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%I #17 Sep 13 2023 09:39:49
%S 1,0,1,0,2,0,3,1,7,3,17,7,43,24,118,74,330,206,888,612,2571,1810,7274,
%T 5552,21099,16334,61252,49025,179239,146048,523455,440980,1554184,
%U 1315927,4572794,3972193,13569220,11873290,40263681,35824869,119901609,107397585
%N The number of sigma-admissible subsets of {1,2,...,n} as defined by Marzuola-Miller.
%C a(n), or Asigma(n), equals the number of sigma-admissible subsets of {1,2,...,n}.
%C Alternate description: (1) Asigma(k) is the same as the number of additive 2-bases for k which are not additive 2-bases for k+1. (2) Asigma(n) is the number of vertices at height n in the rooted tree in figure 5 of [Marzuola-Miller] which spawn only one vertex at height n+1. [Jeremy L. Marzuola (marzuola(AT)math.uni-bonn.de), Aug 08 2009]
%C The number of symmetric numerical sets S with atom monoid A(S) equal to {0,n+1,2n+2,2n+3,2n+4,2n+5,...}
%H Martin Fuller, <a href="/A158449/b158449.txt">Table of n, a(n) for n = 1..65</a>
%H S. R. Finch, <a href="http://www.people.fas.harvard.edu/~sfinch/">Monoids of natural numbers</a> [Broken link]
%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 Martin Fuller, <a href="/A158449/a158449.txt">C program</a>
%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="https://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 A164047 by the formula Asigma(2k+1)' = 2Asigma(2k)'-Asigma(k)
%e a(1)=a(3)=1 since {0,2,4,5,6,7,...} and {0,1,4,5,8,9,10,11,...} are the only sets satisfying the required conditions.
%o (C) See Martin Fuller link
%Y Cf. A066062, A164047.
%K nonn
%O 1,5
%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
%E a(33) onwards from _Martin Fuller_, Sep 13 2023
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