On the sum of a prime and two Fibonacci numbers

Let $\{f_n\}$ be the Fibonacci sequence. For any positive integer $n$, let $r(n)$ be the number of solutions of $n=p+f_{k_1^{2}} +f_{k_{2}^{2}}$, where $p$ is a prime and $k_1, k_2$ are nonnegative integers with $k_1\le k_2$. In this paper, it is proved that $\{ n : r(n)=0\}$ contains an infinite arithmetic progression, and both sets $\{ n : r(n)=1\}$ and $\{ n : r(n)\ge 2\}$ have positive asymptotic densities.