A111361 -id:A111361 - cp's OEIS frontend

A167160 Duplicate of A111361.

0, 0, 0, 0, 1, 0, 1, 1, 2, 1, 5, 2, 8, 5, 12, 8, 25, 13, 30, 23, 51, 33, 76, 51, 109, 78, 144, 106, 218, 150, 274, 212, 382, 279, 499, 366, 650, 493, 815, 623, 1083, 800, 1305, 1020, 1653, 1261, 2045, 1554, 2505, 1946, 3008, 2322, 3713, 2829, 4354, 3418, 5233, 4063

Offset: 2

Jonathan Vos Post, Oct 29 2009

dead

A078666 Number of isomorphism classes of simple quadrangulations of the sphere having n+2 vertices and n faces, minimal degree 3, with orientation-reversing isomorphisms permitted.

Original entry on oeis.org

1, 0, 1, 1, 3, 3, 12, 19, 64, 155, 510, 1514, 5146, 16966, 58782, 203269, 716607, 2536201, 9062402, 32533568, 117498072, 426212952, 1553048548, 5681011890, 20858998805, 76850220654, 284057538480, 1053134292253, 3915683667721

Offset: 6

Slavik V. Jablan and Brendan McKay Feb 06 2003

nonn

Number of basic polyhedra with n vertices.
Initial terms of sequence coincide with A007022. Starting from n=12, to it is added the number of simple 4-regular 4-edge-connected but not 3-connected plane graphs on n nodes (A078672). As a result we obtain the number of basic polyhedra.
a(n) counts 4-valent 4-edge-connected planar maps (or plane graphs on a sphere) up to reflection with no regions bounded by just 2 edges. Conway called such maps "basic polyhedra" and used them in his knot notation. 2-edge-connected maps (which start occurring from n=12) are not taken into account here because they generate only composite knots and links. - Andrey Zabolotskiy, Sep 18 2017

			G.f. = x^6 + x^8 + x^9 + 3*x^10 + 3*x^11 + 12*x^12 + 19*x^13 + 64*x^14 + ...
From _Allan Bickle_, May 13 2024: (Start)
For n=6, the unique graph is the octahedron.
For n=8, the unique graph is the square of an 8-cycle.
For n=9, the unique graph is the dual of the Herschel graph. (End)

J. H. Conway, An enumeration of knots and links and some of their related properties. Computational Problems in Abstract Algebra, Proc. Conf. Oxford 1967 (Ed. J. Leech), 329-358. New York: Pergamon Press, 1970.

G. Brinkmann, S. Greenberg, C. Greenhill, B. D. McKay, R. Thomas, and P. Wollan, Generation of simple quadrangulations of the sphere, Discr. Math., 305 (2005), 33-54.
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]
A. Caudron, Classification des noeuds et des enlacements, Public. Math. d'Orsay 82. Orsay: Univ. Paris Sud, Dept. Math., 1982.
Alain Caudron, Classification des noeuds et des enlacements (Thèse et additifs), Univ. Paris-Sud, 1989 [Scanned copy, included with permission]. Contains additional material.
CombOS - Combinatorial Object Server, generate planar graphs
S. V. Jablan, Ordering Knots
S. V. Jablan, L. M. Radović, and R. Sazdanović, Basic polyhedra in knot theory Kragujevac J. Math., 28 (2005), 155-164.
The Knot Atlas, Conway Notation.
Index entries for sequences related to knots

Cf. A007022, A078672, A113201, A072552, A111361.

Cf. A292515 (abstract planar graphs with same restrictions).

