[seqfan] Re: Floor of Volume of the unit sphere in 2n-dimensional space
Richard Mathar
mathar at strw.leidenuniv.nl
Sun Aug 9 15:19:00 CEST 2009
If we take the formulas of
http://mathworld.wolfram.com/Hypersphere.html
as granted, the volume of the k-dimensional unit sphere is
2*(pi)^(k/2)/Gamma(k/2)/k, so the volumes of the k=2n-dim.
unit spheres are pi^n/Gamma(n)/n = pi^n/n!, which yields the table
k n V(K)
--------------
1 0.5 2.000000 <= A000038
2 1.0 3.141593 <= A000796
3 1.5 4.188790 <= normal 3D, 4*pi/3 A019699
4 2.0 4.934802 <= A102753
5 2.5 5.263789
6 3.0 5.167713
7 3.5 4.724766
8 4.0 4.058712
9 4.5 3.298509
10 5.0 2.550164
11 5.5 1.884104
12 6.0 1.335263
13 6.5 0.910629
14 7.0 0.599265
15 7.5 0.381443
16 8.0 0.235331
17 8.5 0.140981
18 9.0 0.082146
19 9.5 0.046622
20 10.0 0.025807
This seems to agree with the table in
http://en.wikipedia.org/wiki/N-sphere.
I guess that these constants are of greater value than any
truncated integers derived from there, because one would in practise
much more likely have a finite precision constant and would aim
to get a closed form. I haven't tried to see whether the
Plouffe inverter tables have them.
In Maple:
S := proc(n)
2*Pi^(n/2)/GAMMA(n/2) ;
end:
V := proc(n)
S(n)/n ;
end:
Digits := 20 ;
for k from 1 to 20 do
printf("%d %.1f %f\n",k,k/2,evalf(V(k))) ;
od:
Richard Mathar
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