The extended Frobenius problem for Fibonacci sequences incremented by a Fibonacci number

TL;DR

This paper investigates the extended Frobenius problem for Fibonacci sequences shifted by a Fibonacci number, demonstrating that the related numerical semigroups satisfy Wilf's conjecture.

Contribution

It introduces a new class of sequences for the Frobenius problem and proves their associated semigroups meet Wilf's conjecture, extending previous results.

Findings

Numerical semigroups from these sequences satisfy Wilf's conjecture

The extended Frobenius problem is characterized for Fibonacci-based sequences

New insights into the structure of Fibonacci-related numerical semigroups

Abstract

We study the extended Frobenius problem for sequences of the form $\{f_a+f_n\}_{n\in\mathbb{N}}$, where $\{f_n\}_{n\in\mathbb{N}}$ is the Fibonacci sequence and $f_a$ is a Fibonacci number. As a consequence, we show that the family of numerical semigroups associated to these sequences satisfies the Wilf's conjecture.