Consider there recurrence relation $$a_n=\frac{5}{4} a_{n-1} \cos{(\pi\cdot a_{n-1})}.$$
I am trying to find the generating function $A(x)=\sum_{n=1}^\infty a_nx^n.$
I've tried the following:
$$a_n=\frac{5}{4} a_{n-1} \cos{(\pi\cdot a_{n-1})}\\ \implies \sum_{n=1}^\infty a_n x^n = \frac{5}{4}\sum_{n=1}^\infty a_{n-1}\cos{(\pi \cdot a_{n-1})}x^n \\ \implies A(x)-a_0 = \frac{5}{4}x\sum_{n=0}^\infty a_n \cos{(\pi \cdot a_n)} x^n.$$
This is just where I'm stuck. Any ideas?