  For every monic polynomial $f \in \mathbb{Z}[X]$ with $\operatorname{deg}(f)
\geq 1$, 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\{\rho(\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 $a_1 \geq 1$ and $k \geq 4$ there exists $\xi \in \mathbb{R}$
such that the sequence of fractional parts $\big(\!\operatorname{frac}(\xi
\alpha^n)\big)_{n \geq 0}$, where $\alpha := \big(a_1 + \sqrt{a_1^2 + 4}\,\big)
/ 2$, has exactly $k$ limit points. Our proofs are constructive and employ some
results on the existence of special primitive divisors of certain Lehmer
sequences.
