

A333660


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


2



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
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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



