An explicit formula generating the non-Fibonacci numbers

Abstract

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

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