On the number of residues of certain second-order linear recurrences

For every monic polynomial $f\in \mathbb{Z}[X]$ with $\mathrm{deg}(f)\ge 1$, let $\mathcal{L}(f)$ be the set of all linear recurrences with values in $\mathbb{Z}$ and characteristic polynomial $f$, and let \begin{equation*} $\mathcal{R}$(f) := \big{ρ($\mathbf{x}$; m) : $\mathbf{x}$ \in $\mathcal{L}$(f), , m \in $\mathbb{Z}$^+ \big} , \end{equation*} where $\rho (\mathbf{x};m)$ is the number of distinct residues of $\mathbf{x}$ modulo $m$. Dubickas and Novikas proved that $\mathcal{R}(X^2-X-1)={\mathbb{Z}}^{+}$. We generalize this result by showing that $\mathcal{R}(X^2-a_{1}X-1)={\mathbb{Z}}^{+}$ for every nonzero integer $a_1$. As a corollary, we deduce that for all integers $a_1\ge 1$ and $k\ge 4$ there exists $\xi \in \mathbb{R}$ such that the sequence of fractional parts $(\text{\hspace{0.17em}⁣}\mathrm{frac}(\xi {\alpha }^n){)}_{n\ge 0}$, where $\alpha :=(a_1+\sqrt{a_1^2+4})\mathrm{/}2$, has exactly $k$ limit points. Our proofs are constructive and employ some results on the existence of special primitive divisors of certain Lehmer sequences.