A000109 -id:A000109 - cp's OEIS frontend

A049338 Erroneous version of A000109.

1, 1, 1, 2, 5, 14, 50, 233, 1249, 7616

Offset: 3

dead

A000103 Number of n-node triangulations of sphere in which every node has degree >= 4.

0, 0, 1, 1, 2, 5, 12, 34, 130, 525, 2472, 12400, 65619, 357504, 1992985, 11284042, 64719885, 375126827, 2194439398, 12941995397, 76890024027, 459873914230, 2767364341936, 16747182732792

Offset: 4

N. J. A. Sloane

			a(4)=0, a(5)=0 because the tetrahedron and the 5-bipyramid both have vertices of degree 3. a(6)=1 because of the A000109(6)=2 triangulations with 6 nodes (abcdef) the one corresponding to the octahedron (bcde,afec,abfd,acfe,adfb,bedc) has no node of degree 3, whereas the other triangulation (bcdef,afec,abed,ace,adcbf,aeb) has 2 such nodes.

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).

R. Bowen and S. Fisk, Generation of triangulations of the sphere, Math. Comp., 21 (1967), 250-252.
R. Bowen and S. Fisk, Generation of triangulations of the sphere [Annotated scanned copy]
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
R. K. Guy, The Second Strong Law of Small Numbers, Math. Mag, 63 (1990), no. 1, 3-20. [Annotated scanned copy]
D. A. Holton and B. D. McKay, The smallest non-hamiltonian 3-connected cubic planar graphs have 38 vertices, J. Combinat. Theory B vol 45, iss. 3 (1988) 305-319.
D. A. Holton and B. D. McKay, Erratum, J. Combinat. Theory B vol 47, iss. 2 (1989) 248.
J. Lederberg, Dendral-64, II, Report to NASA, Dec 1965 [Annotated scanned copy]
Thom Sulanke, Generating triangulations of surfaces (surftri), (also subpages).
Eric Weisstein's World of Mathematics, Cubic Polyhedral Graph

Cf. all triangulations: A000109, triangulations with minimum degree 5: A081621.

More terms from Hugo Pfoertner, Mar 24 2003

More terms from Herman Jamke (hermanjamke(AT)fastmail.fm) from the Surftri web site, May 05 2007

A005964 Number of trivalent connected (or cubic) planar graphs with 2n nodes.

Original entry on oeis.org

0, 1, 1, 3, 9, 32, 133, 681, 3893, 24809, 169206, 1214462, 9034509, 69093299, 539991437

Offset: 1

N. J. A. Sloane

A. T. Balaban, Enumeration of Cyclic Graphs, pp. 63-105 of A. T. Balaban, ed., Chemical Applications of Graph Theory, Ac. Press, 1976; see p. 92.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

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]
F. C. Bussemaker, S. Cobeljic, L. M. Cvetkovic and J. J. Seidel, Computer investigations of cubic graphs, T.H.-Report 76-WSK-01, Technological University Eindhoven, Dept. Mathematics, 1976.
Jan Goedgebeur and Patric R. J. Ostergard, Switching 3-Edge-Colorings of Cubic Graphs, arXiv:2105:01363 [math.CO], May 2021. See Table 3.
M. Meringer, Tables of Regular Graphs.
Eric Weisstein's World of Mathematics, Connected Graph.
Eric Weisstein's World of Mathematics, Cubic Graph.

Cf. A058378, A000109, A002851, A204186.