Name and offset corrected by Andrey Zabolotskiy, Aug 22 2017

A007022 Number of 4-regular polyhedra with n nodes.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 1, 1, 3, 3, 11, 18, 58, 139, 451, 1326, 4461, 14554, 49957, 171159, 598102, 2098675, 7437910, 26490072, 94944685, 341867921, 1236864842, 4493270976, 16387852863, 59985464681, 220320405895, 811796327750, 3000183106119

Offset: 1

N. J. A. Sloane, Apr 28 1994

Number of simple 4-regular 4-edge-connected 3-connected planar graphs; by Steinitz's theorem, every such graph corresponds to a single planar map up to orientation-reversing isomorphism. Equivalently, number of 3-connected quadrangulations of sphere with orientation-reversing isomorphisms permitted with n faces. - Andrey Zabolotskiy, Aug 22 2017

			For n=6, the sole 6-vertex 4-regular polyhedron is the octahedron. The corresponding 6-face quadrangulation is its dual graph, i. e., the cube graph.
From _Allan Bickle_, May 13 2024: (Start)
For n=8, the unique graph is the square of an 8-cycle.
For n=9, the unique graph is the dual of the Herschel graph. (End)

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Gunnar Brinkmann, Sam Greenberg, Catherine Greenhill, Brendan D. McKay, Robin Thomas, and Paul Wollan, Generation of simple quadrangulations of the sphere, Discr. Math., 305 (2005), 33-54. doi:10.1016/j.disc.2005.10.005
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
Michael B. Dillencourt, Polyhedra of small orders and their Hamiltonian properties, Journal of Combinatorial Theory Series B 66:1 (1996), 87-122.
Slavik V. Jablan, Ljiljana M. Radović, and Radmila Sazdanović, Basic polyhedra in knot theory, Kragujevac J. Math. (2005) Vol. 28, 155-164.
Jorik Jooken, Computer-assisted graph theory: a survey, arXiv:2508.20825 [math.CO], 2025. See Ref. 197 at p. 5.
T. Tarnai, F. Kovács, P. W. Fowler, and S. D. Guest, Wrapping the cube and other polyhedra, Proc. Roy. Soc. A 468(2145) (2012), 2652-2666. DOI: 10.1098/rspa.2012.0116.

Cf. A000944 (all polyhedral graphs), A113204, A078672, A078666 (total number of simple 4-regular 4-edge-connected planar maps, including not 3-connected ones).

Cf. A072552, A078666, A111361, A292515 (4-regular planar graphs with restrictions).

More terms from Hugo Pfoertner, Mar 22 2003
a(29) corrected by Brendan McKay, Jun 22 2006
Leading zeros prepended by Max Alekseyev, Sep 12 2016
Offset corrected by Andrey Zabolotskiy, Aug 22 2017

A167156 Number of n-vertex 4-hedrites.

Original entry on oeis.org

1, 0, 2, 0, 2, 0, 4, 0, 3, 0, 5, 0, 3, 0, 7, 0, 5, 0, 7, 0, 4, 0, 11, 0, 5, 0, 8, 0, 8, 0, 12, 0, 6, 0, 13, 0, 6, 0, 15, 0, 10, 0, 11, 0, 7, 0, 21, 0, 10, 0, 13, 0, 12, 0, 18, 0, 9, 0, 22, 0, 9, 0, 21, 0, 14, 0, 16, 0, 14

Offset: 2

Jonathan Vos Post, Oct 29 2009

nonn

Is this the same as A145393 alternating with zeros? - Andrey Zabolotskiy, Jul 05 2017

			Although every other term is zero, this sequence should be kept, contrary to the usual OEIS rules, to be analogous to the related sequences.

Mathieu Dutour Sikiric, Michel Deza, 4-regular and self-dual analogs of fullerenes, arXiv:0910.5323 [math.GT], 2009.

Cf. A167157, A167158, A167159, A111361.

A167158 Number of n-vertex 6-hedrites.

Original entry on oeis.org

0, 0, 1, 1, 2, 1, 5, 5, 9, 7, 14, 14, 23, 17, 28, 27, 44, 35, 54, 57, 77, 59, 87, 85, 119, 105, 134, 135, 187, 149, 189, 197, 251, 218, 278, 275, 354, 313, 361, 359, 472, 405, 480, 511, 609, 519, 613, 614, 771, 704, 771, 788, 989, 849, 938, 1005, 1175, 1038, 1215

Offset: 2

Jonathan Vos Post, Oct 29 2009

nonn

A k-hedrite is a 4-regular planar graph whose faces have sizes 2, 3 and 4 only and the total number of faces of sizes 2 and 3 is k.

