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A333657 a(n) is the number of convex polyhedra whose faces are regular polygons and whose largest face is an n-gon.

%I A333657 #21 Aug 12 2022 19:22:24
%S A333657 0,0,8,30,37,14,2,9,2,22,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,
%T A333657 2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,
%U A333657 2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2
%N A333657 a(n) is the number of convex polyhedra whose faces are regular polygons and whose largest face is an n-gon.
%H A333657 Peter Kagey, <a href="/A333657/b333657.txt">Table of n, a(n) for n = 1..1000</a>
%H A333657 Mathematics Stack Exchange, <a href="https://math.stackexchange.com/q/3811038/121988">Number of convex polyhedra whose faces are regular polygons and whose largest face is an n-gon</a>
%e A333657 For n = 3, the a(3) = 8 polyhedra consisting of only equilateral triangles are: the tetrahedron, the octahedron, the icosahedron, and the Johnson solids J_12, J_13, J_17, J_51, and J_84.
%e A333657 For n = 8, the a(8) = 9 polyhedra containing an octagonal face but no face with more than eight sides are: the truncated cube, the truncated cuboctahedron, the octagonal prism, the octagonal antiprism, and the Johnson solids J_4, J_19, J_23, J_66, and J_67.
%e A333657 For n > 10, the a(n) = 2 polyhedra are the n-gonal prism and the n-gonal antiprism.
%t A333657 MaxFace[l_] := Max[Length /@ l];
%t A333657 a[n_] := Count[
%t A333657   Join[
%t A333657     MaxFace /@ PolyhedronData["Platonic", "FaceIndices"],
%t A333657     MaxFace /@ PolyhedronData["Archimedean", "FaceIndices"],
%t A333657     MaxFace /@ PolyhedronData["Johnson", "FaceIndices"],
%t A333657     Range[4, n], (*Prisms, including triangular prism, excluding cube*)
%t A333657     Range[4, n]  (*Antiprisms, excluding octahedron*)
%t A333657   ],
%t A333657   n
%t A333657 ]
%Y A333657 Cf. A180916, A299529.
%K A333657 nonn
%O A333657 1,3
%A A333657 _Peter Kagey_, Sep 02 2020