Some arithmetical properties of convergents to algebraic numbers

Let $\xi$ be an irrational algebraic real number and $(p_k / q_k)_{k \ge 1}$ denote the sequence of its convergents. Let $(u_n)_{n \geq 1}$ be a non-degenerate linear recurrence sequence of integers, which is not a polynomial sequence. We show that if the intersection of the sequences $(q_k)_{k \ge 1}$ and $(u_n)_{n \geq 1}$ is infinite, then $\xi$ is a quadratic number. We also discuss several arithmetical properties of the sequence $(q_k)_{k \ge 1}$.