A180916 -id:A180916 - cp's OEIS frontend

A296602 Values of F for which there is a unique convex polyhedron with F faces that are all regular polygons.

4, 19, 23, 25, 29, 31, 33, 35, 39, 41, 43, 45, 49, 51, 53, 55, 57, 59, 61, 63, 65, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 97, 99, 101, 103, 105, 107, 109, 111, 113, 115, 117, 119, 121, 123, 125, 127, 129, 131, 133, 135, 137, 139, 141, 143, 145, 147, 149, 151, 153, 155, 157, 159, 161, 163, 165, 167, 169, 171, 173

Offset: 1

Jonathan Sondow, Jan 28 2018

The main entry for this sequence is A180916.

All terms except 4 are odd, because both the cube and the pentagonal pyramid have 6 faces, and for any even F > 6 both a prism and an antiprism can have F faces. Platonic solids, Archimedean solids, Johnson solids, and prisms account for the missing odd numbers.

			The regular tetrahedron is the only convex polyhedron with 4 faces that are all regular polygons, and no such polyhedron with fewer than 4 faces exists, so a(1) = 4.

Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A180916, A242731, A296603, A296604.

LinearRecurrence[{2, -1}, {4, 19, 23, 25, 29, 31, 33, 35, 39, 41, 43, 45, 49, 51}, 30] (* Georg Fischer, Oct 26 2020 *)

A180916(a(n)) = 1.
From Colin Barker, Jul 05 2020: (Start)
G.f.: x*(4 + 11*x - 11*x^2 - 2*x^3 + 2*x^4 - 2*x^5 + 2*x^8 - 2*x^9 + 2*x^12 - 2*x^13) / (1 - x)^2.
a(n) = 2*a(n-1) - a(n-2) for n>14.
(End)

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

Original entry on oeis.org

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
]

A181708 Numbers of faces of polyhedra made entirely of regular polygons that are not prisms or antiprisms.

Original entry on oeis.org

4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 20, 21, 22, 24, 26, 27, 30, 32, 34, 37, 38, 42, 47, 52, 62, 92

Offset: 1

J. Lowell, Nov 06 2010

The prisms and antiprisms form infinite sequences of polyhedra that contain all integers >= 5 and all even numbers >= 8 respectively.

			19 is not in this sequence because the heptadecagonal prism is the only 19-faced polyhedron made entirely of regular polygons. 36 is not in this sequence because the 34-gonal prism and heptadecagonal antiprism are the only 36-faced polyhedra made entirely of regular polygons.

Cf. A180916

A333660 a(n) is the number of n-vertex convex polyhedra whose faces are regular polygons.

Original entry on oeis.org

0, 0, 0, 1, 2, 3, 3, 6, 5, 7, 4, 10, 1, 6, 5, 6, 0, 6, 0, 8, 1, 4, 1, 8, 4, 2, 0, 3, 0, 9, 0, 3, 0, 2, 3, 2, 0, 2, 0, 5, 0, 2, 0, 2, 1, 2, 0, 3, 0, 5, 0, 2, 0, 2, 4, 2, 0, 2, 0, 10, 0, 2, 0, 2, 1, 2, 0, 2, 0, 4, 0, 2, 0, 2, 1, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2

Offset: 1

Peter Kagey, Sep 02 2020

nonn

Convex polyhedra with whose faces are regular polygons are either Platonic solids, Archimedean solids, prisms, antiprisms, or Johnson solids.

For n > 120, there are two such convex polyhedra for even n, the (n/2)-gonal prism and (n/2)-gonal antiprism, and no polyhedra for odd n.

			For n = 12, the a(12) = 10 convex polyhedra with regular polygonal faces and 12 vertices are: the icosahedron, the truncated tetrahedron, the cuboctahedron, the hexagonal prism, the hexagonal antiprism, and the Johnson solids J_4, J_16, J_27, J_53, and J_88.

Peter Kagey, Table of n, a(n) for n = 1..1000
Wikipedia, List of Johnson Solids

Cf. A180916 (analog for faces), A333661 (analog for edges), A333657.

a[n_] := Count[
  Join[
    PolyhedronData["Platonic", "VertexCount"],
    PolyhedronData["Archimedean", "VertexCount"],
    PolyhedronData["Johnson", "VertexCount"],
    Prepend[Range[10, n, 2], 6], (*Prisms, excluding cube*)
    Range[8, n, 2] (*Antiprisms, excluding octahedron*)
  ],
  n
]

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

Original entry on oeis.org

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, 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, 1, 3, 0, 3, 0, 0, 2, 1, 0, 1, 0, 4, 1, 0, 0, 2, 0, 0, 1

Offset: 1

Peter Kagey, Sep 02 2020

nonn

Convex polyhedra with whose faces are regular polygons are either Platonic solids, Archimedean solids, prisms, antiprisms, or Johnson solids.

			For n = 18, the a(18) = 4 polyhedra are: the truncated tetrahedron, the hexagonal prism, and the Johnson solids J_64 and J_84.
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.
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

Peter Kagey, Table of n, a(n) for n = 1..1000
Wikipedia, List of Johnson Solids

Cf. A180916 (analog for faces), A333660 (analog for vertices), A333657.

a[n_] := Count[
  Join[
   PolyhedronData["Johnson", "EdgeCount"],
   PolyhedronData["Platonic", "EdgeCount"],
   PolyhedronData["Archimedean", "EdgeCount"],
   Prepend[Range[15, n, 3], 9], (*Prisms, excluding cube*)
   Range[16, n, 4] (*Antiprisms, excluding octahedron*)
  ],
  n
]

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A296602 Values of F for which there is a unique convex polyhedron with F faces that are all regular polygons.

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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.

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A181708 Numbers of faces of polyhedra made entirely of regular polygons that are not prisms or antiprisms.

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A333660 a(n) is the number of n-vertex convex polyhedra whose faces are regular polygons.

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Mathematica

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

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Programs

Mathematica