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 W₍-₁+v W₍-₂ (n 2) with W₀=0 and W₁=1. In this paper, we study the p-numerical semigroups of the triple (Wᵢ, W₈+₂, W₈+₊) 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_ with (a₁, a₂, , a_) =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.