A114346 The integer difference between the surface area of the unit sphere in n-1 dimensions and the volume of the unit sphere in n+1 dimensions.

1, 2, 7, 14, 21, 26, 29, 29, 27, 23, 19, 15, 11, 8, 5, 3, 2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset

1,2

Comments

This sequence is important in the n dimensional (topological dimension) theory of particles and has a maximum at n=8.

References

D. M. Y Sommerville, An Introduction to the Geometry of n dimensions, Dover Publications, 1958, pages 136-137.

Links

Table of n, a(n) for n=1..40.

Wikipedia, N-sphere

Formula

Let S(n) = 2*Pi^((n+1)/2)/Gamma((n+1)/2) and V(n) = Pi^(n/2)/Gamma(n/2+1). Then a(n) = floor|S(n-1)-V(n+1)|.

Maple

sar := proc(n)
    2*Pi^((n+1)/2)/GAMMA((n+1)/2) ;
end proc:
vol := proc(n)
    Pi^(n/2)/GAMMA(n/2+1) ;
end proc:
A114346 := proc(n)
    floor(abs(sar(n-1)-vol(n+1))) ;
end proc: # R. J. Mathar, Mar 05 2016

Mathematica

v[n_]=Pi^(n/2)/Gamma[n/2+1] s[n_]=2*Pi^(n/2)/Gamma[n/2] a=Table[Floor[Abs[s[n]-v[n+1]]], {n, 0, 20}]

Crossrefs

Sequence in context: A187142 A263398 A161702 * A369972 A325159 A087324

Adjacent sequences: A114343 A114344 A114345 * A114347 A114348 A114349

Keyword

nonn

Author

Roger L. Bagula, Feb 08 2006

Status

approved