%I #31 Feb 16 2025 08:33:17
%S 0,4,78,399,1263,3084,6395,11848,20212,32377,49348,72250,102328,
%T 140942,189575,249824,323407,412159,518035,643108,789568,959725,
%U 1156007,1380959,1637248,1927657,2255086,2622556,3033205,3490291
%N a(n) = floor(volume of 4-sphere of radius n).
%C The 4-sphere here refers to the geometric sphere, that is, 4 refers to the number of dimensions of the sphere.
%C The general formula for the volume of an n-sphere can be derived using (4)-(10) at the Mathworld link, and some explicit values for higher dimensional spheres are given at the Wikipedia link, section 2.4. Note that Wikipedia uses the topologic definition and calls this 4-sphere a 3-sphere.
%H Clay Math Institute, <a href="http://www.claymath.org/sites/default/files/millenniumprizefull.pdf">Poincaré Conjecture Press Release</a>
%H Mathworld, <a href="https://mathworld.wolfram.com/Hypersphere.html">Hypersphere</a>
%H Mathworld, <a href="https://mathworld.wolfram.com/PoincareConjecture.html">Poincaré's Conjecture</a>
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/N-sphere">N-sphere</a>
%F a(n) = floor(1/2*Pi^2*n^4).
%t Table[Floor[(Pi^2 n^4)/2], {n, 0, 29}]
%o (JavaScript)
%o pi = Math.PI;
%o for (i = 0; i < 60; i++) document.write(Math.floor(pi*pi*i*i*i*i/2) + ", ");
%Y Cf. A066643, A066645.
%K nonn,changed
%O 0,2
%A _Jon Perry_, Jan 26 2013