Extended by Brendan McKay and Gunnar Brinkmann using their program "plantri", Dec 19 2000

A007021 Number of 4-connected simplicial polyhedra with n nodes.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 4, 10, 25, 87, 313, 1357, 6244, 30926, 158428, 836749, 4504607, 24649284, 136610879, 765598927, 4332047595, 24724362117, 142205424580, 823687567019, 4801749063379

Offset: 3

N. J. A. Sloane

Also the number of 4-connected triangulations on n vertices. - Manfred Scheucher, Mar 17 2023

M. B. Dillencourt, Polyhedra of small orders and their Hamiltonian properties. Tech. Rep. 92-91, Info. and Comp. Sci. Dept., Univ. Calif. Irvine, 1992.
H. Heesch, Ein zum Vierfarbensatz aquivalenter Satz der Panisochromie [ A theorem of panisochromaticity equivalent to the four color theorem ], pp. 229-253 of Graph Theory in Memory of G. A. Dirac (Sandbjerg, 1985). Edited by L. D. Andersen et al., Annals of Discrete Mathematics, 41. North-Holland Publishing Co., Amsterdam-New York, 1989.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Gunnar Brinkmann and Brendan McKay, Guide to using plantri
Gunnar Brinkmann and Brendan McKay, Guide to using plantri [Cached copy, with permission]
S. Butler et al., Irreducible Apollonian configurations and packings, Discrete and Computational Geometry, 44 (2010), 487-507.
CombOS - Combinatorial Object Server, generate planar graphs
Paul Jungeblut, Edge Guarding Plane Graphs, Master Thesis, Karlsruhe Institute of Technology (Germany, 2019).
Irene Pivotto and Gordon Royle, Highly-connected planar cubic graphs with few or many Hamilton cycles, arXiv:1901.10683 [math.CO], 2019.
Manfred Scheucher, Hendrik Schrezenmaier, and Raphael Steiner, A Note On Universal Point Sets for Planar Graphs, arXiv:1811.06482 [math.CO], 2018.

Cf. A000109, A007027, A111358.

More terms generated with plantri by Moritz Firsching, Aug 20 2015

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

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.

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.

A081621 Number of n-node triangulations of the sphere with minimal degree 5.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 3, 4, 12, 23, 73, 192, 651, 2070, 7290, 25381, 91441, 329824, 1204737, 4412031, 16248772, 59995535, 222231424, 825028656, 3069993552, 11446245342, 42758608761, 160012226334, 599822851579, 2252137171764, 8469193859271, 31896058068930

Offset: 4

Hugo Pfoertner, Mar 24 2003

nonn

Other face sizes larger than 5 and 6 are allowed and there can be more than 12 vertices with degree 5.
Convex polytopes with minimum degree at least 5. The sequence is extracted from the file more-counts.txt that comes with the plantri distribution.
Grace conjectured that all polyhedra inscribed in the unit sphere with maximal volume are "medial" (all faces triangular and vertex degree either m or m+1 where m < 6 - 12/n < m+1). For n = 12 and n > 13 the medial polyhedra have 12 vertices of degree 5 and n-12 vertices of degree 6. All known numerical solutions of the maximal volume problem (A081314) have this property.
The triangulated arrangements of points on a sphere with icosahedral symmetry given by Hardin, Sloane and Smith are examples for large n.

			With vertices denoted by letters a, b, ... the neighbor lists are for a(14)=1: (bcdef, afghc, abhid, acije, adjkf, aeklgb, bflmh, bgmic, chmnjd, dinke, ejnlf, fknmg, glnih, imlkj).
a(15)=1: (bcdefg, aghic, abijd, acjke, adklf, aelmg, afmhb, bgmni, bhnjc, cinokd, djole, ekomf, flonhg, hmoji, jnmlk); a(16)=3: (bcdef, afghc, abhijd, acjke, adklf, aelmgb, bfmnh, bgnic, chnoj, ciokd, djople, ekpmf, flpng, gmpoih, inpkj, konml), (bcdef, afghc, abhijd, acjke, adklf, aelmgb, bfmnh, bgnic, chnoj, ciopkd, djple, ekpmf, flpong, gmoih, inmpj, jomlk), (bcdef, afghijc, abjkd, ackle, adlmf, aemgb, bfmnh, bgnoi, bhopj, bipkc, cjpld, dkponme, elngf, gmloh, hnlpi, iolkj).

