A114348 The integer difference between (the n-dimensional unit sphere surface area minus the (n+1)-dimensional unit sphere volume) and the (n+2)-dimensional unit sphere volume.

-6, -3, 2, 9, 16, 22, 25, 26, 25, 22, 18, 14, 10, 7, 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, 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

1,1

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..72.

Wikipedia, Hypersphere.

Dominika Závacká, Cristina Dalfó, and Miquel Angel Fiol, Integer sequences from k-iterated line digraphs, CEUR: Proc. 24th Conf. Info. Tech. - Appl. and Theory (ITAT 2024) Vol 3792, 156-161. See p. 161, Table 2.

Formula

Let v(n) = Pi^(n/2)/Gamma(n/2+1) be the volume of the n-dimensional unit sphere and s(n) = 2*Pi^(n/2)/Gamma(n/2) be its surface content. Then a(n) = floor(s(n)-v(n+1)-v(n+2)).

a(n) = 0 for n >= 19. - G. C. Greubel, Feb 06 2021

Mathematica

Table[Floor[ (Pi^(n/2)/2)*( (n*(n+2)-2*Pi)/Gamma[n/2 +2] - 2*Sqrt[Pi]/Gamma[(n+3)/2])], {n, 50}] (* G. C. Greubel, Feb 06 2021 *)

Crossrefs

Cf. A138219.

Sequence in context: A076214 A011488 A021162 * A280680 A272083 A357614

Adjacent sequences: A114345 A114346 A114347 * A114349 A114350 A114351

Keyword

sign,less

Author

Roger L. Bagula, Feb 08 2006; corrected Feb 08 2006

Extensions

Signs reintroduced by R. J. Mathar, Jul 23 2012

Status

approved