I have this function
$$\begin{align}f_N:[-1,1]&\longrightarrow [-1,1] \\ \hspace{-10mm}x &\longrightarrow \frac{1}{N}\sum_{i=0}^N cos(\phi^i \pi x) \end{align}$$
where $\phi=\frac{1+\sqrt 5}{2}$ is the golden ratio.
The image shows a plot of the $f_{10}$ function, the red lines illustrate what I mean by high spikes.
There seems to be a certain pattern that appears, where the number of local minima between two successive "high spikes" follows the Fibonnacci sequence
1, 1, 2, 3, 5, 8, 13
In the same way the number of local maxima between two of the successive high spikes (including the spike itself) also follows the same rule, 1, 1, 2, 3, 5, 8, 13
It also looks like the distance between successive high spikes (length of each red interval) follows a similar property where the total length of two neighboring intervals is the distance of the next.
I also find this for other values of N, I tried values between 8 and 12 and always found this property.
The link between Fibonacci and $\phi$ are well known, this doesn't seem like a coincedence? Spooky or just maths ? Any insight would be appreciated