I have a linear transformation $f: \{0,1 \}^n \to\{0,1\}^n$,
$y_1 = x_n, \; y_2 = x_1 + x_n \bmod 2$, $\;y_3 = x_2, \;...,\; y_n = x_{n-1}$
Find the period of the sequence of sets if $n = 17$ and $\alpha = (1,0,0 ...,0)$:
$$\alpha, \; f(\alpha), \; f(f(\alpha)), \; ... \;$$
I know, that this task can be solved using the theory of linear recurrent. How to solve this another way?