f(N)=1/3*N*(N^2-3N+8) |
answer too. Let F(N)=a*N^3+b*N^2+c*N+d Use F(1)=2, F(2)=4, F(3)=8, F(4)=16 and undetermine coefficient method, solve the system for a,b,c,d |
solution. |
I don't remember what the theorem exactly says, I only got this conclusion from it (but don't take it seriously, maybe my impression is not accurate) : For a k-dim space, let F(N) be the max number of parts that N (k-1)-dim spheres divide the k-dim space into, then F(N) is a polynomial of N and F(N)=O(N^k). In fact, it is true when k=1,2,3 |
http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A046127 Formula: a(n)=n(n^2-3n+8)/3 (n>0). n hyperspheres divide R^k into at most C(n-1,k) + Sum_{i=0..k} C(n,i) regions. |
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