cp's OEIS Frontend

A007022 Number of 4-regular polyhedra with n nodes.

%I A007022 M2290 #77 Sep 01 2025 11:31:50
%S A007022 0,0,0,0,0,1,0,1,1,3,3,11,18,58,139,451,1326,4461,14554,49957,171159,
%T A007022 598102,2098675,7437910,26490072,94944685,341867921,1236864842,
%U A007022 4493270976,16387852863,59985464681,220320405895,811796327750,3000183106119
%N A007022 Number of 4-regular polyhedra with n nodes.
%C A007022 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
%D A007022 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A007022 Gunnar Brinkmann, Sam Greenberg, Catherine Greenhill, Brendan D. McKay, Robin Thomas, and Paul Wollan, <a href="http://people.math.gatech.edu/~thomas/PAP/quad.pdf">Generation of simple quadrangulations of the sphere</a>, Discr. Math., 305 (2005), 33-54. doi:<a href="http://dx.doi.org/10.1016/j.disc.2005.10.005">10.1016/j.disc.2005.10.005</a>
%H A007022 Gunnar Brinkmann and Brendan McKay, <a href="http://users.cecs.anu.edu.au/~bdm/plantri/">plantri and fullgen</a> programs for generation of certain types of planar graph.
%H A007022 Gunnar Brinkmann and Brendan McKay, <a href="/A000103/a000103_1.pdf">plantri and fullgen</a> programs for generation of certain types of planar graph [Cached copy, pdf file only, no active links, with permission]
%H A007022 CombOS - Combinatorial Object Server, <a href="http://combos.org/plantri">generate planar graphs</a>
%H A007022 Michael B. Dillencourt, <a href="http://dx.doi.org/10.1006/jctb.1996.0008">Polyhedra of small orders and their Hamiltonian properties</a>, Journal of Combinatorial Theory Series B 66:1 (1996), 87-122.
%H A007022 Slavik V. Jablan, Ljiljana M. Radović, and Radmila Sazdanović, <a href="http://elib.mi.sanu.ac.rs/files/journals/kjm/28/12.pdf">Basic polyhedra in knot theory</a>, Kragujevac J. Math. (2005) Vol. 28, 155-164.
%H A007022 Jorik Jooken, <a href="https://www.arxiv.org/abs/2508.20825">Computer-assisted graph theory: a survey</a>, arXiv:2508.20825 [math.CO], 2025. See Ref. 197 at p. 5.
%H A007022 T. Tarnai, F. Kovács, P. W. Fowler, and S. D. Guest, <a href="https://doi.org/10.1098/rspa.2012.0116">Wrapping the cube and other polyhedra</a>, Proc. Roy. Soc. A 468(2145) (2012), 2652-2666. DOI: 10.1098/rspa.2012.0116.
%e A007022 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.
%e A007022 From _Allan Bickle_, May 13 2024: (Start)
%e A007022 For n=8, the unique graph is the square of an 8-cycle.
%e A007022 For n=9, the unique graph is the dual of the Herschel graph. (End)
%Y A007022 Cf. A000944 (all polyhedral graphs), A113204, A078672, A078666 (total number of simple 4-regular 4-edge-connected planar maps, including not 3-connected ones).
%Y A007022 Cf. A072552, A078666, A111361, A292515 (4-regular planar graphs with restrictions).
%K A007022 nonn,changed
%O A007022 1,10
%A A007022 _N. J. A. Sloane_, Apr 28 1994
%E A007022 More terms from _Hugo Pfoertner_, Mar 22 2003
%E A007022 a(29) corrected by _Brendan McKay_, Jun 22 2006
%E A007022 Leading zeros prepended by _Max Alekseyev_, Sep 12 2016
%E A007022 Offset corrected by _Andrey Zabolotskiy_, Aug 22 2017