A158448 a(n) equals the number of admissible pairs of subsets of {1,2,...,n} in the notation of Marzuola-Miller.

1, 2, 3, 8, 18, 50, 135, 385, 1065, 3053, 8701, 25579, 73693, 217718, 635220, 1888802

Offset: 1

Author

Steven Finch, Mar 19 2009

Keywords

• nonn
• more

Comments

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.

The number of numerical sets S with atom monoid A(S) equal to {0,n+1, 2n+2,2n+3,2n+4,2n+5,...}

Examples

			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.
		

Links

• S. R. Finch, Monoids of natural numbers
• S. R. Finch, Monoids of natural numbers, March 17, 2009. [Cached copy, with permission of the author]
• 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. A158291, A164048.

Formula

Recursively related to A164048 (call it A'()) by the formula A(2k+1)' = 2A(2k)'-a(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