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

0, 4, 78, 399, 1263, 3084, 6395, 11848, 20212, 32377, 49348, 72250, 102328, 140942, 189575, 249824, 323407, 412159, 518035, 643108, 789568, 959725, 1156007, 1380959, 1637248, 1927657, 2255086, 2622556, 3033205, 3490291

Offset: 0

Author

Jon Perry, Jan 26 2013

Keywords

• nonn

Comments

The 4-sphere here refers to the geometric sphere, that is, 4 refers to the number of dimensions of the sphere.

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.

Links

• Clay Math Institute, Poincaré Conjecture Press Release
• Mathworld, Hypersphere
• Mathworld, Poincaré's Conjecture
• Wikipedia, N-sphere

Crossrefs

Cf. A066643, A066645.

Programs

JavaScript

pi = Math.PI;
for (i = 0; i < 60; i++) document.write(Math.floor(pi*pi*i*i*i*i/2) + ", ");

Mathematica

Table[Floor[(Pi^2 n^4)/2], {n, 0, 29}]

Formula

a(n) = floor(1/2*Pi^2*n^4).