A275162 Decimal expansion of dimension d in which a ball of radius 1/2 has maximum volume.

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.

Links

Table of n, a(n) for n=0..86.

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

Adjacent sequences: A275159 A275160 A275161 * A275163 A275164 A275165

Keyword

nonn,cons

Author

Eric R. Carter, Nov 13 2016

Status

approved