A299529 -id:A299529 - cp's OEIS frontend

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

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, 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, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2

Offset: 1

Peter Kagey, Sep 02 2020

nonn

			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.
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.
For n > 10, the a(n) = 2 polyhedra are the n-gonal prism and the n-gonal antiprism.

Peter Kagey, Table of n, a(n) for n = 1..1000
Mathematics Stack Exchange, Number of convex polyhedra whose faces are regular polygons and whose largest face is an n-gon

Cf. A180916, A299529.

MaxFace[l_] := Max[Length /@ l];
a[n_] := Count[
  Join[
    MaxFace /@ PolyhedronData["Platonic", "FaceIndices"],
    MaxFace /@ PolyhedronData["Archimedean", "FaceIndices"],
    MaxFace /@ PolyhedronData["Johnson", "FaceIndices"],
    Range[4, n], (*Prisms, including triangular prism, excluding cube*)
    Range[4, n]  (*Antiprisms, excluding octahedron*)
  ],
  n
]

A299530 Number of regular-faced convex polyhedra (excluding prisms and antiprisms) with exactly n types of faces.

Original entry on oeis.org

10, 45, 38, 17, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Jonathan Sondow, Feb 11 2018

nonn

The regular-faced convex polyhedra other than prisms and antiprisms are the Platonic, Archimedean, and Johnson solids.

			Each of the five Platonic solids, and each of five Johnson solids, has one type of face, so a(1) = 5 + 5 = 10.
Each of ten Archimedean solids, and each of thirty-five Johnson solids, has two types of faces, so a(2) = 10 + 35 = 45.
Each of three Archimedean solids, and each of thirty-five Johnson solids, has three types of faces, so a(3) = 3 + 35 = 38.
Each of seventeen Johnson solids has four types of faces, so a(4) = 17.

Wikipedia, List of Johnson solids

Cf. A299114, A299529.

a(n) = 0 for n >= 5.

A306949 a(n) is the number of different types of faces of Johnson solid J_n, with solids ordered by indices in Johnson's paper.

Original entry on oeis.org

2, 2, 3, 3, 4, 3, 2, 2, 3, 2, 2, 1, 1, 2, 2, 2, 1, 3, 3, 4, 4, 3, 3, 4, 3, 2, 2, 2, 2, 3, 3, 3, 3, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 2, 2, 3, 3, 2, 2, 2, 1, 3, 3, 3, 3, 3, 3, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 4, 4, 4, 4, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 1, 2, 2, 2, 2, 2, 2, 3, 4

Offset: 1

Felix Fröhlich, Mar 17 2019

A299529(x) equals the number of times the value x occurs as a term in this sequence. In particular, if A299529(x) = 0, then x does not occur in this sequence.

			For n = 5: Johnson solid J_5 is the pentagonal cupola. This solid is bounded by 5 equilateral triangles, 5 squares, 1 pentagon and 1 decagon. Thus, there are 4 types of polygons making up the faces of this solid, hence a(5) = 4.

V. A. Zalgaller, Convex Polyhedra with Regular Faces, in: Seminars in mathematics, Springer, 1969, ISBN 978-1-4899-5671-2.

N. W. Johnson, Convex Polyhedra with Regular Faces, Canadian Journal of Mathematics 18 (1966), 169-200.
Wikipedia, List of Johnson solids
V. A. Zalgaller, Convex Polyhedra with Regular Faces, Zapiski Nauchnykh Seminarov LOMI 2 (1967), 5-221.

Cf. A242731, A242732, A242733, A296603, A296604, A299529.

a(68) corrected and a(88)-a(92) added by Pontus von Brömssen, Mar 13 2021

A343961 a(n) is the number of Johnson solids of unit edge length with a volume V such that n <= V < n+1.

Original entry on oeis.org

10, 15, 9, 9, 5, 1, 3, 1, 5, 3, 1, 1, 2, 2, 1, 2, 2, 1, 0, 0, 1, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 3, 0, 0, 4, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Felix Fröhlich, May 05 2021

nonn

a(n) = 0 for n > 92.

			For n = 6: The Johnson solids with volumes V with 6 <= V < 7 are J_6, J_19 and J_23 with V ~ 6.21, 6.77 and 6.92, respectively, so a(6) = 3.

Wikipedia, List of Johnson solids

Cf. A242731, A242732, A242733, A296603, A296604, A299529, A306949.

cp's OEIS Frontend

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

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A299530 Number of regular-faced convex polyhedra (excluding prisms and antiprisms) with exactly n types of faces.

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A306949 a(n) is the number of different types of faces of Johnson solid J_n, with solids ordered by indices in Johnson's paper.

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Extensions

A343961 a(n) is the number of Johnson solids of unit edge length with a volume V such that n <= V < n+1.

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Author

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Examples

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