cp's OEIS Frontend

A164082 Rounded value of 2^(n-1) times the surface area of the unit sphere in 2n-dimensional space.

%I A164082 #13 Feb 16 2025 08:33:11
%S A164082 6,39,124,260,408,513,537,482,379,264,166,95,50,24,11,5,2,1,0,0,0,0,0,
%T A164082 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
%N A164082 Rounded value of 2^(n-1) times the surface area of the unit sphere in 2n-dimensional space.
%C A164082 The floor of this real sequence is A164081, the ceiling is A164083.
%C A164082 The surface area of the n-dimensional sphere of radius r is n*V_n*r^(n-1); see A072478/ A072479.
%C A164082 There are 18 nonzero terms in this sequence. - _G. C. Greubel_, Sep 11 2017
%D A164082 Conway, J. H. and Sloane, N. J. A. Sphere Packings, Lattices, and Groups, 2nd ed. New York: Springer-Verlag, p. 9, 1993.
%D A164082 Coxeter, H. S. M. Regular Polytopes, 3rd ed. New York: Dover, 1973.
%D A164082 Sommerville, D. M. Y. An Introduction to the Geometry of n Dimensions. New York: Dover, p. 136, 1958.
%H A164082 Eric W. Weisstein, <a href="https://mathworld.wolfram.com/Hypersphere.html">Hypersphere</a>,
%F A164082 a(n) = round(((2*Pi)^n)/(n-1)!).
%e A164082 Table of approximate real values before rounding up or down:
%e A164082 ========================
%e A164082 n ((2*pi)^n) / (n-1)!
%e A164082 1 6.28318531 = A019692
%e A164082 2 39.4784176 = 2*A164102
%e A164082 3 124.025107 = 4*A091925
%e A164082 4 259.757576 = 8*A164109
%e A164082 5 408.026246
%e A164082 6 512.740903
%e A164082 7 536.941018
%e A164082 8 481.957131
%e A164082 9 378.528246
%e A164082 10 264.262568
%e A164082 11 166.041068
%e A164082 12 94.8424365
%e A164082 13 49.6593836
%e A164082 14 24.00147
%e A164082 15 10.7718345
%e A164082 16 4.5120955
%e A164082 17 1.77189576
%e A164082 18 0.654891141
%e A164082 19 0.228600133
%e A164082 20 0.075596684
%e A164082 ========================
%p A164082 A164082 := proc(n) (2*Pi)^n/(n-1)! ; round(%) ; end: seq(A164082(n),n=1..80) ; # _R. J. Mathar_, Sep 09 2009
%t A164082 Table[Round[(2*Pi)^n/(n - 1)!], {n, 1, 20}] (* _G. C. Greubel_, Sep 11 2017 *)
%o A164082 (PARI) for(n=1,20, print1(round((2*Pi)^n/(n-1)!), ", ")) \\ _G. C. Greubel_, Sep 11 2017
%Y A164082 Cf. A072345, A072346, A072478, A072479, A074457, A122510, A154255, A164081, A164083.
%K A164082 nonn
%O A164082 1,1
%A A164082 _Jonathan Vos Post_, Aug 09 2009
%E A164082 Definition corrected by _R. J. Mathar_, Sep 09 2009