A333661 a(n) is the number of convex polyhedra with n edges whose faces are regular polygons.

0, 0, 0, 0, 0, 1, 0, 1, 2, 1, 0, 3, 1, 1, 5, 2, 1, 4, 1, 6, 2, 2, 1, 5, 3, 4, 3, 2, 0, 4, 0, 3, 3, 0, 2, 8, 0, 1, 1, 6, 0, 2, 0, 2, 3, 0, 0, 5, 0, 2, 1, 1, 0, 1, 2, 2, 1, 0, 0, 8, 0, 0, 1, 1, 1, 1, 0, 1, 1, 3, 0, 3, 0, 0, 2, 1, 0, 1, 0, 4, 1, 0, 0, 2, 0, 0, 1

Offset

1,9

Comments

Convex polyhedra with whose faces are regular polygons are either Platonic solids, Archimedean solids, prisms, antiprisms, or Johnson solids.

Links

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

Wikipedia, List of Johnson Solids

Example

For n = 18, the a(18) = 4 polyhedra are: the truncated tetrahedron, the hexagonal prism, and the Johnson solids J_64 and J_84.
For n > 180, the only polyhedra are the prisms and antiprisms. When 3 divides n, there is an (n/3)-gonal prism; when 4 divides n, and there is an (n/4)-gonal antiprism.
Starting at n = 181 the sequence has a 12-term cycle that goes 0,0,1,1,0,1,0,1,1,0,0,2. - J. Lowell, Oct 18 2020

Mathematica

a[n_] := Count[
  Join[
   PolyhedronData["Johnson", "EdgeCount"],
   PolyhedronData["Platonic", "EdgeCount"],
   PolyhedronData["Archimedean", "EdgeCount"],
   Prepend[Range[15, n, 3], 9], (*Prisms, excluding cube*)
   Range[16, n, 4] (*Antiprisms, excluding octahedron*)
  ],
  n
]

Crossrefs

Cf. A180916 (analog for faces), A333660 (analog for vertices), A333657.

Sequence in context: A029312 A287352 A243715 * A143256 A143151 A130106

Adjacent sequences: A333658 A333659 A333660 * A333662 A333663 A333664

Keyword

nonn

Author

Peter Kagey, Sep 02 2020

Status

approved