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=uWn−1+vWn−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,…,aκ with gcd(a1,a2,…,aκ)=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.