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%I #18 May 26 2018 16:25:21
%S 0,0,0,1,5,17,56,182,573,1792,5533,16977,51652,156291,470069,1407264,
%T 4193977,12451760,36838994,108656009,319583578,937634011,2744720126,
%U 8018165821,23379886511,68056985580,197800670948,574068309840,1663907364480,4816910618093,13929036720057
%N Number of connected simple graphs with n vertices, n+1 edges, and vertex degrees no more than 4.
%C Such graphs are also referred to (e.g., by Hendrickson & Parks) as carbon skeletons with two rings, or bicyclic skeletons, although actual number of simple cycles in such graphs can exceed 2 (e.g., in the example). - _Andrey Zabolotskiy_, Nov 24 2017
%C Terms computed with nauty agree at least to a(20) with those computed by formula and sequences A125669, A125670, A125671, A305132. - _Andrew Howroyd_, May 26 2018
%H Andrew Howroyd, <a href="/A112410/b112410.txt">Table of n, a(n) for n = 1..200</a>
%H J. B. Hendrickson and C. A. Parks, <a href="https://doi.org/10.1021/ci00001a018">Generation and Enumeration of Carbon skeletons</a>, J. Chem. Inf. Comput. Sci., 31 (1991), 101-107. See Table 2, column 2 on page 103.
%H Michael A. Kappler, <a href="http://www.daylight.com/meetings/emug04/Kappler/GenSmi.html">GENSMI: Exhaustive Enumeration of Simple Graphs</a>.
%F a(n) = A125669(n) + A125670(n) + A125671(n) + A305132(n). - _Andrew Howroyd_, May 26 2018
%e The only such graph for n = 4 is:
%e o-o
%e |/|
%e o-o
%o (nauty/bash)
%o for n in {4..15}; do geng -c -D4 ${n} $((n+1)):$((n+1)) -u; done # _Andrey Zabolotskiy_, Nov 24 2017
%Y The analogs for n+k edges with k = -1, 0, ..., 7 are: A000602, A036671, this sequence, A112619, A112408, A112424, A112425, A112426, A112442.
%Y Cf. A121941 (any number of edges), A006820 (2n edges).
%Y Cf. A125669, A125670, A125671, A305132.
%K nonn
%O 1,5
%A _Jonathan Vos Post_, Dec 08 2005
%E Corrected offset and new name from _Andrey Zabolotskiy_, Nov 20 2017
%E a(20) corrected by _Andrey Zabolotskiy_ and _Andrew Howroyd_, May 26 2018
%E Terms a(21) and beyond from _Andrew Howroyd_, May 26 2018
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