In 2009, Luca and Nicolae [14] proved that the only Fibonacci numbers whose Euler totient function is another Fibonacci number are 1, 2, and 3. In 2015, Faye and Luca [7] proved that the only Pell numbers whose Euler totient function is another Pell number are 1 and 2. Here we generalise 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 = P un−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.