A121546 - OEIS

%I #14 Jan 05 2016 14:38:48

%S 5,24,56,100,156,225,307,401,508,627,759,904,1061,1231,1413,1607,1815,

%T 2035,2267,2512,2770,3040,3323,3618,3926,4246,4579,4925,5283,5654,

%U 6037,6433,6841,7262,7696,8142,8601,9072,9556,10052,10561,11083,11617,12163

%N a(n) = dimension of the space in which the sphere of radius n is of maximum volume.

%H John Moeller, <a href="https://ontopo.wordpress.com/2009/03/03/reasoning-in-higher-dimensions-hyperspheres/">Reasoning in Higher Dimensions: Hyperspheres</a>, onTopology Blog, 3 March 2009.

%F a(n) >= 6.2835n^2 - 0.009903n - 0.9212 is a lower bound on the real value of the dimension.

%p N:= 10^5: # to get all terms <= N

%p G[1]:= 2/Pi:

%p G[2]:= 3/4:

%p L[2]:= 0:

%p for n from 3 to N do

%p   G[n]:= G[n-2]*(n+1)/n;

%p   L[n]:= floor(G[n]);

%p   if L[n] <> L[n-1] then

%p      A[L[n]]:= n

%p   fi

%p od:

%p seq(A[i],i=1..L[N]); # _Robert Israel_, Jan 05 2016

%t vol[n_,r_]:=If[IntegerQ[n/2],(Pi^(n/2)*r^n)/(n/2)!,(Pi^((n-1)/2)*((n+1)/2)!*2^(n+1)*r^n)/(n+1)!];

%t dim[r_]:=Block[{d=1},While[vol[d,r]<vol[d+1,r],d++];d];

%t dim/@Range@30 (* _Ivan N. Ianakiev_, Dec 27 2015 *)

%o (PARI) V(d,r)=Pi^(d/2)*r^d/gamma(d/2+1)

%o a(n)=my(d=ceil(6.2835*n^2-0.009903*n-0.9212)); while(V(d,n)<V(d-1,n), d--); while(V(d,n)<V(d+1,n), d++); d \\ _Charles R Greathouse IV_, Mar 06 2014

%K nonn

%O 1,1

%A _Sergio Falcon_, Oct 10 2007