login
A333661
a(n) is the number of convex polyhedra with n edges whose faces are regular polygons.
2
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.
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
KEYWORD
nonn
AUTHOR
Peter Kagey, Sep 02 2020
STATUS
approved