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A121546
a(n) = dimension of the space in which the sphere of radius n is of maximum volume.
0
5, 24, 56, 100, 156, 225, 307, 401, 508, 627, 759, 904, 1061, 1231, 1413, 1607, 1815, 2035, 2267, 2512, 2770, 3040, 3323, 3618, 3926, 4246, 4579, 4925, 5283, 5654, 6037, 6433, 6841, 7262, 7696, 8142, 8601, 9072, 9556, 10052, 10561, 11083, 11617, 12163
OFFSET
1,1
LINKS
John Moeller, Reasoning in Higher Dimensions: Hyperspheres, onTopology Blog, 3 March 2009.
FORMULA
a(n) >= 6.2835n^2 - 0.009903n - 0.9212 is a lower bound on the real value of the dimension.
MAPLE
N:= 10^5: # to get all terms <= N
G[1]:= 2/Pi:
G[2]:= 3/4:
L[2]:= 0:
for n from 3 to N do
G[n]:= G[n-2]*(n+1)/n;
L[n]:= floor(G[n]);
if L[n] <> L[n-1] then
A[L[n]]:= n
fi
od:
seq(A[i], i=1..L[N]); # Robert Israel, Jan 05 2016
MATHEMATICA
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)!];
dim[r_]:=Block[{d=1}, While[vol[d, r]<vol[d+1, r], d++]; d];
dim/@Range@30 (* Ivan N. Ianakiev, Dec 27 2015 *)
PROG
(PARI) V(d, r)=Pi^(d/2)*r^d/gamma(d/2+1)
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
CROSSREFS
Sequence in context: A372455 A202326 A085646 * A135703 A258290 A205669
KEYWORD
nonn
AUTHOR
Sergio Falcon, Oct 10 2007
STATUS
approved