A242732 -id:A242732 - cp's OEIS frontend

A384378 Irregular triangle read by rows: T(n,k) is the number of connected subsets of k edges (or polysticks) of the n-th Johnson solid, up to symmetries of that solid; 1 <= n <= 92, 1 <= k <= A242732(n).

2, 4, 7, 12, 10, 6, 2, 1, 2, 4, 9, 17, 28, 25, 16, 7, 2, 1, 4, 7, 20, 47, 123, 274, 531, 779, 758, 504, 241, 87, 22, 4, 1, 4, 7, 20, 51, 144, 382, 990, 2332, 4873, 8546, 11776, 11733, 8529, 4673, 1957, 639, 156, 31, 4, 1

Offset: 1

Pontus von Brömssen and Peter Kagey, May 28 2025

			Triangle begins:
  n\k| 1  2  3  4   5   6   7    8    9   10    11    12   13   14   15  16  17 18 19 20
  ---+----------------------------------------------------------------------------------
  1  | 2  4  7 12  10   6   2    1
  2  | 2  4  9 17  28  25  16    7    2    1
  3  | 4  7 20 47 123 274 531  779  758  504   241    87   22    4    1
  4  | 4  7 20 51 144 382 990 2332 4873 8546 11776 11733 8529 4673 1957 639 156 31  4  1

Pontus von Brömssen, Table of n, a(n) for n = 1..78 (first 5 rows)
Eric Weisstein's World of Mathematics, Johnson Solid.
Wikipedia, List of Johnson solids.

Cf. A242732 (row lengths), A384376, A384379 (row sums), A384380.

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

Original entry on oeis.org

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.

A242733 Number of vertices of Johnson solids in the order given by Johnson.

Original entry on oeis.org

5, 6, 9, 12, 15, 20, 7, 9, 11, 9, 11, 5, 7, 8, 10, 12, 10, 15, 20, 25, 30, 15, 20, 25, 30, 8, 12, 16, 16, 20, 20, 25, 25, 30, 18, 18, 24, 30, 30, 35, 35, 40, 40, 18, 24, 30, 35, 40, 7, 8, 9, 11, 12, 13, 14, 14, 15, 21, 22, 22, 23, 10, 9, 10, 15, 28, 32, 65, 70, 70

Offset: 1

Felix Fröhlich, May 21 2014

Felix Fröhlich, Table of n, a(n) for n = 1..92
Wikipedia, List of Johnson solids

Cf. A242731, A242732.

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

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

A384378 Irregular triangle read by rows: T(n,k) is the number of connected subsets of k edges (or polysticks) of the n-th Johnson solid, up to symmetries of that solid; 1 <= n <= 92, 1 <= k <= A242732(n).

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

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A242733 Number of vertices of Johnson solids in the order given by Johnson.

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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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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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