A103580 - OEIS

A103580

Number of nonempty subsets S of {1,2,3,...,n} that have the property that no element x of S is a nonnegative integer linear combination of elements of S-{x}.

1, 2, 4, 6, 11, 15, 26, 36, 57, 79, 130, 170, 276, 379, 579, 784, 1249, 1654, 2615, 3515, 5343, 7256, 11352, 14930, 23203, 31378, 47510, 63777, 98680, 130502, 201356, 270037, 407428, 548089, 840170, 1110428, 1701871, 2284324, 3440336, 4601655

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

Fausto A. C. Cariboni, Table of n, a(n) for n = 1..100

Sergey Kitaev, Independent Sets on Path-Schemes, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.2.

Sean Li, Counting numerical semigroups by Frobenius number, multiplicity, and depth, arXiv:2208.14587 [math.CO], 2022.

FORMULA

a(n) = A326083(n) - 1. - Gus Wiseman, Jun 07 2019

EXAMPLE

a(4) = 6 because the only permissible subsets are {1}, {2}, {3}, {4}, {2,3}, {3,4}.

From Gus Wiseman, Jun 07 2019: (Start)

The a(1) = 1 through a(6) = 15 nonempty subsets of {1..n} containing none of their own non-singleton nonzero nonnegative linear combinations are:

{1} {1} {1} {1} {1} {1}

{2} {2} {2} {2} {2}

{3} {3} {3} {3}

{2,3} {4} {4} {4}

{2,3} {5} {5}

{3,4} {2,3} {6}

{2,5} {2,3}

{3,4} {2,5}

{3,5} {3,4}

{4,5} {3,5}

{3,4,5} {4,5}

{4,6}

{5,6}

{3,4,5}

{4,5,6}

a(n) is also the number of nonempty subsets of {1..n} containing all of their own nonzero nonnegative linear combinations <= n. For example the a(1) = 1 through a(6) = 15 subsets are:

{1} {2} {2} {3} {3} {4}

{1,2} {3} {4} {4} {5}

{2,3} {2,4} {5} {6}

{1,2,3} {3,4} {2,4} {3,6}

{2,3,4} {3,4} {4,5}

{1,2,3,4} {3,5} {4,6}

{4,5} {5,6}

{2,4,5} {2,4,6}

{3,4,5} {3,4,6}

{2,3,4,5} {3,5,6}

{1,2,3,4,5} {4,5,6}

{2,4,5,6}