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

%I A333661 #31 Oct 23 2020 06:18:55
%S A333661 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,
%T A333661 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,
%U A333661 1,3,0,3,0,0,2,1,0,1,0,4,1,0,0,2,0,0,1
%N A333661 a(n) is the number of convex polyhedra with n edges whose faces are regular polygons.
%C A333661 Convex polyhedra with whose faces are regular polygons are either Platonic solids, Archimedean solids, prisms, antiprisms, or Johnson solids.
%H A333661 Peter Kagey, <a href="/A333661/b333661.txt">Table of n, a(n) for n = 1..1000</a>
%H A333661 Wikipedia, <a href="https://en.wikipedia.org/wiki/List_of_Johnson_solids">List of Johnson Solids</a>
%e A333661 For n = 18, the a(18) = 4 polyhedra are: the truncated tetrahedron, the hexagonal prism, and the Johnson solids J_64 and J_84.
%e A333661 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.
%e A333661 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
%t A333661 a[n_] := Count[
%t A333661   Join[
%t A333661    PolyhedronData["Johnson", "EdgeCount"],
%t A333661    PolyhedronData["Platonic", "EdgeCount"],
%t A333661    PolyhedronData["Archimedean", "EdgeCount"],
%t A333661    Prepend[Range[15, n, 3], 9], (*Prisms, excluding cube*)
%t A333661    Range[16, n, 4] (*Antiprisms, excluding octahedron*)
%t A333661   ],
%t A333661   n
%t A333661 ]
%Y A333661 Cf. A180916 (analog for faces), A333660 (analog for vertices), A333657.
%K A333661 nonn
%O A333661 1,9
%A A333661 _Peter Kagey_, Sep 02 2020