cp's OEIS Frontend

A164083 Ceiling of 2^(n-1) times the surface area of the unit sphere in 2n-dimensional space.

%I A164083 #6 Feb 16 2025 08:33:11
%S A164083 7,40,125,260,409,513,537,482,379,265,167,95,50,25,11,5,2,1,1,1,1,1,1,
%T A164083 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1
%N A164083 Ceiling of 2^(n-1) times the surface area of the unit sphere in 2n-dimensional space.
%C A164083 The rounded values of this real sequence is A164082, the floor is A164081.
%C A164083 The surface area of n-dimensional sphere of radius r is n*V_n*r^(n-1); see A072478/A072479.
%D A164083 Conway, J. H. and Sloane, N. J. A. Sphere Packings, Lattices, and Groups, 2nd ed. New York: Springer-Verlag, p. 9, 1993.
%D A164083 Coxeter, H. S. M. Regular Polytopes, 3rd ed. New York: Dover, 1973.
%D A164083 Sommerville, D. M. Y. An Introduction to the Geometry of n Dimensions. New York: Dover, p. 136, 1958.
%H A164083 Eric W. Weisstein, <a href="https://mathworld.wolfram.com/Hypersphere.html">Hypersphere</a>,
%F A164083 a(n) = ceiling(((2*pi)^n)/(n-1)!).
%e A164083 Table of approximate real values before rounding up.
%e A164083 ========================
%e A164083 n ((2*pi)^n) / (n-1)!
%e A164083 1 6.28318531 = A019692
%e A164083 2 39.4784176 = 2*A164102
%e A164083 3 124.025107 = 4*A091925
%e A164083 4 259.757576 = 8*A164109
%e A164083 5 408.026246
%e A164083 6 512.740903
%e A164083 7 536.941018
%e A164083 8 481.957131
%e A164083 9 378.528246
%e A164083 10 264.262568
%e A164083 11 166.041068
%e A164083 12 94.8424365
%e A164083 13 49.6593836
%e A164083 14 24.00147
%e A164083 15 10.7718345
%e A164083 16 4.5120955
%e A164083 17 1.77189576
%e A164083 18 0.654891141
%e A164083 19 0.228600133
%e A164083 20 0.075596684
%e A164083 ========================
%t A164083 Table[Ceiling[(2Pi)^n/(n-1)!],{n,60}] (* _Harvey P. Dale_, Jul 30 2020 *)
%Y A164083 Cf. A072345, A072346, A072478, A072479, A074457, A122510, A154255, A164081, A164082.
%K A164083 nonn
%O A164083 1,1
%A A164083 _Jonathan Vos Post_, Aug 09 2009
%E A164083 Definition corrected - _R. J. Mathar_, Sep 09 2009