cp's OEIS Frontend

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

%I A242816 #22 Feb 16 2025 08:33:22
%S A242816 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,
%T A242816 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,
%U A242816 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
%N A242816 Decimal expansion of the expected number of returns to the origin of a random walk on an 8-d lattice.
%D A242816 Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.9 Polya's random walk constants, p. 323.
%H A242816 Marc Mezzarobba, <a href="/A242816/b242816.txt">Table of n, a(n) for n = 1..10000</a>
%H A242816 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PolyasRandomWalkConstants.html">Pólya's Random Walk Constants</a>.
%F A242816 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.
%F A242816 m(d) = Integral_{t>0} exp(-t)*BesselI(0,t/d)^d dt where BesselI(0,x) is the zeroth modified Bessel function.
%F A242816 Equals 1/(1 - A086236). - _Amiram Eldar_, Aug 28 2020
%e A242816 1.0786470120...
%p A242816 m8:= int(exp(-t)*BesselI(0, t/8)^8, t=0..infinity):
%p A242816 s:= convert(evalf(m8, 120), string):
%p A242816 map(parse, subs("."=NULL, [seq(i, i=s)]))[]; # _Alois P. Heinz_, May 23 2014
%t A242816 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
%Y A242816 Cf. A086230, A086231, A086232, A086233, A086234, A086235, A086236, A242812, A242813, A242814, A242815.
%K A242816 nonn,cons
%O A242816 1,3
%A A242816 _Jean-François Alcover_, May 23 2014
%E A242816 More terms from _Alois P. Heinz_, May 23 2014