A242731 -id:A242731 - cp's OEIS frontend

A384380 Irregular triangle read by rows: T(n,k) is the number of connected subsets of k faces (or polyforms) of the n-th Johnson solid, up to symmetries of that solid; 1 <= n <= 92, 1 <= k <= A242731(n).

2, 2, 3, 2, 1, 2, 2, 3, 3, 2, 1, 4, 4, 9, 14, 14, 9, 4, 1, 4, 4, 12, 20, 32, 30, 23, 11, 4, 1, 4, 4, 13, 29, 54, 75, 75, 55, 31, 12, 4, 1, 5, 5, 17, 43, 118, 285, 595, 992, 1320, 1348, 1045, 603, 262, 86, 22, 5, 1, 3, 4, 6, 7, 6, 3, 1, 3, 4, 7, 12, 17, 16, 9, 3, 1

Offset: 1

Pontus von Brömssen and Peter Kagey, May 28 2025

Two faces are connected if they share an edge.

Equivalently, T(n,k) is the number of connected induced k-vertex subgraphs of the 1-skeleton of the dual of the n-th Johnson solid, up to symmetries of that dual.

			Triangle begins:
  n\k| 1  2  3  4   5   6   7    8    9   10   11   12   13  14  15 16 17 18
  ---+----------------------------------------------------------------------
   1 | 2  2  3  2   1
   2 | 2  2  3  3   2   1
   3 | 4  4  9 14  14   9   4    1
   4 | 4  4 12 20  32  30  23   11    4    1
   5 | 4  4 13 29  54  75  75   55   31   12    4    1
   6 | 5  5 17 43 118 285 595  992 1320 1348 1045  603  262  86  22  5  1
   7 | 3  4  6  7   6   3   1
   8 | 3  4  7 12  17  16   9    3    1
   9 | 3  4  7 13  24  35  36   22    9    3    1
  10 | 4  4  8 15  28  47  81  102   87   45   16    4    1
  11 | 4  4  8 17  35  71 139  252  378  429  340  183   67  18   4  1
  12 | 1  2  2  3   1   1
  13 | 1  2  2  5   6  10   7    5    1    1
  14 | 2  3  5  7  10   9   6    2    1
  15 | 2  3  5 11  19  31  38   38   20   10    2    1
  16 | 2  3  5 11  23  45  82  126  154  130   77   30   10   2   1
  17 | 2  3  4 10  16  35  61  120  180  237  194  117   40  13   2  1
  18 | 6  7 17 36  81 165 300  386  337  197   82   25    6   1
  19 | 6  7 20 44 121 290 701 1403 2359 3047 2975 2110 1106 435 131 31  6  1

Pontus von Brömssen, Table of n, a(n) for n = 1..341 (first 24 rows)
Eric Weisstein's World of Mathematics, Johnson Solid.
Wikipedia, List of Johnson solids.

Cf. A242731 (row lengths), A384376, A384378, A384381 (row sums).

A242732 Number of edges of Johnson solids in the order given by Johnson.

Original entry on oeis.org

8, 10, 15, 20, 25, 35, 12, 16, 20, 20, 25, 9, 15, 15, 20, 25, 24, 27, 36, 45, 55, 33, 44, 55, 65, 14, 24, 32, 32, 40, 40, 50, 50, 60, 36, 36, 48, 60, 60, 70, 70, 80, 80, 42, 56, 70, 80, 90, 13, 17, 21, 19, 23, 22, 26, 26, 30, 35, 40, 40, 45, 20, 15, 18, 27, 48

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, A242733.

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.

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.

A296604 Number of Johnson solids with n faces.

Original entry on oeis.org

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

Offset: 1

Jonathan Sondow, Jan 28 2018

nonn

Sum(n>0, a(n)) = 92, the number of Johnson solids, as conjectured by Johnson and proved by Zalgaller.

a(n) > 0 if and only if n is a member of A296603.

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

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, A296603.

a(62) = 5.

a(n) = 0 for n > 62.

A343954 a(n) is the number of triangular faces of Johnson solid J_n.

