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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 regular-faced 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

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Last modified June 25 07:53 EDT 2019. Contains 324347 sequences. (Running on oeis4.)