A333660 a(n) is the number of n-vertex convex polyhedra whose faces are regular polygons.

0, 0, 0, 1, 2, 3, 3, 6, 5, 7, 4, 10, 1, 6, 5, 6, 0, 6, 0, 8, 1, 4, 1, 8, 4, 2, 0, 3, 0, 9, 0, 3, 0, 2, 3, 2, 0, 2, 0, 5, 0, 2, 0, 2, 1, 2, 0, 3, 0, 5, 0, 2, 0, 2, 4, 2, 0, 2, 0, 10, 0, 2, 0, 2, 1, 2, 0, 2, 0, 4, 0, 2, 0, 2, 1, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2

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.

For n > 120, there are two such convex polyhedra for even n, the (n/2)-gonal prism and (n/2)-gonal antiprism, and no polyhedra for odd n.

Examples

			For n = 12, the a(12) = 10 convex polyhedra with regular polygonal faces and 12 vertices are: the icosahedron, the truncated tetrahedron, the cuboctahedron, the hexagonal prism, the hexagonal antiprism, and the Johnson solids J_4, J_16, J_27, J_53, and J_88.
		

Links

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

Crossrefs

Cf. A180916 (analog for faces), A333661 (analog for edges), A333657.

Programs

Mathematica

a[n_] := Count[
  Join[
    PolyhedronData["Platonic", "VertexCount"],
    PolyhedronData["Archimedean", "VertexCount"],
    PolyhedronData["Johnson", "VertexCount"],
    Prepend[Range[10, n, 2], 6], (*Prisms, excluding cube*)
    Range[8, n, 2] (*Antiprisms, excluding octahedron*)
  ],
  n
]