Original entry on oeis.org

4, 5, 4, 4, 5, 10, 4, 4, 5, 12, 15, 6, 10, 6, 8, 10, 16, 4, 4, 5, 10, 16, 20, 25, 30, 4, 8, 8, 8, 10, 10, 15, 15, 20, 8, 8, 8, 10, 10, 15, 15, 20, 20, 20, 24, 30, 35, 40, 6, 10, 14, 4, 8, 4, 8, 8, 12, 5, 10, 10, 15, 10, 5, 7, 8, 12, 16, 25, 30, 30, 35, 20, 20, 20, 20, 15, 15, 15, 15, 10, 10, 10, 5, 12, 24, 12, 16, 16, 18, 20, 8, 13

Offset: 1

Felix Fröhlich, May 05 2021

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

Cf. A242731, A343955, A343956, A343957, A343958, A343959.

a(n) <= A242731(n).

a(74)-a(92) from Pontus von Brömssen, May 27 2025

A343955 a(n) is the number of square faces of Johnson solid J_n.

Original entry on oeis.org

1, 0, 3, 5, 5, 0, 3, 5, 5, 1, 0, 0, 0, 3, 4, 5, 0, 9, 13, 15, 10, 3, 5, 5, 0, 4, 6, 10, 10, 10, 10, 5, 5, 0, 12, 12, 18, 20, 20, 15, 15, 10, 10, 6, 10, 10, 5, 0, 2, 1, 0, 4, 3, 5, 4, 4, 3, 0, 0, 0, 0, 0, 0, 0, 3, 5, 10, 5, 10, 10, 15, 30, 30, 30, 30, 25, 25, 25, 25, 20, 20, 20, 15, 0, 2, 2, 1, 2, 3, 4, 2, 3

Offset: 1

Felix Fröhlich, May 05 2021

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

Cf. A242731, A343954, A343956, A343957, A343958, A343959.

a(n) < A242731(n).

a(78)-a(92) from Pontus von Brömssen, May 27 2025

A343956 a(n) is the number of pentagonal faces of Johnson solid J_n.

Original entry on oeis.org

0, 1, 0, 0, 1, 6, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 6, 0, 0, 1, 6, 0, 0, 0, 0, 2, 2, 7, 7, 12, 0, 0, 0, 2, 2, 7, 7, 12, 12, 0, 0, 2, 7, 12, 0, 0, 0, 2, 2, 0, 0, 0, 0, 11, 10, 10, 9, 2, 3, 3, 0, 0, 0, 1, 2, 2, 3, 12, 12, 12, 12, 11, 11, 11, 11, 10, 10, 10, 9, 0, 0, 0, 0, 0, 0, 0, 4, 3

Offset: 1

Felix Fröhlich, May 05 2021

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

Cf. A242731, A343954, A343955, A343957, A343958, A343959.

a(n) < A242731(n).

a(82)-a(92) from Pontus von Brömssen, May 27 2025

A343957 a(n) is the number of hexagonal faces of Johnson solid J_n.

Original entry on oeis.org

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

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, A343958, A343959.

a(n) < A242731(n).

a(88)-a(92) from Pontus von Brömssen, May 27 2025

A343958 a(n) is the number of octagonal faces of Johnson solid J_n.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 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, 5, 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

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, A343959.

a(n) < A242731(n).

a(88)-a(92) from Pontus von Brömssen, May 27 2025

cp's OEIS Frontend

A384380 Irregular triangle read by rows: T(n,k) is the number of connected subsets of k faces (or polyforms) of the n-th Johnson solid, up to symmetries of that solid; 1 <= n <= 92, 1 <= k <= A242731(n).

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A242732 Number of edges 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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A296603 Number of faces a Johnson solid can have.

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A296604 Number of Johnson solids with n faces.

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A343954 a(n) is the number of triangular faces of Johnson solid J_n.

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A343955 a(n) is the number of square faces of Johnson solid J_n.

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A343956 a(n) is the number of pentagonal faces of Johnson solid J_n.

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A343957 a(n) is the number of hexagonal faces of Johnson solid J_n.

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A343958 a(n) is the number of octagonal faces of Johnson solid J_n.

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