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

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.