On the primitive divisors of the recurrent sequence $$u_{n+1}=(4rm{cos}^2(2pi/7)-1)it{u}_{n}-u_{n-rm{1}}$$ with applications to group theory

Consider the sequence of algebraic integers u_n given by the starting values u₀ = 0, u₁ = 1 and the recurrence (u_{n+1}=(4rm{cos}^2(2pi/7)-1)it{u}_{n}-u_{n-rm{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 (mathbb{Z}[2rm{cos}(2pi/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 (x^2=y^3=(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.