1,1

Let G, the group of rotations in 4 dimensional space, act on the set of n! paintings of each convex regular 4-polytopes having n cells. There are n! fixed points in the action table since the only element in G that leaves a painting fixed is the identity element. The order of G is A273509/2. So by Burnside's Lemma a(n)=n!/|G|. a(5) = 120!/7200 and a(6) = 600!/72000 and they are too large to display.

See A198861 for the Platonic solids which are the analogs of the regular polyhedra in three dimensions.

a(6) = 17577...66368*10^146 has 1405 digits. - Georg Fischer, Jun 16 2025

Georg Fischer, Table of n, a(n) for n = 1..5

Wikipedia, Regular 4-polytope

a(n) = 2*A063924(n)! / A273509(n). [Corrected by Georg Fischer, Jun 16 2025]

The second of these six 4-polytopes (in sequence of cell count) is the 4-cube (with 8 cells). It has |G| = 192 rotations with n = 8. Hence a(2) = 8!/192 = 210.

{5!/60, 8!/192, 16!/192, 24!/576, 120!/7200, 600!/7200};

{5!/60, 8!/192, 16!/192, 24!/576, 120!/7200, 600!/7200}

Cf. A063924, A098427, A198861, A273509.

Sequence in context: A092700 A178388 A056065 * A050445 A167833 A167838

Adjacent sequences: A317975 A317976 A317977 * A317979 A317980 A317981

nonn,fini

Frank M Jackson, Aug 12 2018

approved