A158449 The number of sigma-admissible subsets of {1,2,...,n} as defined by Marzuola-Miller.

1, 0, 1, 0, 2, 0, 3, 1, 7, 3, 17, 7, 43, 24, 118, 74, 330, 206, 888, 612, 2571, 1810, 7274, 5552, 21099, 16334, 61252, 49025, 179239, 146048, 523455, 440980, 1554184, 1315927, 4572794, 3972193, 13569220, 11873290, 40263681, 35824869, 119901609, 107397585

Offset: 1

Author

Steven Finch, Mar 19 2009

Keywords

• nonn

Comments

a(n), or Asigma(n), equals the number of sigma-admissible subsets of {1,2,...,n}.

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]

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,...}

Examples

			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.
		

Links

• Martin Fuller, Table of n, a(n) for n = 1..65
• S. R. Finch, Monoids of natural numbers [Broken link]
• S. R. Finch, Monoids of natural numbers, March 17, 2009. [Cached copy, with permission of the author]
• Martin Fuller, C program
• J. Marzuola and A. Miller, Counting Numerical Sets with No Small Atoms, arXiv:0805.3493 [math.CO], 2008.
• J. Marzuola and A. Miller, Counting numerical sets with no small atoms, J. Combin. Theory A 117 (6) (2010) 650-667.

Crossrefs

Cf. A066062, A164047.

Programs

C

See Martin Fuller link

Formula

Recursively related to A164047 by the formula Asigma(2k+1)' = 2Asigma(2k)'-Asigma(k)

Extensions

Definition rephrased by Jeremy L. Marzuola (marzuola(AT)math.uni-bonn.de), Aug 08 2009

Edited by R. J. Mathar, Aug 31 2009

a(33) onwards from Martin Fuller, Sep 13 2023