The extended Frobenius problem for Lucas series incremented by a Lucas number

TL;DR

This paper investigates the Frobenius problem extended to Lucas number sequences and demonstrates that the associated numerical semigroups satisfy Wilf's conjecture, contributing to the understanding of Frobenius numbers in Lucas sequences.

Contribution

It introduces the extended Frobenius problem for Lucas sequences and proves that related numerical semigroups adhere to Wilf's conjecture, a significant open problem.

Findings

Numerical semigroups from Lucas sequences satisfy Wilf's conjecture.

Extended Frobenius problem formulated for Lucas series.

Results contribute to Frobenius problem and semigroup theory.

Abstract

We study the extended Frobenius problem for sequences of the form $\{l_a\}\cup\{l_a+l_n\}_{n\in\mathbb{N}}$ and $\{l_a+l_n\}_{n\in\mathbb{N}}$, where $\{l_n\}_{n\in\mathbb{N}}$ is the Lucas series and $l_a$ is a Lucas number. As a consequence, we show that the families of numerical semigroups associated to both sequences satisfy the Wilf's conjecture.