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A333660 a(n) is the number of n-vertex 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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified October 1 23:41 EDT 2022. Contains 357173 sequences. (Running on oeis4.)