A058787 -id:A058787 - cp's OEIS frontend

A212438 Irregular triangle read by rows: T(n,k) is the number of polyhedra with n faces and k vertices (n >= 4, k=4..2n-4).

1, 0, 1, 1, 0, 1, 2, 2, 2, 0, 0, 2, 8, 11, 8, 5, 0, 0, 2, 11, 42, 74, 76, 38, 14, 0, 0, 0, 8, 74, 296, 633, 768, 558, 219, 50, 0, 0, 0, 5, 76, 633, 2635, 6134, 8822, 7916, 4442, 1404, 233, 0, 0, 0, 0, 38, 768, 6134, 25626, 64439, 104213, 112082, 79773, 36528, 9714, 1249

Offset: 4

N. J. A. Sloane, May 16 2012

Because of duality, T(n,k) = T(k,n). - Ivan Neretin, May 25 2016

The number of edges is n+k-2. - Andrew Howroyd, Mar 27 2021

			Triangle begins:
1
0 1 1
0 1 2  2  2
0 0 2  8 11   8    5
0 0 2 11 42  74   76   38   14
0 0 0  8 74 296  633  768  558  219   50
0 0 0  5 76 633 2635 6134 8822 7916 4442 1404 233
...

Andrew Howroyd, Table of n, a(n) for n = 4..199 (rows 4..17)
Gunnar Brinkmann and Brendan McKay, Fast generation of planar graphs (expanded edition), Table 9-11.
G. P. Michon, Counting Polyhedra

A049337, A058787, A212438 are all versions of the same triangle.
Row sums (the same as column sums) are A000944.
Main diagonal is A002856.
Cf. A002840 (by edges), A239893.

Terms a(53) and beyond from Andrew Howroyd, Mar 27 2021

A049337 Triangle read by rows: T(n,k) is the number of 3-connected planar graphs (or polyhedra) with n >= 1 nodes and 0 <= k <= C(n,2) edges.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 2, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 8, 11, 8, 5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 11, 42, 74, 76, 38, 14, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 8, 74, 296, 633, 768, 558, 219, 50

Offset: 1

Brendan McKay

			Triangle begins
  0;
  0,0;
  0,0,0,0;
  0,0,0,1,0,0,0;
  0,0,0,0,0,1,1,0,0,0;
  0,0,0,0,0,0,1,2,2,...;
  ...
From _Hugo Pfoertner_, Nov 24 2020: (Start)
Transposed table:
.
                              Nodes                        Sums
       4  5  6   7   8    9    10     11     12    13  14 |A002840
  Edges-+--+--+---+---+----+-----+------+------+-----+---+|-------
   6 | 1  .  .   .   .    .     .      .      .     .   . |      1
   7 | .  .  .   .   .    .     .      .      .     .   . |      0
   8 | .  1  .   .   .    .     .      .      .     .   . |      1
   9 | .  1  1   .   .    .     .      .      .     .   . |      2
  10 | .  .  2   .   .    .     .      .      .     .   . |      2
  11 | .  .  2   2   .    .     .      .      .     .   . |      4
  12 | .  .  2   8   2    .     .      .      .     .   . |     12
  13 | .  .  .  11  11    .     .      .      .     .   . |     22
  14 | .  .  .   8  42    8     .      .      .     .   . |     58
  15 | .  .  .   5  74   74     5      .      .     .   . |    158
  16 | .  .  .   .  76  296    76      .      .     .   . |    448
  17 | .  .  .   .  38  633   633     38      .     .   . |   1342
  18 | .  .  .   .  14  768  2635    768     14     .   . |   4199
  19 | .  .  .   .   .  538  6134   6134    558     .   . |  13384
  20 | .  .  .   .   .  219  8822  25626   8822   219   . |  43708
  21 | .  .  .   .   .   50  7916  64439  64439  7916  50 | 144810
  .. | .  .  .   .   .    .    ..     ..     ..    ..  .. |     ..
     ---+--+--+---+---+----+-----+------+-------+----+---+
  Sums 1  2  7  34 257 2606 32300 440564 6384634 .. A000944
(End)

Sean A. Irvine, Table of n, a(n) for n = 1..469; rows 1..14 flattened

Cf. A000944, A002840, A021103, A003094, A049334.
A049337, A058787, A212438 are all versions of the same triangle.
Cf. A058788.

