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%I #30 Jun 04 2020 09:42:13
%S 1,0,0,0,0,1,11,9,7,5,6,7,10,11,14,18,22,26,34,40,50,61,74,89,111,131,
%T 159,192,231,274,332,392,469,557,661,780,928,1088,1285,1511,1776,2076,
%U 2439,2843,3324,3873,4511,5238,6096,7057,8183,9466
%N Number of unlabeled connected graphs with n vertices such that every vertex has exactly 2 vertices at distance 2.
%C Gaar and Krenn call these graphs 2-metamour-regular.
%H E. Gaar and D. Krenn, <a href="http://arxiv.org/abs/2005.14121">Metamour-regular Polyamorous Relationships and Graphs</a>, arXiv:2005.14121 [math.CO], 2020.
%F a(n) = p_3(n) + 1 for n >= 9 with p_3(n) being the number of integer partitions of n with parts at least 3 (A008483).
%e For n = 8 vertices, there exist the connected 2-metamour-regular graphs
%e - c(C_8), c(C_5) join c(C_3), c(C_4) join c(C_4),
%e - C_8 and
%e - 3 exceptional graphs,
%e where C_i is the cycle graph on i vertices, and c(G) is the complement graph of G.
%e Therefore the unlabeled total is a(8) = 7.
%o (SageMath) [(len(Partitions(n, min_part=3)) if n >= 6 else 0)
%o + (1 if n >= 5 else 0)
%o + {0: 1, 6: 8, 7: 6, 8: 3}.get(n, 0)
%o for n in srange(52)]
%o (PARI) a(n)=if(n<9,[1, 0, 0, 0, 0, 1, 11, 9, 7, 5][n+1], numbpart(n)-numbpart(n-1)-numbpart(n-2)+numbpart(n-3)+1) \\ _Charles R Greathouse IV_, Apr 22 2020
%Y Cf. A008483.
%K nonn
%O 0,7
%A _Daniel Krenn_, Apr 21 2020
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