A167156 -id:A167156 - cp's OEIS frontend

A145393 Number of inequivalent sublattices of index n in square lattice, where two sublattices are considered equivalent if one can be rotated or reflected to give the other, with that rotation or reflection preserving the parent square lattice.

1, 2, 2, 4, 3, 5, 3, 7, 5, 7, 4, 11, 5, 8, 8, 12, 6, 13, 6, 15, 10, 11, 7, 21, 10, 13, 12, 18, 9, 22, 9, 21, 14, 16, 14, 29, 11, 17, 16, 29, 12, 28, 12, 25, 23, 20, 13, 39, 16, 27, 20, 29, 15, 34, 20, 36, 22, 25, 16, 50, 17, 26, 29, 38, 24, 40, 18, 36, 26, 40

Offset: 1

N. J. A. Sloane, Feb 23 2009

nonn

From Andrey Zabolotskiy, Mar 12 2018: (Start)
If reflections are not allowed, we get A145392. If any rotations and reflections are allowed, we get A054346.
The parent lattice of the sublattices under consideration has Patterson symmetry group p4mm, and two sublattices are considered equivalent if they are related via a symmetry from that group [Rutherford]. For other 2D Patterson groups, the analogous sequences are A000203 (p2), A069734 (p2mm), A145391 (c2mm), A145392 (p4), A145394 (p6), A003051 (p6mm).
Rutherford says at p. 161 that a(n) != A054346(n) only when A002654(n) > 2, but actually these two sequence differ at other terms, too, for example, at n = 30 (see illustration). (End)

Andrey Zabolotskiy, Table of n, a(n) for n = 1..10000
Amihay Hanany, Domenico Orlando, and Susanne Reffert, Sublattice counting and orbifolds, High Energ. Phys., 2010 (2010), 51, arXiv.org:1002.2981 [hep-th] (see table 6 and fig. 2).
John S. Rutherford, Sublattice enumeration. IV. Equivalence classes of plane sublattices by parent Patterson symmetry and colour lattice group type, Acta Cryst. (2009). A65, 156-163. [See Table 2; beware the typo in a(5).]
Andrey Zabolotskiy, Sublattices of the square lattice (illustrations for n = 1..6, 15, 25)
Index entries for sequences related to sublattices
Index entries for sequences related to square lattice

Cf. A054345, A054346, A167156, A008621.
Cf. A000203, A069734, A145391, A145392, A145394, A003051, A002324, A002654, A069735, A145390.
Cf. A019590, A157228, A157226, A157230, A157231, A053866, A025441.

terms = 70;
CoefficientList[Sum[(1/((1-x^m)(1-x^(4m)))-1), {m, 1, terms}] + O[x]^(terms + 1), x] // Rest (* Jean-François Alcover, Aug 05 2018 *)

a(n) = (A000203(n) + A002654(n) + A069735(n) + A145390(n))/4. [Rutherford] - N. J. A. Sloane, Mar 13 2009
G.f.: Sum_{ m>=1 } (1/((1-x^m)(1-x^(4m))) - 1). [Hanany, Orlando & Reffert, eq. (6.8)] - Andrey Zabolotskiy, Jul 05 2017
a(n) = Sum_{ m: m^2|n } A019590(n/m^2) + A157228(n/m^2) + A157226(n/m^2) + A157230(n/m^2) + A157231(n/m^2) = A053866(n) + A025441(n) + Sum_{ m: m^2|n } A157226(n/m^2) + A157230(n/m^2) + A157231(n/m^2). [Rutherford] - Andrey Zabolotskiy, May 07 2018
a(n) = Sum_{ d|n } A008621(d) = Sum_{ d|n } (1 + floor(d/4)). [From the above-given g.f.] - Andrey Zabolotskiy, Jul 17 2019

New name from Andrey Zabolotskiy, Mar 12 2018

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

Original entry on oeis.org

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

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

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

A145393 Number of inequivalent sublattices of index n in square lattice, where two sublattices are considered equivalent if one can be rotated or reflected to give the other, with that rotation or reflection preserving the parent square lattice.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Extensions

A167158 Number of n-vertex 6-hedrites.

Views

Author

Keywords

Comments

Links

Crossrefs

Extensions

A167159 Number of n-vertex 7-hedrites.

Views

Author

Keywords

Comments

Links

Crossrefs

Extensions

A167157 Number of n-vertex 5-hedrites.

Views

Author

Keywords

Comments

Links

Crossrefs

Extensions

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

Views

Author

Keywords

Comments

Links

Crossrefs

Formula

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

Views

Author

Keywords

Links

Crossrefs

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

Views

Author

Keywords

Links

Crossrefs