A121546 a(n) = dimension of the space in which the sphere of radius n is of maximum volume.

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

Table of n, a(n) for n=1..44.

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

Adjacent sequences: A121543 A121544 A121545 * A121547 A121548 A121549

Keyword

nonn

Author

Sergio Falcon, Oct 10 2007

Status

approved