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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. 8
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

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

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

REFERENCES

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 10, Eq. 19.

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..52.

Eric Weisstein's World of Mathematics, Ball

Eric Weisstein's World of Mathematics, Hypersphere

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

Index to sequences with linear recurrences with constant coefficients, signature (0,3,0,-2).

FORMULA

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). [Colin Barker, Sep 04 2012]

EXAMPLE

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

MATHEMATICA

f[n_] := Pi^(n/2 - Floor[n/2])*n/(n/2)!; Table[ Numerator[ f[n]], {n, 0, 52}]

CROSSREFS

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

Sequence in context: A122977 A003980 A132801 * A190014 A100577 A018818

Adjacent sequences:  A072475 A072476 A072477 * A072479 A072480 A072481

KEYWORD

nonn,frac,easy

AUTHOR

N. J. A. Sloane, Aug 02 2002

EXTENSIONS

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

STATUS

approved

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Last modified December 17 21:13 EST 2014. Contains 252040 sequences.