A000944 -id:A000944 - cp's OEIS frontend

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

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

Gerard P. Michon, Nov 29 2000

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).

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

G. P. Michon, Counting Polyhedra

Cf. A000109, A002856, A000944, A002840, A058786, A058788, A001651.

A049337, A058787, A212438 are all versions of the same triangle.

A279015 Greatest possible number of diagonals of a polyhedron having n faces.

Original entry on oeis.org

0, 0, 4, 10, 20, 34, 52, 73, 100, 128, 162, 199, 240, 285, 334, 387, 444, 505, 570, 639, 712, 789, 870, 955, 1044, 1137, 1234, 1335, 1440, 1549, 1662, 1779, 1900, 2025, 2154, 2287, 2424, 2565, 2710, 2859, 3012, 3169, 3330, 3495, 3664, 3837, 4014, 4195, 4380, 4569

Offset: 4

Vladimir Letsko, Dec 03 2016

Also the greatest possible number of diagonals of a simple polyhedron with n faces. In other words, a polyhedron with n faces having the greatest possible number of diagonals must be a simple one.

			a(6)=4 because 6 is the greatest possible number of diagonals of a hexahedron.

Colin Barker, Table of n, a(n) for n = 4..1000
D. Bremner, V. Klee, Inner Diagonals of Convex Polytopes, Journal of Combinatorial Theory, Series A, Volume 87, Issue 1, July 1999, Pages 175-197.
Vladimir Letsko, Mathematical Marathon, Problem 219 (in Russian)
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000944, A279019, A279022.

F:=n->piecewise(4<=n and n<=5,0,6<=n and n<=10,2*n^2-20*n+52,n=11,73,n=13,128,n=12 or n>=14,2*n^2-21*n+64);

Drop[#, 4] &@ CoefficientList[Series[x^6*(4 - 2 x + 2 x^2 - x^5 + 3 x^6 - 5 x^7 + 5 x^8 - 3 x^9 + x^10)/(1 - x)^3, {x, 0, 53}], x] (* Michael De Vlieger, Dec 05 2016 *)

concat(vector(2), Vec(x^6*(4 - 2*x + 2*x^2 - x^5 + 3*x^6 - 5*x^7 + 5*x^8 - 3*x^9 + x^10) / (1 - x)^3 + O(x^30))) \\ Colin Barker, Dec 05 2016

a(n) = 2*n^2 - 21*n + 64 for n=12 or n>=14.
From Colin Barker, Dec 05 2016: (Start)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>8.
G.f.: x^6*(4 - 2*x + 2*x^2 - x^5 + 3*x^6 - 5*x^7 + 5*x^8 - 3*x^9 + x^10) / (1 - x)^3.
(End)

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.

A005841 Number of 4-dimensional polytopes with n vertices.

Original entry on oeis.org

1, 4, 31, 1294, 274148

Offset: 5

N. J. A. Sloane

H. T. Croft, K. J. Falconer and R. K. Guy, Unsolved Problems in Geometry, B15.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Lukas Finschi, A Graph Theoretical Approach for Reconstruction and Generation of Oriented Matroids, A dissertation submitted to the Swiss Federal Institute of Technology, Zurich for the degree of Doctor of Mathematics, 2001. See Table 7.5.
Moritz Firsching, The complete enumeration of 4-polytopes and 3-spheres with nine vertices, arXiv:1803.05205 [math.MG], 2018.
Fukuda, Komei; Miyata, Hiroyuki; Moriyama, Sonoko. Complete Enumeration of Small Realizable Oriented Matroids. Discrete Comput. Geom. 49 (2013), no. 2, 359--381. MR3017917. Also arXiv:1204.0645. - From _N. J. A. Sloane_, Feb 16 2013

Cf. A000944.

a(9), from the paper by M. Firshing, added by Jud McCranie, Jan 30 2020

A093478 Number of regular (finite but not necessarily convex) polytopes of full rank in n-dimensional space, or -1 if the number is infinite.

Original entry on oeis.org

1, 1, -1, 18, 34, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6

Offset: 0

N. J. A. Sloane, May 22 2004

sign

P. McMullen and E. Schulte, Abstract Regular Polytopes, Encyclopedia of Mathematics and its Applications, Vol. 92, Cambridge University Press, Cambridge, 2002.

P. McMullen, Regular polytopes of full rank, Discrete Comput. Geom. 32 (2004), no. 1, 1-35.
Index entries for sequences related to polytopes

Cf. A093479, A060296, A000943, A000944, A053016, A063927.

A093479 Number of regular (infinite) apeirotopes of full rank in n-dimensional space.

