login
A275162
Decimal expansion of dimension d in which a ball of radius 1/2 has maximum volume.
0
4, 7, 6, 5, 8, 2, 5, 8, 2, 3, 0, 6, 0, 8, 5, 2, 9, 5, 2, 0, 7, 6, 1, 5, 7, 6, 8, 8, 5, 8, 8, 2, 3, 2, 4, 0, 3, 0, 1, 6, 4, 5, 5, 1, 5, 1, 8, 0, 4, 9, 7, 5, 6, 9, 3, 1, 9, 5, 9, 5, 1, 7, 2, 3, 7, 2, 4, 1, 2, 7, 3, 1, 0, 1, 1, 4, 1, 5, 0, 1, 1, 8, 6, 2, 1, 6, 6
OFFSET
0,1
COMMENTS
The definition of hypervolume for a ball of radius r, generalized to continuous dimension d, is given by ((Pi^(d/2))*(r^d))/Gamma((d/2) + 1). Assigning r = 1/2, the d > 0 which maximizes this formula is the non-integral real number 0.4765825... whose digits form this sequence.
FORMULA
Maximizing ((Pi^(d/2))*((1/2)^d))/Gamma((d/2) + 1) for d>0 we obtain a volume of 1.0386933280526... when d equals the positive real root of the derivative: ((2^(-1-d))*(Pi^(d/2))*((log(4*Pi) + PolyGamma(0, 1+d/2))))/(Gamma(1+d/2)). - Corrected by Eric R. Carter, May 09 2019
EXAMPLE
d = 0.47658258230608529520761576885882324030164...
MATHEMATICA
RealDigits[d/.FindRoot[Log[4/Pi] + PolyGamma[0, 1 + d/2], {d, 1}, WorkingPrecision -> 200]][[1]]
CROSSREFS
Cf. A074455.
Sequence in context: A056849 A116081 A361351 * A105228 A199550 A292510
KEYWORD
nonn,cons
AUTHOR
Eric R. Carter, Nov 13 2016
STATUS
approved