How to probe the recursiveness order of a sequence $\{S_n\}$ whose generating function is known:

$$ \sum_{n\geq0} S_n z^n= \frac{4 z \left(\sqrt{49 z^2-18 z+1}+7 z-1\right)}{\sqrt{49 z^2-18 z+1} \left(\sqrt{49 z^2-18 z+1}+15 z-1\right) \left(3 \sqrt{49 z^2-18 z+1}+21 z-1\right)}+\frac{3}{3-44 z} $$

And even more, as the following is proven.

$ \textbf{Conjecture}$ : $\{S_n\}$ satisfy the following recurrence

$$ 0= P_0(n) S_n+ P_1(n) S_{n-1}+ P_2(n) S_{n-2}+ P_3(n) S_{n-3}+ P_4(n) S_{n-4}$$

where $P_i(n) \in \mathbb{R}[n]$ are polynomials of degree 2.

1more comment