OFFSET
1,12
COMMENTS
A numerical semigroup is a subset S of N, the nonnegative integers, that is closed under addition, contains the element 0 and such that N-S is finite. The maximum element of N-S is called the Frobenius number of S. The least positive integer belonging to S is called the multiplicity of S. The number of numerical semigroups of Frobenius number n is A124506(n).
LINKS
Maria Bras-Amorós and Vicenç Torra, Table of n, a(n) for n = 1..5050
Maria Bras-Amorós and Vicenç Torra, Multiparameter counting of numerical semigroups: recurrences and leaf-discriminating trees, arXiv:2607.23111 [cs.DM], 2026. See p. 27.
FORMULA
If (n+2)/3 <= k < n-1, then T(n,k) = 2*T(n-2,k-1).
EXAMPLE
Triangle begins
.n\k.|..2....3....4....5....6....7....8....9...10
= = = = = = = = = = = = = = = = = = = = = = = = =
..1..|..1
..2..|..0....1
..3..|..1....0....1
..4..|..0....1....0....1
..5..|..1....2....1....0....1
..6..|..0....0....2....1....0....1
..7..|..1....2....4....2....1....0....1
..8..|..0....2....0....4....2....1....0....1
..9..|..1....0....4....8....4....2....1....0....1
...
T(5,3) = 2: The two numerical semigroups of Frobenius number 5 and multiplicity 3 are S = {0,3,6,...} = N - {1,2,4,5} and S = {0,3,4,6,...} = N - {1,2,5}.
CROSSREFS
KEYWORD
AUTHOR
Maria Bras-Amorós and Vicenç Torra, Jul 22 2026
STATUS
approved