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Definition: Fibonacci Sequence
Fibonacci sequence $F_n $ is a sequence defined as follow:
$$F_n = \begin{cases} 1 &(n=1)\br 1 &(n=2)\br F_{n-1} + F_{n-2} &(n \geq 3) \end{cases}$$
Definition: Fibonacci Search
Fibonacci search, similar to Golden section search, is a search method that can narrow down the minimizer $x^*$ of a unimodal function $f: \R \to \R $ over a closed interval $[a_0, b_0]$.
Unlike golden section search, Fibonacci search uses not a fixed $\rho$, but a sequence of $\rho$ based on how precise the result needs to be.
$N$ steps of reduction using the Fibonacci search method reduces the range by the factor
$$\frac{ 1 + 2 \epsilon }{ F_{N+1}}$$
in the worst case.
Thus, to reduce the range from $\pare{b_0 - a_0} $ to $\delta$. We need
$$\frac{ 1 + 2 \epsilon }{ F_{N+1}} \leq \frac{ b_0 - a_0 }{ \delta }$$
i.e.
$$ F_{N+1} \geq \frac{ \pare{1 + 2 \epsilon} \delta }{ b_0 - a_0 } $$