Original entry on oeis.org

0, 1, 6, 8, 18, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8

Offset: 0

N. J. A. Sloane, May 22 2004

nonn

P. McMullen, Regular polytopes of full rank, lecture at The Coxeter Legacy meeting, Toronto, 2004.
P. McMullen and E. Schulte, Abstract Regular Polytopes, Encyclopedia of Mathematics and its Applications, Vol. 92, Cambridge University Press, Cambridge, 2002.
P. McMullen and E. Schulte, Paper to appear in Discrete and Computational Geometry, 2004.

Cf. A093478, A060296, A000943, A000944, A053016, A063927.

A187927 Number of embeddings on the sphere of 2-connected planar graphs, minimum vertex degree 3, with n nodes.

Original entry on oeis.org

2, 13, 163, 2067, 30953, 486674, 7957459, 133344454, 2280001754, 39648557743, 699731146514, 12511186297320

Offset: 6

Stuart E Anderson, Mar 16 2011

The graphs are exactly 2-connected, not at least 2-connected. The graphs were enumerated using plantri (by B.D. McKay & G. Brinkmann) for the purpose of finding compound perfect squared squares.

Stuart Anderson, Compound Perfect Squared Squares
S. E. Anderson, Compound Perfect Squared Squares of the Order Twenties, 2013.
Gunnar Brinkmann and Brendan McKay, Plantri Guide
Gunnar Brinkmann and Brendan McKay, Guide to using plantri [Cached copy, with permission]

Cf. A034889, A181340, A187928, A000944.

plantri
```
plantri -p -c2 -m3 -x -u -v n  ; or
```

a(15)-a(17) from Lorenz Milla, Oct 08 2013

A002841 Number of 3-connected self-dual planar graphs with 2n edges.

Original entry on oeis.org

1, 1, 2, 6, 16, 50, 165, 554, 1908, 6667, 23556, 84048, 302404, 1095536, 3993623

Offset: 3

N. J. A. Sloane

Also number of self-dual polyhedra with n+1 vertices (and n+1 faces). - Franklin T. Adams-Watters, Dec 18 2006

M. B. Dillencourt, Polyhedra of small orders and their Hamiltonian properties. Tech. Rep. 92-91, Info. and Comp. Sci. Dept., Univ. Calif. Irvine, 1992.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

C. J. Bouwkamp & N. J. A. Sloane, Correspondence, 1971
M. B. Dillencourt, Polyhedra of small orders and their Hamiltonian properties, Journal of Combinatorial Theory, Series B, Volume 66, Issue 1, January 1996, Pages 87-122. See bottom of Table IV on page 98.
P. J. Federico, Enumeration of polyhedra: the number of 9-hedra, J. Combin. Theory, 7 (1969), 155-161.
House of Graphs, Planar graphs
Eric Weisstein's World of Mathematics, Self-Dual Graph

Cf. A000944.

Definition corrected by Gordon F. Royle, Dec 15 2005

a(14)-a(17) added by Jan Goedgebeur, Sep 16 2021

A049340 Triangle read by rows: T(n,k) is the number of planar graphs with n >= 1 nodes and 0 <= k <= binomial(n,2) edges, all degrees even.

Original entry on oeis.org

1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 0, 0, 1, 1, 1, 1, 1, 0, 0, 0, 1, 0, 0, 1, 1, 1, 3, 2, 2, 1, 1, 0, 1, 0, 0, 0, 1, 0, 0, 1, 1, 1, 3, 4, 4, 6, 5, 5, 3, 2, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 1, 3, 4, 7, 9, 15, 17, 22, 14, 16, 5, 4, 0, 1, 0, 0, 0

Offset: 1

Brendan McKay

			Triangle begins:
  1;
  1, 0;
  1, 0, 0, 1;
  1, 0, 0, 1, 1, 0, 0;
  1, 0, 0, 1, 1, 1, 1, 1, 0, 0, 0;
  1, 0, 0, 1, 1, 1, 3, 2, 2, 1, 1, 0, 1, 0, 0, 0;
  ...

Sean A. Irvine, Table of n, a(n) for n = 1..231, rows 1..11 flattened
Sean A. Irvine, Java program (github)

Rows sums give A049339.

Cf. A000944, A021103, A003094, A049334.

Entry revised by Sean A. Irvine, Jul 29 2021

cp's OEIS Frontend

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

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A279015 Greatest possible number of diagonals of a polyhedron having n faces.

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Maple

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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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A005841 Number of 4-dimensional polytopes with n vertices.

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A093478 Number of regular (finite but not necessarily convex) polytopes of full rank in n-dimensional space, or -1 if the number is infinite.

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A093479 Number of regular (infinite) apeirotopes of full rank in n-dimensional space.

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A187927 Number of embeddings on the sphere of 2-connected planar graphs, minimum vertex degree 3, with n nodes.

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plantri

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A002841 Number of 3-connected self-dual planar graphs with 2n edges.

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