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
KEYWORD
nonn
AUTHOR
Peter Kagey, Sep 02 2020
STATUS
approved