p-Numerical Semigroups of the Triples of the Sequence $(a^n-b^n)/(a-b)$

Abstract

For a non-negative integer p, we give explicit formulas for the p-Frobenius number and the p-genus of numerical semigroups of $(\nu _n,\nu _{n+1},\nu _{n+2})$, where $\nu _n=(a^n-b^n)/(a-b)$ with $\gcd (a,b)=1$ and $a>b>1$. Here, the p-numerical semigroup $S_p$ is the set of integers whose non-negative integral linear combinations of given positive integers are expressed more than p ways. When $p=0$, $S_0$ with the 0-Frobenius number and the 0-genus is the original numerical semigroup $S_0$ with the Frobenius number and the genus. Symmetric properties of numerical semigroups are important to characterize the numerical semigroup. In recent works, some closed formulas of p-Frobenius numbers have been successfully given, but no symmetric property has been found. We give a symmetric property of this numerical semigroup.