  We consider the family of Lucas sequences uniquely determined by
$U_{n+2}(k)=(4k+2)U_{n+1}(k) -U_n(k),$ with initial values $U_0(k)=0$ and
$U_1(k)=1$ and $k\ge 1$ an arbitrary integer. For any integer $n\ge 1$ the
discriminator function $\mathcal{D}_k(n)$ of $U_n(k)$ is defined as the
smallest integer $m$ such that $U_0(k),U_1(k),\ldots,U_{n-1}(k)$ are pairwise
incongruent modulo $m$. Numerical work of Shallit on $\mathcal{D}_k(n)$
suggests that it has a relatively simple characterization. In this paper we
will prove that this is indeed the case by showing that for every $k\ge 1$
there is a constant $n_k$ such that ${\mathcal D}_{k}(n)$ has a simple
characterization for every $n\ge n_k$. The case $k=1$ turns out to be
fundamentally different from the case $k>1$.
