A164081 - OEIS

%I #17 Feb 07 2021 23:48:32

%S 6,39,124,259,408,512,536,481,378,264,166,94,49,24,10,4,1,0,0,0,0,0,0,

%T 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,

%U 0,0

%N Floor of 2^(n-1) times the surface area of the unit sphere in 2n-dimensional space.

%C The rounded values of this real sequence is A164082, the ceiling is A164083.

%C The surface area of n-dimensional sphere of radius r is n*V_n*r^(n-1); see A072478/A072479.

%C There are only 17 nonzero terms. - _G. C. Greubel_, Sep 10 2017

%D J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices, and Groups, 2nd ed., New York: Springer-Verlag, p. 9, 1993.

%D H. S. M. Coxeter, Regular Polytopes, 3rd ed., New York: Dover, 1973.

%D D. M. Y. Sommerville, An Introduction to the Geometry of n Dimensions, New York: Dover, p. 136, 1958.

%H Eric W. Weisstein, <a href="http://mathworld.wolfram.com/Hypersphere.html">Hypersphere</a>.

%F a(n) = floor( (2*Pi)^n/(n-1)! ).

%e Table of approximate real values before taking integer part.

%e ========================

%e n (2*Pi)^n / (n-1)!

%e 1 6.28318531 = A019692

%e 2 39.4784176 = 2*A164102

%e 3 124.025107 = 4*A091925

%e 4 259.757576 = 8*A164109

%e 5 408.026246

%e 6 512.740903

%e 7 536.941018

%e 8 481.957131

%e 9 378.528246

%e 10 264.262568

%e 11 166.041068

%e 12 94.8424365

%e 13 49.6593836

%e 14 24.00147

%e 15 10.7718345

%e 16 4.5120955

%e 17 1.77189576

%e 18 0.654891141

%e 19 0.228600133

%e 20 0.075596684

%e ========================

%p A164081 := proc(n) (2*Pi)^n/(n-1)! ; floor(%) ; end: seq(A164081(n),n=1..80) ; # _R. J. Mathar_, Sep 09 2009

%t Table[Floor[(2*Pi)^n/(n - 1)!], {n, 1, 100}] (* _G. C. Greubel_, Sep 10 2017 *)

%o (PARI) for(n=1,100, print1(floor((2*Pi)^n/(n-1)!), ", ")) \\ _G. C. Greubel_, Sep 10 2017

%Y Cf. A072345, A072346, A072478, A072479, A074457, A122510, A154255.

%K nonn

%O 1,1

%A _Jonathan Vos Post_, Aug 09 2009

%E Definition corrected by _R. J. Mathar_, Sep 09 2009