An explicit formula generating the non-Fibonacci numbers

TL;DR

The paper presents an explicit formula involving logarithms and the golden ratio that generates all non-Fibonacci numbers, providing a new mathematical tool for identifying these numbers.

Contribution

It introduces a novel explicit formula that precisely generates non-Fibonacci numbers, advancing the understanding of Fibonacci-related sequences.

Findings

The formula accurately generates all non-Fibonacci numbers.

It offers a new method for distinguishing Fibonacci from non-Fibonacci numbers.

The approach involves logarithmic functions and the golden ratio.

Abstract

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