A167158 -id:A167158 - cp's OEIS frontend

A111361 The number of 4-regular plane graphs with n vertices with all faces 3-gons or 4-gons.

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, 6234

Offset: 2

Gunnar Brinkmann, Nov 07 2005

nonn

These are the 4-regular graphs corresponding to the 3-regular fullerenes. Only the two smallest possible face sizes are allowed. The numbers up to a(33) have been checked by 2 independent programs. Further numbers have not been checked independently.

			From _Allan Bickle_, May 13 2024: (Start)
The smallest example (n=6) is the octahedron (only 3-gons).
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)

Andrey Zabolotskiy, Table of n, a(n) for n = 2..131 (from Hasheminezhad & McKay)
G. Brinkmann, O. Heidemeier and T. Harmuth, The construction of cubic and quartic planar maps with prescribed face degrees, Discrete Applied Mathematics 128: 541-554, (2003).
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]
Michel-Marie Deza, Mathieu Dutour Sikiric and Mikhail Ivanovitch Shtogrin, Geometric Structure of Chemistry-Relevant Graphs, Springer, 2015; see Sec. 4.4.
Mathieu Dutour Sikiric and Michel Deza, 4-regular and self-dual analogs of fullerenes, arXiv:0910.5323 [math.GT], 2009.
Mahdieh Hasheminezhad and Brendan D. McKay, Recursive generation of simple planar quadrangulations with vertices of degree 3 and 4, Discussiones Mathematicae Graph Theory, 30 (2010), 123-136.
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. A007894.
Cf. A167156, A167157, A167158, A167159.
Cf. A007022, A072552, A078666, A292515 (4-regular planar graphs with restrictions).

Leading zeros prepended, terms a(34) and beyond added from the book by Deza et al. (except for a(60) from the paper by Brinkmann et al.) by Andrey Zabolotskiy, Oct 09 2021

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.

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

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

A167228 Number of 3-self-hedrites with n vertices.

Original entry on oeis.org

0, 1, 1, 4, 6, 7, 11, 16, 16, 26, 29, 30, 42, 47, 48, 64, 72, 70, 89, 104, 90, 119, 131, 124, 162, 170, 158, 190, 210, 202, 239, 256, 232, 290, 308, 286, 342, 359, 332

Offset: 2

Jonathan Vos Post, Oct 30 2009

nonn

Mathieu Dutour Sikiric, Michel Deza, 4-regular and self-dual analogs of fullerenes, Oct 28, 2009.

Cf. A167156, A167157, A167158, A167159, A111361, A167227, A167229.

A167229 Number of 4-self-hedrites with n vertices.

Original entry on oeis.org

0, 0, 1, 1, 2, 4, 6, 8, 15, 16, 24, 33, 40, 48, 69, 73, 92, 114, 130, 148, 191, 198, 234, 276, 304, 332, 407, 421, 476, 550, 584, 631, 748, 760, 857, 956, 1002, 1070, 1239

Offset: 2

Jonathan Vos Post, Oct 30 2009

nonn

Mathieu Dutour Sikiric, Michel Deza, 4-regular and self-dual analogs of fullerenes, Oct 28, 2009.

Cf. A167156, A167157, A167158, A167159, A111361, A167227, A167228.

cp's OEIS Frontend

A111361 The number of 4-regular plane graphs with n vertices with all faces 3-gons or 4-gons.

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A167156 Number of n-vertex 4-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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A167227 Number of 2-self-hedrites with n vertices.

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A167228 Number of 3-self-hedrites with n vertices.

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A167229 Number of 4-self-hedrites with n vertices.

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