A296604 -id:A296604 - cp's OEIS frontend

A242731 Number of faces of Johnson solids in the order given by Johnson.

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

Offset: 1

Felix Fröhlich, May 21 2014

For the distinct terms sorted, see A296603. For the number of Johnson solids with n faces, see A296604. - Jonathan Sondow, Jan 29 2018

Felix Fröhlich, Table of n, a(n) for n = 1..92
Norman W. Johnson, Convex Polyhedra with Regular Faces, Canadian Journal of Mathematics, 18 (1966), 169-200.
Wikipedia, List of Johnson solids
Victor A. Zalgaller, Convex Polyhedra with Regular Faces, Zap. Nauchn. Sem. LOMI, 1967, Volume 2. Pages 5-221 (Mi znsl1408).

Cf. A242732, A242733, A296602, A296603, A296604.

A296603 Number of faces a Johnson solid can have.

Original entry on oeis.org

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

Offset: 1

Jonathan Sondow, Jan 28 2018

Distinct terms in A242731, sorted.

n is a member if and only if A296604(n) > 0.

			The square pyramid is the Johnson solid with the fewest faces, namely, 5, so a(1) = 5.

Norman W. Johnson, Convex Polyhedra with Regular Faces, Canadian Journal of Mathematics, 18 (1966), 169-200.
Eric Weisstein's World of Mathematics, Johnson Solid.
Wikipedia, List of Johnson solids.
Victor A. Zalgaller, Convex Polyhedra with Regular Faces, Zap. Nauchn. Sem. LOMI, 1967, Volume 2. Pages 5-221 (Mi znsl1408).

Cf. A181708, A242731, A296602, A296604.

A180916 Number of convex polyhedra with n faces that are all regular polygons.

Original entry on oeis.org

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

Offset: 1

J. Lowell, Sep 23 2010

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

			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

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

Cf. A296602, A296603, A296604.

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 *)

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

More terms from J. Lowell, Feb 28 2011

Corrected by T. D. Noe, Mar 04 2011

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

Original entry on oeis.org

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)

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

A242731 Number of faces of Johnson solids in the order given by Johnson.

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A296603 Number of faces a Johnson solid can have.

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A180916 Number of convex polyhedra with n faces that are all regular polygons.

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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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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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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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