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%I #59 Mar 19 2020 16:00:04
%S 1,2,6,20,91,509,3608,31856,340416,4269971,61133757,978098997,
%T 17228295555,330552900516,6853905618223,152626436936272,
%U 3631575281503404,91928898608055819,2466448432564961852,69907637101781318907
%N Number of isomorphism classes of connected 3-regular (trivalent, cubic) loopless multigraphs of order 2n.
%C a(n) is also the number of isomorphism classes of connected 3-regular simple graphs of order 2n with possibly loops. - _Nico Van Cleemput_, Jun 04 2014
%C There are no graphs of order 2n+1 satisfying the condition above. - _Natan Arie Consigli_, Dec 20 2019
%D A. T. Balaban, Enumeration of Cyclic Graphs, pp. 63-105 of A. T. Balaban, ed., Chemical Applications of Graph Theory, Ac. Press, 1976; see p. 92 [gives incorrect a(6)].
%D CRC Handbook of Combinatorial Designs, 1996, p. 651 [or: 2006, table 4.40].
%H Jan-Peter Börnsen, Anton E. M. van de Ven, <a href="https://arxiv.org/abs/1807.04817">Tangent Developable Orbit Space of an Octupole</a>, arXiv:1807.04817 [hep-th], 2018.
%H G. Brinkmann, N. Van Cleemput, T. Pisanski, <a href="http://dx.doi.org/10.1016/j.tcs.2012.01.018">Generation of various classes of trivalent graphs</a>, Theoretical Computer Science 502, 2013, pp.16-29.
%H R. J. Mathar, <a href="/A000421/a000421.pdf">Cubic multigraphs A000421</a>
%H Brendan McKay and others, <a href="http://pallini.di.uniroma1.it/">Nauty Traces</a>
%F Inverse Euler transform of A129416. - _Andrew Howroyd_, Mar 19 2020
%e From _Natan Arie Consigli_, Dec 20 2019: (Start)
%e a(1) = 1: with two nodes the only viable option is the triple edged path multigraph.
%e a(2) = 4: with four nodes we have two cases: the tetrahedral graph and the square graph with single and double edges on opposite sides.
%e (End)
%o (nauty/bash) for n in {1..10}; do geng -cqD3 $[2*$n] | multig -ur3; done # _Sean A. Irvine_, Sep 24 2015
%Y Column k=3 of A328682 (table of k-regular n-node multigraphs).
%Y Cf. A129416, A005967 (loops allowed), A129417, A129419, A129421, A129423, A129425, A002851 (no multiedges).
%K nonn,hard,more
%O 1,2
%A _N. J. A. Sloane_
%E More terms from _Brendan McKay_, Apr 15 2007
%E a(13)-a(20) from _Andrew Howroyd_, Mar 19 2020
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