p-Numerical Semigroups of Triples from the Three-Term Recurrence Relations - Jiaxin Mu; Takao Komatsu (Axioms)
Many people, including Horadam, have studied the numbers <semantics>Wn</semantics>, satisfying the recurrence relation <semantics>Wn=uWn−1+vWn−2</semantics> (<semantics>n≥2</semantics>) with <semantics>W0=0</semantics> and <semantics>W1=1</semantics>. In this paper, we study the p-numerical semigroups of the triple <semantics>(Wi,Wi+2,Wi+k)</semantics> for integers <semantics>i,k(≥3)</semantics>. For a nonnegative integer p, the p-numerical semigroup <semantics>Sp</semantics> is defined as the set of integers whose nonnegative integral linear combinations of given positive integers <semantics>a1,a2,…,aκ</semantics> with <semantics>gcd(a1,a2,…,aκ)=1</semantics> are expressed in more than p ways. When <semantics>p=0</semantics>, <semantics>S=S0</semantics> is the original numerical semigroup. The largest element and the cardinality of <semantics>N0∖Sp</semantics> are called the p-Frobenius number and the p-genus, respectively.