A333660 - OEIS

%I #26 Sep 04 2020 03:13:10

%S 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,

%T 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,

%U 0,4,0,2,0,2,1,2,0,2,0,2,0,2,0,2,0,2

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

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

%C 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.

%H Peter Kagey, <a href="/A333660/b333660.txt">Table of n, a(n) for n = 1..1000</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/List_of_Johnson_solids">List of Johnson Solids</a>

%e 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.

%t a[n_] := Count[

%t   Join[

%t     PolyhedronData["Platonic", "VertexCount"],

%t     PolyhedronData["Archimedean", "VertexCount"],

%t     PolyhedronData["Johnson", "VertexCount"],

%t     Prepend[Range[10, n, 2], 6], (*Prisms, excluding cube*)

%t     Range[8, n, 2] (*Antiprisms, excluding octahedron*)

%t   ],

%t   n

%t ]

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

%K nonn

%O 1,5

%A _Peter Kagey_, Sep 02 2020