

A180916


Number of convex polyhedra with n faces that are all regular polygons.


2



0, 0, 0, 1, 2, 3, 2, 7, 3, 6, 4, 7, 3, 13, 2, 5, 4, 6, 1, 9, 2, 6, 1, 4, 1, 8, 4, 2, 1, 3, 1, 10, 1, 3, 1, 2, 4, 3, 1, 2, 1, 9, 1, 2, 1, 2, 2, 2, 1, 2, 1, 9, 1, 2, 1, 2, 1, 2, 1, 2, 1, 9, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 2, 1, 2
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

1,5


COMMENTS

For all n > 92, the sequence is identical to A000034 because for large n only prisms (even and odd n) and antiprisms (even n) are convex and have regular polygonal faces. The MathWorld article about Johnson Solids is very informative about this topic.
In a regularfaced polyhedron, any two faces with the same number of edges are congruent. (Proof: As the two faces are regular polygons, it suffices to show their edges have the same length. But as all faces are regular polygons and the polyhedron is connected, all edges have the same length.)  Jonathan Sondow, Feb 11 2018


LINKS

Table of n, a(n) for n=1..100.
Eric W. Weisstein, MathWorld: Cube
Eric W. Weisstein, MathWorld: Pentagonal Pyramid
Eric W. Weisstein, MathWorld: Dipyramid
Eric W. Weisstein, MathWorld: Pentagonal Prism
Eric W. Weisstein, MathWorld: Elongated Triangular Pyramid
Eric W. Weisstein, MathWorld: Hexagonal Pyramid
Eric W. Weisstein, MathWorld: Johnson Solid


FORMULA

a(A296602(n)) = 1.  Jonathan Sondow, Jan 29 2018


EXAMPLE

a(6) = 3 because the cube, pentagonal pyramid, and triangular bipyramid all qualify. a(7) = 2 because only the pentagonal prism and elongated triangular pyramid qualify; the hexagonal pyramid is impossible with equilateral triangles


MATHEMATICA

f = Tally[Join[PolyhedronData["Platonic", "FaceCount"], PolyhedronData["Archimedean", "FaceCount"], PolyhedronData["Johnson", "FaceCount"], {PolyhedronData[{"Prism", 3}, "FaceCount"]}]]; f2 = Transpose[f]; cnt = Table[0, {n, 100}]; cnt[[f2[[1]]]] = f2[[2]]; Do[cnt[[n]]++, {n, 7, 100}] (* add prisms *); Do[ cnt[[n]]++, {n, 10, 100, 2}] (* add antiprisms *); cnt (* T. D. Noe, Mar 04 2011 *)


CROSSREFS

Cf. A296602, A296603, A296604.
Sequence in context: A144456 A262427 A266258 * A319374 A205687 A118007
Adjacent sequences: A180913 A180914 A180915 * A180917 A180918 A180919


KEYWORD

nice,nonn


AUTHOR

J. Lowell, Sep 23 2010


EXTENSIONS

More terms from J. Lowell, Feb 28 2011
Corrected by T. D. Noe, Mar 04 2011


STATUS

approved