Missing zeros inserted by Sean A. Irvine, Jul 29 2021

A058786 Number of n-hedra with 2n-5 vertices or 3n-7 edges (the vertices of these are all of degree 3, except one which is of degree 4). Alternatively, the number of polyhedra with n vertices whose faces are all triangular, except one which is tetragonal.

Original entry on oeis.org

1, 2, 8, 38, 219, 1404, 9714, 70454, 527235, 4037671, 31477887, 249026400, 1994599707, 16147744792, 131959532817, 1087376999834, 9027039627035, 75441790558926, 634311771606750, 5362639252793358, 45565021714371644, 388937603694422120, 3333984869758146814

Offset: 5

Gerard P. Michon, Nov 29 2000

			a(5)=1 because the square pyramid is the only pentahedron with 5=2*5-5 vertices (or 8=3*5-7 edges). Alternatively, a(5)=1 because the square pyramid is the only polyhedron with 5 vertices whose faces are all triangles with only one tetragonal exception.

Andrew Howroyd, Table of n, a(n) for n = 5..500
CombOS - Combinatorial Object Server, generate planar graphs
G. P. Michon, Counting Polyhedra

Column k=4 of A342053.

Cf. A000109, A002856, A000944, A002840, A058787, A058788, A049337.

A342053ColSeq(25,4) \\ See links in A342053. - Andrew Howroyd, Feb 28 2021

Terms a(19) and beyond from Andrew Howroyd, Feb 27 2021

A058788 Triangle T(n,k) = number of polyhedra (triconnected planar graphs) with n edges and k vertices (or k faces), where (n/3+2) <= k <= (2n/3). Note that there is no such k when n=7.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 2, 2, 8, 2, 11, 11, 8, 42, 8, 5, 74, 74, 5, 76, 296, 76, 38, 633, 633, 38, 14, 768, 2635, 768, 14, 558, 6134, 6134, 558, 219, 8822, 25626, 8822, 219, 50, 7916, 64439, 64439, 7916, 50, 4442, 104213, 268394, 104213, 4442, 1404, 112082, 709302, 709302, 112082, 1404, 233, 79773, 1263032, 2937495, 1263032, 79773, 233, 36528, 1556952, 8085725, 8085725, 1556952, 36528, 9714, 1338853, 15535572, 33310550

Offset: 6

Gerard P. Michon, Nov 29 2000

Rows are of lengths 1,0,1,2,1,2,3,2,3,4,3,4,5,4,5,6,5, ... n-1-2*floor((n+2)/3). See A008611. Note the zero length, which means that there are no polyhedra with n=7 edges.

			There are 768 different polyhedra with 18 edges and 9 or 11 faces.

G. P. Michon, Counting Polyhedra

Cf. A000109, A002856, A000944, A002840, A058786, A058787, A049337, A008611.

A058789 Number of polyhedra with n faces and n+1 vertices (or n vertices and n+1 faces).

Original entry on oeis.org

0, 1, 2, 11, 74, 633, 6134, 64439, 709302, 8085725, 94713809, 1134914458, 13865916560, 172301697581, 2173270387051

Offset: 4

Gerard P. Michon, Nov 30 2000

Through a(18) the only primes are 2, 11, and 64439. - Jonathan Vos Post, Apr 23 2011

			a(5)=1 because the triangular prism is the only pentahedron with 6 vertices.

G. P. Michon, Counting Polyhedra

Cf. A002856, A000109, A058786, A058787, A058788.

cp's OEIS Frontend

A212438 Irregular triangle read by rows: T(n,k) is the number of polyhedra with n faces and k vertices (n >= 4, k=4..2n-4).

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A049337 Triangle read by rows: T(n,k) is the number of 3-connected planar graphs (or polyhedra) with n >= 1 nodes and 0 <= k <= C(n,2) edges.

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A058786 Number of n-hedra with 2n-5 vertices or 3n-7 edges (the vertices of these are all of degree 3, except one which is of degree 4). Alternatively, the number of polyhedra with n vertices whose faces are all triangular, except one which is tetragonal.

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A058788 Triangle T(n,k) = number of polyhedra (triconnected planar graphs) with n edges and k vertices (or k faces), where (n/3+2) <= k <= (2n/3). Note that there is no such k when n=7.

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A058789 Number of polyhedra with n faces and n+1 vertices (or n vertices and n+1 faces).

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