The Euler totient function on Lucas sequences

In 2009, Luca and Nicolae [φ(Fm) = Fn, Integers 9 (2009) A30] proved that the only Fibonacci numbers whose Euler totient function is another Fibonacci number are 1, 2, and 3. In 2015, Faye and Luca [Pell numbers whose Euler function is a Pell number, Publ. Inst. Math. 101(115) (2017) 231-245] proved that the only Pell numbers whose Euler totient function is another Pell number are 1 and 2. Here, we add to these two results and prove that for any fixed natural number P ≥ 3, if we define the sequence (un)n as u0 = 0, u1 = 1, and un = Pun-1 + un-2 for all n ≥ 2, then the only solution to the Diophantine equation φ(un) = um is φ(u1) = φ(1) = 1 = u1.