A333661 a(n) is the number of convex polyhedra with n edges whose faces are regular polygons.

0, 0, 0, 0, 0, 1, 0, 1, 2, 1, 0, 3, 1, 1, 5, 2, 1, 4, 1, 6, 2, 2, 1, 5, 3, 4, 3, 2, 0, 4, 0, 3, 3, 0, 2, 8, 0, 1, 1, 6, 0, 2, 0, 2, 3, 0, 0, 5, 0, 2, 1, 1, 0, 1, 2, 2, 1, 0, 0, 8, 0, 0, 1, 1, 1, 1, 0, 1, 1, 3, 0, 3, 0, 0, 2, 1, 0, 1, 0, 4, 1, 0, 0, 2, 0, 0, 1

Offset: 1

Author

Peter Kagey, Sep 02 2020

Keywords

• nonn

Comments

Convex polyhedra with whose faces are regular polygons are either Platonic solids, Archimedean solids, prisms, antiprisms, or Johnson solids.

Examples

			For n = 18, the a(18) = 4 polyhedra are: the truncated tetrahedron, the hexagonal prism, and the Johnson solids J_64 and J_84.
For n > 180, the only polyhedra are the prisms and antiprisms. When 3 divides n, there is an (n/3)-gonal prism; when 4 divides n, and there is an (n/4)-gonal antiprism.
Starting at n = 181 the sequence has a 12-term cycle that goes 0,0,1,1,0,1,0,1,1,0,0,2. - _J. Lowell_, Oct 18 2020
		

Links

• Peter Kagey, Table of n, a(n) for n = 1..1000
• Wikipedia, List of Johnson Solids

Crossrefs

Cf. A180916 (analog for faces), A333660 (analog for vertices), A333657.

Programs

Mathematica

a[n_] := Count[
  Join[
   PolyhedronData["Johnson", "EdgeCount"],
   PolyhedronData["Platonic", "EdgeCount"],
   PolyhedronData["Archimedean", "EdgeCount"],
   Prepend[Range[15, n, 3], 9], (*Prisms, excluding cube*)
   Range[16, n, 4] (*Antiprisms, excluding octahedron*)
  ],
  n
]