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

6, 39, 124, 260, 408, 513, 537, 482, 379, 264, 166, 95, 50, 24, 11, 5, 2, 1, 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

Offset: 1

Author

Jonathan Vos Post, Aug 09 2009

Keywords

• nonn

Comments

The floor of this real sequence is A164081, the ceiling is A164083.

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

There are 18 nonzero terms in this sequence. - G. C. Greubel, Sep 11 2017

Examples

			Table of approximate real values before rounding up or down:
========================
n ((2*pi)^n) / (n-1)!
1 6.28318531 = A019692
2 39.4784176 = 2*A164102
3 124.025107 = 4*A091925
4 259.757576 = 8*A164109
5 408.026246
6 512.740903
7 536.941018
8 481.957131
9 378.528246
10 264.262568
11 166.041068
12 94.8424365
13 49.6593836
14 24.00147
15 10.7718345
16 4.5120955
17 1.77189576
18 0.654891141
19 0.228600133
20 0.075596684
========================
		

References

• Conway, J. H. and Sloane, N. J. A. Sphere Packings, Lattices, and Groups, 2nd ed. New York: Springer-Verlag, p. 9, 1993.
• Coxeter, H. S. M. Regular Polytopes, 3rd ed. New York: Dover, 1973.
• Sommerville, D. M. Y. An Introduction to the Geometry of n Dimensions. New York: Dover, p. 136, 1958.

Links

• Eric W. Weisstein, Hypersphere,

Crossrefs

Cf. A072345, A072346, A072478, A072479, A074457, A122510, A154255, A164081, A164083.

Programs

Maple

A164082 := proc(n) (2*Pi)^n/(n-1)! ; round(%) ; end: seq(A164082(n),n=1..80) ; # R. J. Mathar, Sep 09 2009

Mathematica

Table[Round[(2*Pi)^n/(n - 1)!], {n, 1, 20}] (* G. C. Greubel, Sep 11 2017 *)

PARI

for(n=1,20, print1(round((2*Pi)^n/(n-1)!), ", ")) \\ G. C. Greubel, Sep 11 2017

Formula

a(n) = round(((2*Pi)^n)/(n-1)!).

Extensions

Definition corrected by R. J. Mathar, Sep 09 2009