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%I #10 Jan 30 2020 08:31:01
%S 1,2,3,8,18,50,135,385,1065,3053,8701,25579,73693,217718,635220,
%T 1888802
%N a(n) equals the number of admissible pairs of subsets of {1,2,...,n} in the notation of Marzuola-Miller.
%C Alternate description: a(n) is the number of vertices at height n in the rooted tree in figure 4 of [Marzuola-Miller] which spawn only three vertices at height n+1.
%C The number of numerical sets S with atom monoid A(S) equal to {0,n+1, 2n+2,2n+3,2n+4,2n+5,...}
%H S. R. Finch, <a href="http://www.people.fas.harvard.edu/~sfinch/">Monoids of natural numbers</a>
%H S. R. Finch, <a href="/A066062/a066062.pdf">Monoids of natural numbers</a>, March 17, 2009. [Cached copy, with permission of the author]
%H J. Marzuola and A. Miller, <a href="http://arxiv.org/abs/0805.3493">Counting numerical sets with no small atoms</a>, arXiv:0805.3493 [math.CO], 2008.
%H J. Marzuola and A. Miller, <a href="http://doi.org/10.1016/j.jcta.2010.03.002">Counting numerical sets with no small atoms</a>, J. Combin. Theory A 117 (6) (2010) 650-667.
%F Recursively related to A164048 (call it A'()) by the formula A(2k+1)' = 2A(2k)'-a(k).
%e a(3)=3 since {0,4,8,9,10,11,...}, {0,1,4,5,8,9,10,11,...} and {0,1,2, 4,5,6,8,9,10,11,...} are the only three sets satisfying the required conditions.
%Y Cf. A158291, A164048.
%K nonn,more
%O 1,2
%A _Steven Finch_, Mar 19 2009
%E Definition rephrased by Jeremy L. Marzuola (marzuola(AT)math.uni-bonn.de), Aug 08 2009
%E Edited by _R. J. Mathar_, Aug 31 2009
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