Diophantine representations of linear recurrent sequences. II

Direct constructions of diophantine representations of linear recurrent sequences are considered. Diophantine representations of the sets of values of third-order sequences with negative discriminant are found. As an auxiliary problem we study the structure of the multiplicative group of the ring $\mathbb Z[\lambda]$, where $\lambda$ is an invertible algebraic number (unit) in a real quadratic field or in a cubic field of a negative discriminant. Tge index of the subgroup $\langle\pm\lambda^n\mid n\in\mathbf Z\rangle$ in the group $(\mathbf Z[\lambda])^*$ and the generator of $(\mathbf Z[\lambda])^*$ are evaluated explicitly.