|
%I #35 Jun 21 2018 05:28:23
%S 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,
%T 8192,1,16384,1,32768,1,65536,1,131072,1,262144,1,524288,1,1048576,1,
%U 2097152,1,4194304,1,8388608,1,16777216,1,33554432,1,67108864,1
%N 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.
%C Answer to question of how to extend the sequence 0, 2, 2 Pi r, 4 Pi r^2, 2 Pi^2 r^3, ...
%C Volume of n-dimensional sphere of radius r is V_n*r^n - see A072345/A072346.
%C a(2*n-1) = 2^n and for n>2 a(2*n)=1.
%C Denominator of the rational coefficient of integral_{x>0} exp(-x^2)*x^n. - _Jean-François Alcover_, Apr 23 2013
%C From _Ilya Gutkovskiy_, Aug 02 2016: (Start)
%C Numerator of n/Gamma(n/2+1).
%C 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)
%D N. Cakic, D. Letic, B. Davidovic, The Hyperspherical functions of a derivative, Abstr. Appl. Anal. (2010) 364292 doi:10.1155/2010/364292
%D J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 10, Eq. 19.
%H Colin Barker, <a href="/A072478/b072478.txt">Table of n, a(n) for n = 0..1000</a>
%H Dusko Letic, Nenad Cakic, Branko Davidovic and Ivana Berkovic, <a href="http://dx.doi.org/10.1186/1687-1847-2012-22">Orthogonal and diagonal dimension fluxes of hyperspherical function</a>, Advances in Difference Equations 2012, 2012:22. - _N. J. A. Sloane_, Sep 04 2012
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Ball.html">Ball</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Hypersphere.html">Hypersphere</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Four-DimensionalGeometry.html">Four-Dimensional Geometry</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,3,0,-2).
%F From _Colin Barker_, Sep 04 2012: (Start)
%F a(n) = 3*a(n-2)-2*a(n-4) for n>4.
%F G.f.: x*(2+2*x-2*x^2-4*x^3-x^5+2*x^7) / (1-3*x^2+2*x^4).
%F (End)
%F From _Colin Barker_, Aug 01 2016: (Start)
%F a(n) = (1+(-1)^n-2^((1+n)/2)*(-1+(-1)^n))/2 for n>4.
%F a(n) = 1 for n>4 and even.
%F a(n) = 2^((n+1)/2) for n>4 and odd.
%F (End)
%e Sequence of S_n's begins 0, 2, 2, 4, 2, 8/3, 1, 16/15, 1/3, 32/105, 1/12, 64/945, ...
%t f[n_] := Pi^(n/2 - Floor[n/2])*n/(n/2)!; Table[ Numerator[ f[n]], {n, 0, 52}]
%t 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 *)
%t LinearRecurrence[{0,3,0,-2},{0,2,2,4,2,8,1,16,1},60] (* _Harvey P. Dale_, May 30 2018 *)
%o (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
%Y Cf. A072479. A072478(n)/A072479(n) = n*A072345(n)/A072346(n).
%K nonn,frac,easy
%O 0,2
%A _N. J. A. Sloane_, Aug 02 2002
%E More terms from _Robert G. Wilson v_, Aug 18 2002
|
