A333657 -id:A333657 - cp's OEIS frontend

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

Peter Kagey, Sep 02 2020

nonn

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.

			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.

Peter Kagey, Table of n, a(n) for n = 1..1000
Wikipedia, List of Johnson Solids

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

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
]

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

Original entry on oeis.org

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

Peter Kagey, Sep 02 2020

nonn

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

			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

Peter Kagey, Table of n, a(n) for n = 1..1000
Wikipedia, List of Johnson Solids

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

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
]

cp's OEIS Frontend

A333660 a(n) is the number of n-vertex convex polyhedra whose faces are regular polygons.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica