An explicit formula generating the non-Fibonacci numbers

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