cp's OEIS Frontend

A242815 Decimal expansion of the expected number of returns to the origin of a random walk on a 7-d lattice.

%I A242815 #22 Feb 16 2025 08:33:22
%S A242815 1,0,9,3,9,0,6,3,1,5,5,8,7,8,4,7,9,9,6,6,8,3,2,7,1,8,2,3,5,5,9,0,1,9,
%T A242815 8,6,3,7,1,1,2,8,9,9,7,7,1,6,4,9,6,1,1,5,4,4,9,1,6,8,9,0,7,3,8,8,6,1,
%U A242815 2,6,5,4,5,7,0,5,0,8,0,5,2,2,8,4,4,8,9,5,1,9,1,9,7,2,9,8,5,5,9,8,7,5,7,2,9,9
%N A242815 Decimal expansion of the expected number of returns to the origin of a random walk on a 7-d lattice.
%D A242815 Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.9 Polya's random walk constants, p. 323.
%H A242815 Marc Mezzarobba, <a href="/A242815/b242815.txt">Table of n, a(n) for n = 1..10000</a>
%H A242815 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PolyasRandomWalkConstants.html">Pólya's Random Walk Constants</a>.
%F A242815 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 A242815 m(d) = Integral_{t>0} exp(-t)*BesselI(0,t/d)^d dt where BesselI(0,x) is the zeroth modified Bessel function.
%F A242815 Equals 1/(1 - A086235). - _Amiram Eldar_, Aug 28 2020
%e A242815 1.09390631558784799668327...
%p A242815 m7:= int(exp(-t)*BesselI(0, t/7)^7, t=0..infinity):
%p A242815 s:= convert(evalf(m7, 120), string):
%p A242815 map(parse, subs("."=NULL, [seq(i, i=s)]))[]; # _Alois P. Heinz_, May 23 2014
%t A242815 d = 7; 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 A242815 Cf. A086230, A086231, A086232, A086233, A086234, A086235, A086236, A242812, A242813, A242814, A242816.
%K A242815 nonn,cons
%O A242815 1,3
%A A242815 _Jean-François Alcover_, May 23 2014
%E A242815 More terms from _Alois P. Heinz_, May 23 2014