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

1,5

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.

Links

Peter Kagey, Table of n, a(n) for n = 1..1000

Wikipedia, List of Johnson Solids

Example

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.

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
]

Crossrefs

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

Sequence in context: A023821 A262332 A262240 * A187754 A347732 A075258

Adjacent sequences: A333657 A333658 A333659 * A333661 A333662 A333663

Keyword

nonn

Author

Peter Kagey, Sep 02 2020

Status

approved