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.