Andrey Zabolotskiy, Table of n, a(n) for n = 2..70
Mathieu Dutour Sikiric and Michel Deza, 4-regular and self-dual analogs of fullerenes, arXiv:0910.5323 [math.GT], 2009.

Cf. A167156, A167157, A167159, A111361.

New name from Andrey Zabolotskiy, Jul 05 2017

A167159 Number of n-vertex 7-hedrites.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 1, 1, 3, 4, 5, 7, 9, 12, 18, 22, 25, 36, 46, 48, 62, 76, 88, 107, 126, 142, 179, 198, 216, 257, 304, 329, 382, 431, 483, 547, 601, 643, 764, 838, 889, 998, 1134, 1197, 1324, 1435, 1574, 1751, 1874, 1963, 2247, 2419, 2511, 2735, 3041, 3187, 3453

Offset: 2

Jonathan Vos Post, Oct 29 2009

nonn

A k-hedrite is a 4-regular planar graph whose faces have sizes 2, 3 and 4 only and the total number of faces of sizes 2 and 3 is k.

Andrey Zabolotskiy, Table of n, a(n) for n = 2..70
Mathieu Dutour Sikiric and Michel Deza, 4-regular and self-dual analogs of fullerenes, arXiv:0910.5323 [math.GT], 2009.

Cf. A167156, A167157, A167158, A111361.

New name from Andrey Zabolotskiy, Jul 05 2017

A167157 Number of n-vertex 5-hedrites.

Original entry on oeis.org

0, 1, 0, 1, 2, 3, 1, 2, 3, 5, 3, 4, 7, 10, 6, 6, 7, 12, 9, 8, 15, 20, 11, 12, 16, 21, 18, 16, 24, 32, 24, 18, 26, 37, 23, 24, 38, 45, 37, 30, 33, 52, 44, 34, 56, 69, 45, 40, 54, 66, 58, 48, 66, 92, 68, 49, 71, 98, 70, 63, 96, 104, 92, 74, 80, 122, 98, 72, 120

Offset: 2

Jonathan Vos Post, Oct 29 2009

nonn

A k-hedrite is a 4-regular planar graph whose faces have sizes 2, 3 and 4 only and the total number of faces of sizes 2 and 3 is k.

Mathieu Dutour Sikiric, Michel Deza, 4-regular and self-dual analogs of fullerenes, arXiv:0910.5323 [math.GT], 2009.

Cf. A167156, A167158, A167159, A111361.

New name from Andrey Zabolotskiy, Jul 05 2017

A072552 Number of connected planar regular graphs of degree 4 with n nodes.

Original entry on oeis.org

1, 0, 1, 1, 3, 3, 13, 21, 68, 166, 543, 1605, 5413, 17735, 61084, 210221, 736287

Offset: 6

Markus Meringer (meringer(AT)uni-bayreuth.de), Aug 05 2002

Numbers were obtained using the graph generator GENREG in combination with a test for planarity implemented by M. Raitner.

			From _Allan Bickle_, May 13 2024: (Start)
For n=6, the unique graph is the octahedron.
For n=8, the unique graph is the square of an 8-cycle.
For n=9, the unique graph is the dual of the Herschel graph. (End)

Markus Meringer, GENREG
Markus Meringer, GenReg, Generation of regular graphs.
Markus Meringer, Tables of Regular Graphs
M. Raitner, Test for planarity [broken link]
Robert E. Tuzun and Adam S. Sikora, Verification Of The Jones Unknot Conjecture Up To 22 Crossings, Journal of Knot Theory and Its Ramifications (2018) 1840009, arXiv:1606.06671 [math.GT], 2016-2020 (see table 2).

Cf. A005964, A006820, A078666, A292515 (4-edge-connected graphs only).

Cf. A007022, A111361 (other 4-regular planar graphs).

a(19)-a(22) from Andrey Zabolotskiy, Mar 21 2018 from Tuzun & Sikora.

A167227 Number of 2-self-hedrites with n vertices.

