p-Numerical Semigroups of Triples from the Three-Term Recurrence Relations

Many people, including Horadam, have studied the numbers Wₙ, satisfying the recurrence relation Wₙ=u Wₙ₋₁+v Wₙ₋₂ (n≥ 2) with W₀=0 and W₁=1. In this paper, we study the p-numerical semigroups of the triple (Wᵢ,Wᵢ₊₂,Wᵢ₊ₖ) for integers i,k(≥ 3). For a nonnegative integer p, the p-numerical semigroup Sₚ is defined as the set of integers whose nonnegative integral linear combinations of given positive integers a₁,a₂,,a_κ with (a₁,a₂,,a_κ)=1 are expressed in more than p ways. When $p=0$, S=S₀ is the original numerical semigroup. The largest element and the cardinality of N₀ Sₚ are called the p-Frobenius number and the p-genus, respectively.