A007022 -id:A007022 - cp's OEIS frontend

A113204 Same as A007022, but with orientation-reversing isomorphisms forbidden.

0, 0, 0, 0, 0, 1, 0, 1, 1, 4, 3, 15, 25, 92, 234, 803, 2469, 8512, 28290, 98148, 338673, 1188338, 4180854, 14840031, 52904562, 189724510, 683384218, 2472961423, 8984888982, 32772085447, 119963084542, 440623586740, 1623555117611

Offset: 1

N. J. A. Sloane, Jan 07 2006

nonn

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. 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

Cf. A113201, A078666, A007022, A002880, A113202, A113205.

a(29) corrected by Brendan McKay, Jun 22 2006
Leading zeros prepended by Max Alekseyev, Sep 13 2016
Offset corrected by Andrey Zabolotskiy, Feb 09 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

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

A002880 Number of 3-connected nets with n edges.

Original entry on oeis.org

1, 0, 1, 1, 2, 2, 9, 11, 37, 79, 249, 671, 2182, 6692, 22131, 72405, 243806, 822788, 2815119, 9679205, 33551192, 116900081, 409675567, 1442454215, 5102542680, 18124571838, 64634480340, 231334873091, 830828150081, 2993489821771

Offset: 6

N. J. A. Sloane

Also, the number of 3-connected quadrangulations without separating 4-cycles (up to orientation) with n faces. - Andrey Zabolotskiy, Sep 20 2019

			G.f. = x^6 + x^8 + x^9 + 2*x^10 + 2*x^11 + 9*x^12 + 11*x^13 + 37*x^14 + ...

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).

C. J. Bouwkamp & N. J. A. Sloane, Correspondence, 1971
J. A. D. Cameron, Searching for Squared Squares, USYD Master of Science Thesis (1976). Rare Books & Special Collections Fisher Library, Sydney University.
J. A. D. Cameron, Table 7.2 - listing of tri-connected planar graphs by edge, from the thesis - this was the first count of order 20 (22131). Photo by Stuart E Anderson.
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]
CombOS - Combinatorial Object Server, generate planar graphs
M. B. Dillencourt, Polyhedra of small orders and their Hamiltonian properties, Journal of Combinatorial Theory Series B 66:1 (1996), 87-122.
P. J. Federico, Enumeration of polyhedra: the number of 9-hedra, J. Combin. Theory, 7 (1969), 155-161.
N. D. Kazarinoff and R. Weitzenkamp, Squaring rectangles and squares, Amer. Math. Monthly, 80 (1973), 877-888.

Cf. A002840, A002841, A113201, A078666, A007022, A113205, A338511.

A113201 Number of isomorphism classes of simple quadrangulations of the sphere having n vertices and n-2 faces, with orientation-reversing isomorphisms permitted.

Original entry on oeis.org

1, 1, 2, 3, 9, 18, 62, 198, 803, 3378, 15882, 77185, 393075, 2049974, 10938182, 59312272, 326258544, 1815910231, 10213424233, 57974895671, 331820721234, 1913429250439, 11109119321058, 64901418126997

Offset: 4

N. J. A. Sloane, Jan 07 2006

nonn

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]
J. Cantarella, H. Chapman, M. Mastin, Knot Probabilities in Random Diagrams, arXiv preprint arXiv:1512.05749 [math.GT], 2015. See Tables 1.
CombOS - Combinatorial Object Server, generate planar graphs
Paul Jungeblut, Edge Guarding Plane Graphs, Master Thesis, Karlsruhe Institute of Technology (Germany, 2019).
Sage, Common Graphs

Cf. A078666, A007022, A002880.

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.

A113205 Same as A002880, but with orientation-reversing isomorphisms forbidden.

Original entry on oeis.org

1, 0, 1, 1, 3, 2, 12, 16, 59, 133, 445, 1248, 4162, 13014, 43474, 143304, 484444, 1639388, 5617205, 19332596, 67048051, 233691112, 819121608, 2884443024, 10204104900, 36247138920, 129264732757, 462661038926, 1661637913984

Offset: 6

N. J. A. Sloane, Jan 07 2006

nonn

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]
CombOS - Combinatorial Object Server, generate planar graphs

Cf. A113201, A078666, A007022, A002880, A113202, A113203, A113204.

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

A113202 Number of isomorphism classes of simple quadrangulations of the sphere having n vertices and n-2 faces, with orientation-reversing isomorphisms forbidden.

Original entry on oeis.org

1, 1, 2, 3, 10, 21, 83, 298, 1339, 6049, 29765, 148842, 770267, 4054539, 21743705, 118237471, 651370528, 3628421181, 20416662314, 115919209155, 663548898942, 3826577783917, 22217382001865, 129800215435088

Offset: 4

N. J. A. Sloane, Jan 07 2006

nonn

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]
CombOS - Combinatorial Object Server, generate planar graphs

Cf. A113201, A078666, A007022, A002880, A113203, A113204, A113205.

A078672 Number of simple 4-regular 4-edge-connected but not 3-connected plane graphs on n nodes.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 1, 6, 16, 59, 188, 685, 2412, 8825, 32110, 118505, 437526, 1624492, 6043496, 22553387, 84345031, 316183706, 1187740914, 4471145942, 16864755973, 63737132585, 241337964503, 915500561602

Offset: 6

Slavik V. Jablan and Brendan McKay, Feb 06 2003

			The first such graph has 12 nodes. It is called 12E [Jablan, Radović & Sazdanović, Fig. 2; or Caudron, p. 308c] and looks like that:
    ___________
   /           \
  / O---O   O---O
  |/|\ /|\ /|\ /|
  O | O | O | O |
  |\|/ \|/ \|/ \|
  \ O---O   O---O
   \___________/

A. Caudron, Classification des noeuds et des enlacements, Public. Math. d'Orsay 82. Orsay: Univ. Paris Sud, Dept. Math., 1982.
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.

Cf. A007022, A078666.

a(n) = A078666(n) - A007022(n).

a(23)-a(34) from Sean A. Irvine, Jul 09 2025

cp's OEIS Frontend

A113204 Same as A007022, but with orientation-reversing isomorphisms forbidden.

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

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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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A002880 Number of 3-connected nets with n edges.

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A113201 Number of isomorphism classes of simple quadrangulations of the sphere having n vertices and n-2 faces, with orientation-reversing isomorphisms permitted.

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

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A113205 Same as A002880, but with orientation-reversing isomorphisms forbidden.

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

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A113202 Number of isomorphism classes of simple quadrangulations of the sphere having n vertices and n-2 faces, with orientation-reversing isomorphisms forbidden.

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A078672 Number of simple 4-regular 4-edge-connected but not 3-connected plane graphs on n nodes.

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