Original entry on oeis.org

1, 1, 2, 2, 3, 3, 3, 3, 5, 4, 4, 6, 5, 5, 8, 5, 6, 8, 6, 8, 10, 7, 7, 10, 10, 8, 12, 10, 9, 14, 9, 9, 14, 10, 14, 16, 11, 11, 16

Offset: 2

Jonathan Vos Post, Oct 30 2009

A 2-self-hedrite is a self-dual plane multigraph such that each its face has 4 sides except for 2 faces with 2 sides. - Andrey Zabolotskiy, Dec 16 2021

Mathieu Dutour Sikiric and Michel Deza, 4-regular and self-dual analogs of fullerenes, arXiv:0910.5323 [math.GT], 2009.

Cf. A167156, A167157, A167158, A167159, A111361, A167228, A167229.

It appears that a(n+1) = A167156(2*n) - A167156(n) [discovered using Sequence Machine]. An equivalent assertion is that if a plane multigraph and its dual both have only 4-gonal faces except for 2 2-gonal ones, then they are isomorphic. - Andrey Zabolotskiy, Dec 16 2021

A292515 Number of 4-regular 4-edge-connected planar simple graphs on n vertices.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 1, 1, 3, 3, 12, 19, 63, 153, 499, 1473, 4974, 16296, 56102, 192899, 674678, 2381395, 8468424

Offset: 1

Andrey Zabolotskiy, Sep 18 2017

The difference between this sequence and A078666 arises because the latter lists not abstract planar graphs but plane graphs (on the sphere, with the same restrictions). Among A078666(14)=64 plane graphs there is 1 pair of isomorphic graphs, namely graphs No. 63 and 64 (hereafter the enumeration of plane graphs from the LinKnot Mathematica package is used, see The Knot Atlas link), hence a(14)=64-1=63. Among 155 plane graphs on 15 vertices, the isomorphic pairs are (143, 149) and (153, 155), hence a(15)=155-2=153. The 11 isomorphic pairs of plane graphs on 16 vertices are: (456, 492), (459, 493), (464, 496), (465, 501), (466, 468), (470, 487), (473, 503), (477, 488), (478, 479), (486, 497), (498, 504).

Tuzun and Sikora say that such planar graphs constitute the set of 4-edge-connected basic Conway polyhedra, and indeed it suffices to consider any one embedding of each of these graphs into sphere or plane to list all prime knots. However, usually the set of Conway polyhedra is identified with the set of plane graphs instead (see A078666 and references therein), which is necessary to list or encode all prime knot diagrams (on the sphere).

			From _Allan Bickle_, May 13 2024: (Start)
For n=6, the unique graph is the octahedron.
For n=8, the unique graph is the square of an 8-cycle.
For n=9, the unique graph is the dual of the Herschel graph. (End)

The Knot Atlas, Conway Notation.
Robert E. Tuzun and Adam S. Sikora, Verification Of The Jones Unknot Conjecture Up To 22 Crossings, Journal of Knot Theory and Its Ramifications (2018) 1840009, arXiv:1606.06671 [math.GT], 2016-2020 (see table 2).
Robert E. Tuzun and Adam S. Sikora, Verification Of The Jones Unknot Conjecture Up To 24 Crossings, arXiv:2003.06724 [math.GT], 2020 (see table 1).

Cf. A007022, A072552, A078666, A111361 (4-regular planar graphs with restrictions).

a(23)-a(24) added from Tuzun & Sikora (2020) by Andrey Zabolotskiy, Apr 27 2020

cp's OEIS Frontend

A167160 Duplicate of A111361.

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A078666 Number of isomorphism classes of simple quadrangulations of the sphere having n+2 vertices and n faces, minimal degree 3, with orientation-reversing isomorphisms permitted.

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A007022 Number of 4-regular polyhedra with n nodes.

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A167156 Number of n-vertex 4-hedrites.

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A167158 Number of n-vertex 6-hedrites.

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A167159 Number of n-vertex 7-hedrites.

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A167157 Number of n-vertex 5-hedrites.

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A072552 Number of connected planar regular graphs of degree 4 with n nodes.

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A167227 Number of 2-self-hedrites with n vertices.

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A292515 Number of 4-regular 4-edge-connected planar simple graphs on n vertices.

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