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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.
6
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
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
KEYWORD
nonn,cons
AUTHOR
EXTENSIONS
More terms from Alois P. Heinz, May 23 2014
STATUS
approved