Aug 28 2020

Extensions

More terms from Alois P. Heinz, May 23 2014