An explicit formula generating the non-Fibonacci numbers

We show among others that the formula: $$\lfloor n + \log_{\Phi}\{\sqrt{5}(\log_{\Phi}(\sqrt{5}n) + n) -5 + \frac{3}{n}\} - 2 \rfloor (n \geq 2),$$ (where $\Phi$ denotes the golden ratio and $\lfloor \rfloor$ denotes the integer part) generates the non-Fibonacci numbers.