Consider the sequence of algebraic integers u_n given by the starting values u₀ = 0, u₁ = 1 and the recurrence u_{n + 1}= (4 cos²(2 π/ 7) − 1) u_n−u_{n − 1}. We prove that for any n ∉ {1, 2, 3, 5, 8, 12, 18, 28, 30} the n-th term of the sequence has a primitive divisor in Z[2cos(2π/7)]. As a consequence we deduce that for any suffciently large n there exists a prime power q such that the group PSL₂(q) can be generated by a pair x, y with x²= y³= (xy) ⁷= 1 and the order of the commutator [x, y] is exactly n. The latter result answers in affrmative a question of Holt and Plesken.