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]
G. Brinkmann and B. D. McKay, Construction of planar triangulations with minimum degree 5, Discr. Math. 301 (2005), 147-163.
CombOS - Combinatorial Object Server, generate planar graphs.
D. W. Grace, Search for largest polyhedra, Math. Comp. 17, 197-199 (1963).
R. H. Hardin, N. J. A. Sloane and W. D. Smith, Spherical Codes with Icosahedral Symmetry.
Hugo Pfoertner, Icosahedral best coverings.
Sage, Common Graphs.
Thom Sulanke, Generating triangulations of surfaces (surftri), (also subpages).

Cf. A000109, A000103, A081314.

More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), May 05 2007

a(41) computed with plantri by Jan Goedgebeur, Dec 03 2021

A108239 Number of triangulated surfaces with n vertices.

Original entry on oeis.org

1, 1, 3, 9, 43, 655, 42426, 11590894, 12561206794

Offset: 4

Ralf Stephan, Jun 17 2005

F. H. Lutz, Enumeration and random realization of triangulated surfaces, arXiv:math/0610022 [math.CO], 2006-2007.
F. H. Lutz, The Manifold Page.
T. Sulanke and F. H. Lutz, Isomorphism free lexicographic enumeration of triangulated surfaces and 3-manifolds, arXiv:math/0610022 [math.CO], 2006-2007.

Cf. A000109.

a(11)-a(12) from Herman Jamke (hermanjamke(AT)fastmail.fm), Nov 08 2006

A111358 Numbers of planar triangulations with minimum degree 5 and without separating 3- or 4-cycles - that is 3- or 4-cycles where the interior and exterior contain at least one vertex.

Original entry on oeis.org

1, 0, 1, 1, 3, 4, 12, 23, 71, 187, 627, 1970, 6833, 23384, 82625, 292164, 1045329, 3750277, 13532724, 48977625, 177919099, 648145255, 2368046117, 8674199554, 31854078139, 117252592450, 432576302286, 1599320144703, 5925181102878

Offset: 12

Gunnar Brinkmann, Nov 07 2005

nonn

A006791 and this sequence are the same sequence. The correspondence is just that these objects are planar duals of each other. But the offset and step are different: if the cubic graph has 2*n vertices, the dual triangulation has n+2 vertices. - Brendan McKay, May 24 2017

Also the number of 5-connected triangulations on n vertices. - Manfred Scheucher, Mar 17 2023

			The icosahedron is the smallest triangulation with minimum degree 5 and it doesn't contain any separating 3- or 4-cycles. Examples can easily be seen as 2D and 3D pictures using the program CaGe cited above.

G. Brinkmann, CaGe.
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]
G. Brinkmann and Brendan D. McKay, Construction of planar triangulations with minimum degree 5 , Disc. Math. vol 301, iss. 2-3 (2005) 147-163.
D. A. Holton and B. D. McKay, The smallest non-hamiltonian 3-connected cubic planar graphs have 38 vertices, J. Combinat. Theory B vol 45, iss. 3 (1988) 305-319.
D. A. Holton and B. D. McKay, Erratum, J. Combinat. Theory B vol 47, iss. 2 (1989) 248.

Cf. A081621, A007894, A006791, A000109, A007021, A361578.

cp's OEIS Frontend

A049338 Erroneous version of A000109.

Views

Author

Keywords

A000103 Number of n-node triangulations of sphere in which every node has degree >= 4.

Views

Author

Keywords

Examples

References

Links

Crossrefs

Extensions

A005964 Number of trivalent connected (or cubic) planar graphs with 2n nodes.

Views

Author

Keywords

References

Links

Crossrefs

Extensions

A007021 Number of 4-connected simplicial polyhedra with n nodes.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Extensions

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

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.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

Extensions

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A081621 Number of n-node triangulations of the sphere with minimal degree 5.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Extensions

A108239 Number of triangulated surfaces with n vertices.

Views

Author

Keywords

Links

Crossrefs

Extensions

A111358 Numbers of planar triangulations with minimum degree 5 and without separating 3- or 4-cycles - that is 3- or 4-cycles where the interior and exterior contain at least one vertex.

Views

Author

Keywords

Comments

Examples

Links