cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-6 of 6 results.

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).

Original entry on oeis.org

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

Views

Author

N. J. A. Sloane, May 16 2012

Keywords

Comments

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

Examples

			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
...

Links

Crossrefs

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.

Extensions

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

A058787 Triangle T(n,k) = number of polyhedra (triconnected planar graphs) with n faces and k vertices, where (n/2+2) <= k <= (2n+8).

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 2, 2, 8, 11, 8, 5, 2, 11, 42, 74, 76, 38, 14, 8, 74, 296, 633, 768, 558, 219, 50, 5, 76, 633, 2635, 6134, 8822, 7916, 4442, 1404, 233, 38, 768, 6134, 25626, 64439, 104213, 112082, 79773, 36528, 9714, 1249, 14, 558, 8822, 64439, 268394, 709302
Offset: 4

Views

Author

Gerard P. Michon, Nov 29 2000

Keywords

Comments

Rows are of lengths 1, 2, 4, 5, 7, 8, 10, 11, 13, 14, ... floor(3n/2)-5. See A001651 (this is the sequence of integers not divisible by 3).

Examples

			There are 38 polyhedra with 9 faces and 11 vertices, or with 11 faces and 9 vertices.

Links

Crossrefs

Cf. A000109, A002856, A000944, A002840, A058786, A058788, A001651.
A049337, A058787, A212438 are all versions of the same triangle.

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

Views

Author

Gerard P. Michon, Nov 29 2000

Keywords

Examples

			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.

Links

Crossrefs

Column k=4 of A342053.
Cf. A000109, A002856, A000944, A002840, A058787, A058788, A049337.

Programs

Extensions

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

Views

Author

Gerard P. Michon, Nov 29 2000

Keywords

Comments

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.

Examples

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

Links

Crossrefs

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

A342057 Number of polyhedra with n faces and n vertices up to orientation preserving isomorphisms.

Original entry on oeis.org

1, 1, 3, 11, 69, 541, 5096, 50586, 534292, 5865150, 66582243, 776705379, 9274453627, 112984297173
Offset: 4

Views

Author

Andrew Howroyd, Mar 27 2021

Keywords

Crossrefs

Main diagonal of A239893.
Cf. A002856.

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

Views

Author

Gerard P. Michon, Nov 30 2000

Keywords

Comments

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

Examples

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

Links

Crossrefs

Cf. A002856, A000109, A058786, A058787, A058788.
Showing 1-6 of 6 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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