A242731 -id:A242731 - cp's OEIS frontend

A343959 a(n) is the number of decagonal faces of Johnson solid J_n.

0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 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, 11, 10, 10, 9, 0, 0, 0, 0, 1, 1, 1, 1, 2, 2, 2, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Felix Fröhlich, May 05 2021

Eric Weisstein's World of Mathematics, Johnson Solid.
Wikipedia, List of Johnson solids.

Cf. A242731, A343954, A343955, A343956, A343957, A343958.

a(n) < A242731(n).

a(87)-a(92) from Pontus von Brömssen, May 27 2025

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)

A343209 Number of spanning trees of the graph of the n-th Johnson solid.

Original entry on oeis.org

45, 121, 1815, 24000, 297025, 78250050, 361, 3509, 30976, 27216, 403202, 75, 1805, 1728, 31500, 508805, 207368, 1609152, 227402340, 29821320745, 8223103375490, 37158912, 15482880000, 5996600870820, 1702422879696000, 1176, 324900, 29859840, 30950832, 2518646460

Offset: 1

Pontus von Brömssen, Apr 08 2021

Terms are taken from the paper by Horiyama and Shoji, verified by Pontus von Brömssen.

			The gyrobifastigium (J26) has a(26) = 1176 spanning trees.

Pontus von Brömssen, Table of n, a(n) for n = 1..92
Takashi Horiyama and Wataru Shoji, The number of different unfoldings of polyhedra, The 29th European Workshop on Computational Geometry, Technische Universität Braunschweig 2013, 143-146. [Apparently, two pairs of Johnson solids have switched numbers in Table 2, namely J32 <-> J33 and J40 <-> J41.]
Wikipedia, List of Johnson solids

Cf. A242731, A343210, A343213.

A343210 Number of nonisomorphic unfoldings of the n-th Johnson solid.

Original entry on oeis.org

8, 15, 308, 3030, 29757, 7825005, 63, 448, 3116, 3421, 40321, 9, 99, 156, 2010, 25574, 13041, 268260, 28427091, 2982139245, 822310337549, 6193152, 1935360000, 599660087082, 170242287969600, 152, 27195, 1867560, 1934427, 125939163, 132627603, 74520844992

Offset: 1

Pontus von Brömssen, Apr 08 2021

Unfoldings with overlaps are allowed.

Terms are taken from the paper by Horiyama and Shoji.

			The gyrobifastigium (J26) has a(26) = 152 nonisomorphic unfoldings.

Pontus von Brömssen, Table of n, a(n) for n = 1..92
Takashi Horiyama and Wataru Shoji, The number of different unfoldings of polyhedra, The 29th European Workshop on Computational Geometry, Technische Universität Braunschweig 2013, 143-146. [Apparently, two pairs of Johnson solids have switched numbers in Table 2, namely J32 <-> J33 and J40 <-> J41.]
Wikipedia, List of Johnson solids

Cf. A242731, A343209.

A343211 Number of (undirected) Hamiltonian cycles of the graph of the n-th Johnson solid.

Original entry on oeis.org

4, 5, 7, 11, 16, 90, 6, 16, 30, 80, 240, 6, 30, 12, 52, 160, 268, 67, 225, 716, 3550, 794, 6228, 44092, 194620, 9, 96, 396, 361, 1350, 1296, 6560, 6520, 32560, 708, 718, 6033, 45625, 45856, 221970, 221680, 1083340, 1082370, 8422, 162301, 2751301, 12817980

Offset: 1

Pontus von Brömssen, Apr 08 2021

			The gyrobifastigium (J26) has a(26) = 9 Hamiltonian cycles.

Pontus von Brömssen, Table of n, a(n) for n = 1..92
Wikipedia, List of Johnson solids

Cf. A242731, A268283.

A299114 Number of sides of a face of an Archimedean solid.

Original entry on oeis.org

3, 4, 5, 6, 8, 10

Offset: 1

Jonathan Sondow, Feb 02 2018

Values of n for which the regular n-gon is a face of some Archimedean solid.

Remarkably, the same is true for Johnson solids. Indeed, before Johnson (1966) and Zalgaller (1967) classified the 92 Johnson solids, Grünbaum and Johnson (1965) proved that the only polygons that occur as faces of a non-uniform regular-faced convex polyhedron (i.e., a Johnson solid) are triangles, squares, pentagons, hexagons, octagons, and decagons.

Branko Grünbaum, Norman Johnson, The faces of a regular-faced polyhedron, J. Lond. Math. Soc. 40, 577-586 (1965).
Norman W. Johnson, Convex Polyhedra with Regular Faces, Canadian Journal of Mathematics, 18 (1966), 169-200.
Joseph Malkevitch, Regular-Faced Polyhedra: Remembering Norman Johnson, AMS Feature Column, Jan. 2018.
Eric Weisstein's World of Mathematics, Archimedean 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. A092536, A092537, A092538, A242731, A242732, A242733.

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

A343212 Order of the symmetry group of the n-th Johnson solid.

Original entry on oeis.org

8, 10, 6, 8, 10, 10, 6, 8, 10, 8, 10, 12, 20, 12, 16, 20, 16, 6, 8, 10, 10, 6, 8, 10, 10, 8, 12, 16, 16, 20, 20, 10, 10, 20, 12, 12, 16, 20, 20, 10, 10, 20, 20, 6, 8, 10, 5, 10, 4, 4, 12, 4, 4, 4, 8, 4, 12, 10, 20, 4, 6, 4, 6, 6, 6, 8, 16, 10, 20, 4, 6, 10, 20, 4, 6, 10, 10, 2, 2, 20, 4, 2, 6, 8, 16, 4, 2, 4, 4, 8, 8, 6

Offset: 1

Pontus von Brömssen, Apr 08 2021

			The gyrobifastigium (J26) has a(26) = 8 symmetries.

Wikipedia, List of Johnson solids

Cf. A098427, A242731.

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

A343959 a(n) is the number of decagonal faces of Johnson solid J_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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A343209 Number of spanning trees of the graph of the n-th Johnson solid.

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A343210 Number of nonisomorphic unfoldings of the n-th Johnson solid.

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A343211 Number of (undirected) Hamiltonian cycles of the graph of the n-th Johnson solid.

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A299114 Number of sides of a face of an Archimedean solid.

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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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A343212 Order of the symmetry group of the n-th Johnson solid.

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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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