A253882
Number of 3-connected planar triangulations of the sphere with n vertices up to orientation preserving isomorphisms.
Original entry on oeis.org
1, 1, 2, 6, 17, 73, 389, 2274, 14502, 97033, 672781, 4792530, 34911786, 259106122, 1954315346, 14949368524, 115784496932, 906736988527, 7171613842488, 57231089062625, 460428456484557, 3731572377382341, 30447133566946517, 249968326771680542, 2063931874299323140
Offset: 4
- Andrew Howroyd, Table of n, a(n) for n = 4..500
- CombOS - Combinatorial Object Server, generate planar graphs
- Pascal Honvault, Equivalent classes of degree sequences for triangulated polyhedra and their convex realization, Contributions to Disc. Math. (2021) Vol. 16, No. 1, 128-137.
- Pascal Honvault, Local geometry of polyhedra, hal-03744217 [math], 2022.
- The House of Graphs, Planar graphs
-
a(n)={if(n<3, 0, (2*binomial(4*(n-3)+1, n-3)/((n-2)*(3*n-7))
+ 3*sumdiv(n-2, d, if(d>=2, my(s=(n-2)/d); eulerphi(d)*binomial(4*s,s))/4)
+ if(n%2==1, my(s=(n-3)/2); 3*binomial(4*s,s)*(2*s+1)/(3*s+1))
+ if(n%3==1, my(s=(n-4)/3); 8*binomial(4*s,s)*(4*s+1)/(3*s+1))
+ if(n%3==0, my(s=(n-3)/3); 2*binomial(4*s,s)) )/(6*(n-2)))} \\ Andrew Howroyd, Mar 02 2021
Name clarified and terms a(24) and beyond from
Andrew Howroyd, Mar 02 2021
A119501
Number of isomorphism classes of 3-connected simple planar graphs (convex polytopes) where isomorphism does not allow reflection.
Original entry on oeis.org
0, 0, 0, 1, 2, 8, 45, 419, 4798, 62754, 872411, 12728018, 192324654, 2991463239, 47663036427, 775158142233, 12831576165782
Offset: 1
- Gunnar Brinkmann and Brendan McKay, plantri and fullgen programs for generation of certain types of planar graph.
- Gunnar Brinkmann and Brendan McKay, plantri and fullgen programs for generation of certain types of planar graph [Cached copy, pdf file only, no active links, with permission]
- CombOS - Combinatorial Object Server, generate planar graphs
A140800
Total number of vertices in all finite n-dimensional convex regular polytopes, or 0 if the number is infinite.
Original entry on oeis.org
1, 2, 0, 50, 773, 48, 83, 150, 281, 540, 1055, 2082, 4133, 8232, 16427, 32814, 65585, 131124, 262199, 524346, 1048637, 2097216, 4194371, 8388678, 16777289, 33554508, 67108943, 134217810, 268435541, 536871000, 1073741915, 2147483742
Offset: 0
a(0) = 1 because the 0-D regular polytope is the point.
a(1) = 2 because the only regular 1-D polytope is the line segment, with 2 vertices, one at each end.
a(2) = 0, indicating infinity, because the regular k-gon has k vertices.
a(3) = 50 (4 for the tetrahedron, 6 for the octahedron, 8 for the cube, 12 for the icosahedron, 20 for the dodecahedron) = the sum of A053016.
a(4) = 773 = 5 + 8 + 16 + 24 + 120 + 600 = sum of A063924.
For n>4 there are only the three regular n-dimensional polytopes, the simplex with n+1 vertices, the hypercube with 2^n vertices and the hyperoctahedron = cross polytope = orthoplex with 2*n vertices, for a total of A086653(n) + 1 = 2^n + 3*n + 1 (again restricted to n > 4).
- H. S. M. Coxeter, Regular Polytopes, 3rd ed., Dover, NY, 1973.
- Branko Grunbaum, Convex Polytopes, second edition (first edition (1967) written with the cooperation of V. L. Klee, M. Perles and G. C. Shephard; second edition (2003) prepared by V. Kaibel, V. L. Klee and G. M. Ziegler), Graduate Texts in Mathematics, Vol. 221, Springer 2003.
- P. McMullen and E. Schulte, Abstract Regular Polytopes, Encyclopedia of Mathematics and its Applications, Vol. 92, Cambridge University Press, Cambridge, 2002.
Cf.
A000943,
A000944,
A019503,
A053016,
A060296,
A063924,
A063925,
A063926,
A063927,
A065984,
A086653,
A093478,
A093479,
A105230,
A105231.
-
LinearRecurrence[{4, -5, 2}, {1, 2, 0, 50, 773, 48, 83, 150}, 32] (* Georg Fischer, May 03 2019 *)
A308489
Number of 5-regular polyhedra with 2n nodes.
Original entry on oeis.org
1, 0, 1, 1, 6, 14, 96, 518, 3917, 29821, 240430, 1957382, 16166596
Offset: 6
- Mahdieh Hasheminezhad, Brendan D. McKay, and Tristan Reeves, Recursive generation of 5-regular planar graphs, WALCOM: Algorithms and Computation: Third International Workshop, WALCOM 2009, Kolkata, India, February 18-20, 2009, Proceedings, Springer, 2009, pp. 129-140.
A361578
Number of 5-connected polyhedra (or 5-connected simple planar graphs) with n nodes.
Original entry on oeis.org
1, 0, 1, 1, 5, 8, 30, 85, 382, 1550, 7352
Offset: 12
- M. Kirchweger, M. Scheucher, and S. Szeider, SAT-Based Generation of Planar Graphs, in preparation.
Cf.
A049373 (planar graphs with minimum degree~5) and
A111358 (5-connected planar trianguations)
A378074
Number of embeddings on the sphere of 2-connected homeomorphically irreducible planar graphs with n nodes.
Original entry on oeis.org
0, 0, 0, 1, 2, 9, 47, 420, 4673, 63253, 927238, 14342093, 229607392, 3776227106, 63482545872, 1087322656758, 18927037827561
Offset: 1
A006867
Number of irreducible polyhedral graphs with n faces.
Original entry on oeis.org
1, 2, 5, 20, 107, 826, 7703, 81231, 914973, 10772406
Offset: 4
- M. B. Dillencourt, personal communication.
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
A006869
Number of distinct vertex-degree sequences of n-faced polyhedral graphs.
Original entry on oeis.org
1, 2, 7, 18, 52, 133, 330, 762, 1681
Offset: 4
From _Andrey Zabolotskiy_, Jun 15 2022: (Start)
All A000944(6) = 7 topologically distinct hexahedra have distinct vertex-degree sequences, so a(6) = 7.
There are A000944(7) = 34 heptahedra (polyhedral graphs with 7 faces), but some of them have identical vertex-degree sequences. See Wikipedia for these a(7) = 18 vertex-degree sequences (or, equivalently by polyhedron duality, sets of faces). (End)
- M. B. Dillencourt, personal communication.
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
A361371
Number of weakly 3-connected simple planar digraphs with n unlabeled nodes.
Original entry on oeis.org
42, 2688, 316208
Offset: 4
- M. Kirchweger, M. Scheucher, and S. Szeider, SAT-Based Generation of Planar Graphs, in preparation.
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