p-Numerical Semigroups of Generalized Fibonacci Triples

Abstract

For a nonnegative integer p, we give explicit formulas for the p-Frobenius number and the p-genus of generalized Fibonacci numerical semigroups. Here, the p-numerical semigroup Sp is defined as the set of integers whose nonnegative integral linear combinations of given positive integers a1,a2,…,ak are expressed in more than p ways. When p=0, S0 with the 0-Frobenius number and the 0-genus is the original numerical semigroup with the Frobenius number and the genus. In this paper, we consider the p-numerical semigroup involving Jacobsthal polynomials, which include Fibonacci numbers as special cases. We can also deal with the Jacobsthal–Lucas polynomials, including Lucas numbers accordingly. An application on the p-Hilbert series is also provided. There are some interesting connections between Frobenius numbers and geometric and algebraic structures that exhibit symmetry properties.

The p-Numerical Semigroup of the Triple of Arithmetic Progressions

“…For two variables, it is still easy to find explicit formulas of g p (a, b), n p (a, b), and s p (a, b). However, for three or more variables, no explicit formula had been found, but finally, in 2022, we succeeded in obtaining closed formulas for some special cases, including the triplets of triangular numbers [9], repunits [10], Fibonacci [11], and Jacobsthal numbers [12,13].…”