A072478 Surface area of n-dimensional sphere of radius r is n*V_n*r^(n-1) = n*Pi^(n/2)*r^(n-1)/(n/2)! = S_n*Pi^floor(n/2)*r^(n-1); sequence gives numerator of S_n.

0, 2, 2, 4, 2, 8, 1, 16, 1, 32, 1, 64, 1, 128, 1, 256, 1, 512, 1, 1024, 1, 2048, 1, 4096, 1, 8192, 1, 16384, 1, 32768, 1, 65536, 1, 131072, 1, 262144, 1, 524288, 1, 1048576, 1, 2097152, 1, 4194304, 1, 8388608, 1, 16777216, 1, 33554432, 1, 67108864, 1

Offset: 0

Author

N. J. A. Sloane, Aug 02 2002

Keywords

• nonn
• frac
• easy

Comments

Answer to question of how to extend the sequence 0, 2, 2 Pi r, 4 Pi r^2, 2 Pi^2 r^3, ...

Volume of n-dimensional sphere of radius r is V_n*r^n - see A072345/A072346.

a(2*n-1) = 2^n and for n>2 a(2*n)=1.

Denominator of the rational coefficient of integral_{x>0} exp(-x^2)*x^n. - Jean-François Alcover, Apr 23 2013

From Ilya Gutkovskiy, Aug 02 2016: (Start)

Numerator of n/Gamma(n/2+1).

More generally, the ordinary generating function for the surface area of the n-dimensional sphere of radius r is 2*x*(1 + exp(Pi*r^2*x^2)*Pi*r*x + exp(Pi*r^2*x^2)*Pi*r*erf(sqrt(Pi)*r*x)*x) = 2*x + 2*Pi*r*x^2 + 4*Pi*r^2*x^3 + 2*Pi^2*r^3*x^4 + (8*Pi^2*r^4/3)*x^5 + Pi^3*r^5*x^6 + ... (End)

Examples

			Sequence of S_n's begins 0, 2, 2, 4, 2, 8/3, 1, 16/15, 1/3, 32/105, 1/12, 64/945, ...
		

References

• N. Cakic, D. Letic, B. Davidovic, The Hyperspherical functions of a derivative, Abstr. Appl. Anal. (2010) 364292 doi:10.1155/2010/364292
• J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 10, Eq. 19.

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• Dusko Letic, Nenad Cakic, Branko Davidovic and Ivana Berkovic, Orthogonal and diagonal dimension fluxes of hyperspherical function, Advances in Difference Equations 2012, 2012:22. - _N. J. A. Sloane_, Sep 04 2012
• Eric Weisstein's World of Mathematics, Ball
• Eric Weisstein's World of Mathematics, Hypersphere
• Eric Weisstein's World of Mathematics, Four-Dimensional Geometry
• Index entries for linear recurrences with constant coefficients, signature (0,3,0,-2).

Crossrefs

Cf. A072479. A072478(n)/A072479(n) = n*A072345(n)/A072346(n).

Programs

Mathematica

f[n_] := Pi^(n/2 - Floor[n/2])*n/(n/2)!; Table[ Numerator[ f[n]], {n, 0, 52}]
CoefficientList[Series[x (2 + 2 x - 2 x^2 - 4 x^3 - x^5 + 2 x^7)/(1 - 3 x^2 + 2 x^4), {x, 0, 52}], x] (* Michael De Vlieger, Aug 01 2016 *)
LinearRecurrence[{0,3,0,-2},{0,2,2,4,2,8,1,16,1},60] (* Harvey P. Dale, May 30 2018 *)

PARI

concat(0, Vec(x*(2+2*x-2*x^2-4*x^3-x^5+2*x^7)/(1-3*x^2+2*x^4) + O(x^100))) \\ Colin Barker, Aug 01 2016

Formula

From Colin Barker, Sep 04 2012: (Start)

a(n) = 3*a(n-2)-2*a(n-4) for n>4.

G.f.: x*(2+2*x-2*x^2-4*x^3-x^5+2*x^7) / (1-3*x^2+2*x^4).

(End)

From Colin Barker, Aug 01 2016: (Start)

a(n) = (1+(-1)^n-2^((1+n)/2)*(-1+(-1)^n))/2 for n>4.

a(n) = 1 for n>4 and even.

a(n) = 2^((n+1)/2) for n>4 and odd.

(End)

Extensions

More terms from Robert G. Wilson v, Aug 18 2002