>Are there references in the literature that deal with the probability of finding an integer sequence in a linear homogeneous recurrence with constant coefficients $ \in \mathbb{Z}$? (or provides a way to count them?). If the linear recurrence is given by
$$\sum_{k=0}^{d}a_kx_{n+k}=0 \quad (x_0,...,x_{d-1})=(b_1,.., b_{d-1}) \in \mathbb{Z} \; \forall \;b_{i} $$
Are there references for the following probability or cardinal?
$$P(\textrm{drawing (uniformly) random}\;a_{k}\; \textrm{with} -m\leq a_{k}\leq m\;: \; x_{n} \in \mathbb{Z} \;\forall n )$$
$$\#\{-m\leq a_{k}\leq m, \; a_{k} \in \mathbb{Z}:  x_{n} \in \mathbb{Z} \;\forall n \}$$

The closest research I have found is related to generate [Binet's formulas](https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.388.6853&rep=rep1&type=pdf) for recurrences of degree $d$. But I haven't seen the specific question before.