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A210519 a(n) = floor(volume of 4-sphere of radius n).

%I A210519 #31 Feb 16 2025 08:33:17
%S A210519 0,4,78,399,1263,3084,6395,11848,20212,32377,49348,72250,102328,
%T A210519 140942,189575,249824,323407,412159,518035,643108,789568,959725,
%U A210519 1156007,1380959,1637248,1927657,2255086,2622556,3033205,3490291
%N A210519 a(n) = floor(volume of 4-sphere of radius n).
%C A210519 The 4-sphere here refers to the geometric sphere, that is, 4 refers to the number of dimensions of the sphere.
%C A210519 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 A210519 Clay Math Institute, <a href="http://www.claymath.org/sites/default/files/millenniumprizefull.pdf">Poincaré Conjecture Press Release</a>
%H A210519 Mathworld, <a href="https://mathworld.wolfram.com/Hypersphere.html">Hypersphere</a>
%H A210519 Mathworld, <a href="https://mathworld.wolfram.com/PoincareConjecture.html">Poincaré's Conjecture</a>
%H A210519 Wikipedia, <a href="http://en.wikipedia.org/wiki/N-sphere">N-sphere</a>
%F A210519 a(n) = floor(1/2*Pi^2*n^4).
%t A210519 Table[Floor[(Pi^2 n^4)/2], {n, 0, 29}]
%o A210519 (JavaScript)
%o A210519 pi = Math.PI;
%o A210519 for (i = 0; i < 60; i++) document.write(Math.floor(pi*pi*i*i*i*i/2) + ", ");
%Y A210519 Cf. A066643, A066645.
%K A210519 nonn
%O A210519 0,2
%A A210519 _Jon Perry_, Jan 26 2013