  In 2009, Luca and Nicolae proved that the only Fibonacci numbers whose Euler
totient function is another Fibonacci number are $1,2$, and $3$. In 2015, Faye
and Luca 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 $P\geq 3$, if we define the sequence
$\left(u_n\right)_n$ as $u_0=0$, $u_1=1$, and $u_n=Pu_{n-1}+u_{n-2}$ for all
$n\geq 2$, then the only solution to the Diophantine equation
$\varphi\left(u_n\right)=u_m$ is $\varphi\left(u_1\right)=\varphi(1)=1=u_1$.
