A072345 Volume of n-dimensional sphere of radius r is V_n*r^n = Pi^(n/2)*r^n/(n/2)! = C_n*Pi^floor(n/2)*r^n; sequence gives numerator of C_n.

1, 2, 1, 4, 1, 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, 134217728, 1, 268435456

Offset

0,2

Comments

Answer to question of how to extend the sequence 1, 2 r, Pi r^2, 4 Pi r^3 / 3, Pi^2 r^4 / 2, ...

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

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. 9, Eq. 17.

Dusko Letic, Nenad Cakic, Branko Davidovic and Ivana Berkovic, Orthogonal and diagonal dimension fluxes of hyperspherical function, Advances in Difference Equations 2012, 2012:22; http://www.advancesindifferenceequations.com/content/2012/1/22. - From N. J. A. Sloane, Sep 04 2012

Links

Table of n, a(n) for n=0..55.

Eric Weisstein's World of Mathematics, Hypersphere

Eric Weisstein's World of Mathematics, Ball

Eric Weisstein's World of Mathematics, Four-Dimensional Geometry

Index entries for linear recurrences with constant coefficients, signature (0,3,0,-2).

Formula

1 if n even, 2^((n+1)/2) if n odd.

a(n) = 3*a(n-2)-2*a(n-4). G.f.: (1+2*x-2*x^2-2*x^3)/((1-x)*(1+x)*(1-2*x^2)). [Colin Barker, Sep 04 2012]

a(n) = 2^((n+1)/2)*(1-(-1)^n)/2+(1+(-1)^n)/2. - Wesley Ivan Hurt, Jan 10 2017

E.g.f.: sqrt(2)*sinh(sqrt(2)*x) + cosh(x). - Ilya Gutkovskiy, Mar 16 2017

Example

Sequence of C_n's begins 1, 2, 1, 4/3, 1/2, 8/15, 1/6, 16/105, 1/24, 32/945, 1/120, 64/10395, ...

Maple

seq(seq(k^n, k=1..2), n=1..28); # Zerinvary Lajos, Jun 29 2007

Mathematica

f[n_] := Pi^(n/2 - Floor[n/2])/(n/2)!; Table[ Numerator[ f[n]], {n, 0, 55} ]
Riffle[2^Range[30], 1, {1, -1, 2}] (* or *) LinearRecurrence[{0, 3, 0, -2}, {1, 2, 1, 4}, 60] (* Harvey P. Dale, Oct 16 2013 *)
CoefficientList[ Series[(-2x^3 - 2x^2 + 2x + 1)/(2x^4 - 3x^2 + 1), {x, 0, 56}],
x] (* Robert G. Wilson v, Jul 31 2018 *)

Prog

(Magma) [2^((n+1) div 2)*(1-(-1)^n)/2+(1+(-1)^n)/2 : n in [0..100]]; // Wesley Ivan Hurt, Jan 10 2017

Crossrefs

Cf. A072346, A072478, A072479.

Sequence in context: A325348 A307683 A248058 * A200583 A366842 A115120

Adjacent sequences: A072342 A072343 A072344 * A072346 A072347 A072348

Keyword

nonn,frac,easy

Author

N. J. A. Sloane, Jul 31 2002

Status

approved