On the number of residues of certain second-order linear recurrences (Q6639465)

Let \(f\) be a monic integer polynomial \(x^k-a_1x^{k-1}-\dots-a_k\). Consider the linear recurrence given by some initial values \(x_0,\dots,x_{k-1} \in {\mathbb Z}\) and\N\[\Nx_n=a_1x_{n-1}+\dots+a_kx_{n-k}\N\]\Nfor \(n=k,k+1,k+2,\dots\). Let \(R(f)\) be the number of distinct positive integers such that \(d \in R(f)\) if and only if for some \(m \in {\mathbb N}\) there exist \(x_0,\dots,x_{k-1} \in {\mathbb Z}\) such that the number of distinct residues \(x_n \pmod m\) as \(n\) runs through all non-negative integers is exactly \(d\). The problem of determining the set \(R(f)\) has been raised by the reviewer in [Arch. Math., Brno 42, No. 2, 151--158 (2006; Zbl 1164.11026)]. Then, the reviewer and \textit{A. Novikas} [J. Number Theory 215, 120--137 (2020; Zbl 1473.11039)] showed that \(R(x^2-x-1)={\mathbb N}\). Now, the authors prove that, more generally, for each nonzero \(a \in {\mathbb Z}\) we have \(R(x^2-ax-1)={\mathbb N}\). (For \(a=0\) it is easy to verify that \(R(x^2-1)=\{1,2\}\).) Their proof is constructive. Furthermore, in passing, they also give a simpler proof in the case \(a=1\).