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

%I A242814 #19 Feb 16 2025 08:33:22
%S A242814 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,
%T A242814 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,
%U A242814 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
%N A242814 Decimal expansion of the expected number of returns to the origin of a random walk on a 6-d lattice.
%D A242814 Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.9 Polya's random walk constants, p. 323.
%H A242814 Marc Mezzarobba, <a href="/A242814/b242814.txt">Table of n, a(n) for n = 1..10000</a>
%H A242814 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PolyasRandomWalkConstants.html">Pólya's Random Walk Constants</a>.
%F A242814 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 A242814 m(d) = integral_(t>0) exp(-t)*BesselI(0,t/d)^d dt where BesselI(0,x) is the zeroth modified Bessel function.
%F A242814 Equals 1/(1 - A086234). - _Amiram Eldar_, Aug 28 2020
%e A242814 1.1169633732...
%p A242814 m6:= int(exp(-t)*BesselI(0, t/6)^6, t=0..infinity):
%p A242814 s:= convert(evalf(m6, 120), string):
%p A242814 map(parse, subs("."=NULL, [seq(i, i=s)]))[]; # _Alois P. Heinz_, May 23 2014
%t A242814 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
%o A242814 (PARI) intnumosc(t=0,exp(-t)*besseli(0,t/6)^6,12*Pi) \\ _Charles R Greathouse IV_, Oct 23 2023
%Y A242814 Cf. A086230, A086231, A086232, A086233, A086234, A086235, A086236, A242812, A242813.
%K A242814 nonn,cons
%O A242814 1,4
%A A242814 _Jean-François Alcover_, May 23 2014
%E A242814 More terms from _Alois P. Heinz_, May 23 2014