A242814 Decimal expansion of the expected number of returns to the origin of a random walk on a 6-d lattice.

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

Offset

1,4

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.

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 - A086234). - Amiram Eldar, Aug 28 2020

Example

1.1169633732...

Maple

m6:= int(exp(-t)*BesselI(0, t/6)^6, t=0..infinity):
s:= convert(evalf(m6, 120), string):
map(parse, subs("."=NULL, [seq(i, i=s)]))[]; # Alois P. Heinz, May 23 2014

Mathematica

d = 6; 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, 8]& // First

Prog

(PARI) intnumosc(t=0, exp(-t)*besseli(0, t/6)^6, 12*Pi) \\ Charles R Greathouse IV, Oct 23 2023

Crossrefs

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

Sequence in context: A330594 A371534 A196462 * A239808 A198144 A154394

Adjacent sequences: A242811 A242812 A242813 * A242815 A242816 A242817

Keyword

nonn,cons

Author

Jean-François Alcover, May 23 2014

Extensions

More terms from Alois P. Heinz, May 23 2014

Status

approved