On the discriminator of Lucas sequences

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 $$k>1$$ .