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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 $$\sum_{k=0}^{d}a_kx_{n+k}=0 \quad (x_0,...,x_{d-1})=(b_1,.., b_{d-1}) \in \mathbb{Z} \; \forall \;b_{i} \quad n \in \mathbb{N}_{0} $$ Are there references for the following probability or cardinal? $$P(\textrm{drawing (uniformly) random}\;a_{k}\; \textrm{with} -m\leq a_{k}\leq m\;: \; x_{n} \in \mathbb{Z} \;\;\forall n \in \mathbb{N}_{0})$$ $$\#\{-m\leq a_{k}\leq m, \; a_{k} \in \mathbb{Z}: x_{n} \in \mathbb{Z} \;\;\forall n \in \mathbb{N}_{0} \}$$

The closest research I have found is related to generate Binet's formulas for recurrences of degree $d$ and this paper of Pemantle and Wilf for general bounded non-decreasing integer sequences. But I haven't seen the specific question before.

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 $$\sum_{k=0}^{d}a_kx_{n+k}=0 \quad (x_0,...,x_{d-1})=(b_0,.., b_{d-1}) \in \mathbb{Z} \; \forall \;b_{i} \quad n \in \mathbb{N}_{0} $$ Are there references for the following probability or cardinal? $$P(\textrm{drawing (uniformly) random}\;a_{k}\; \textrm{with} -m\leq a_{k}\leq m\;: \; x_{n} \in \mathbb{Z} \;\;\forall n \in \mathbb{N}_{0})$$ $$\#\{-m\leq a_{k}\leq m, \; a_{k} \in \mathbb{Z}: x_{n} \in \mathbb{Z} \;\;\forall n \in \mathbb{N}_{0} \}$$

The closest research I have found is related to generate Binet's formulas for recurrences of degree $d$ and this paper of Pemantle and Wilf for general bounded non-decreasing integer sequences. But I haven't seen the specific question before.

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 for recurrences of degree $d$ and this paper of Pemantle and Wilf for general bounded non-decreasing integer sequences. But I haven't seen the specific question before.

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 $$\sum_{k=0}^{d}a_kx_{n+k}=0 \quad (x_0,...,x_{d-1})=(b_1,.., b_{d-1}) \in \mathbb{Z} \; \forall \;b_{i} \quad n \in \mathbb{N}_{0} $$ Are there references for the following probability or cardinal? $$P(\textrm{drawing (uniformly) random}\;a_{k}\; \textrm{with} -m\leq a_{k}\leq m\;: \; x_{n} \in \mathbb{Z} \;\;\forall n \in \mathbb{N}_{0})$$ $$\#\{-m\leq a_{k}\leq m, \; a_{k} \in \mathbb{Z}: x_{n} \in \mathbb{Z} \;\;\forall n \in \mathbb{N}_{0} \}$$

The closest research I have found is related to generate Binet's formulas for recurrences of degree $d$ and this paper of Pemantle and Wilf for general bounded non-decreasing integer sequences. But I haven't seen the specific question before.

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 for recurrences of degree $d$. But I haven't seen the specific question before.

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 for recurrences of degree $d$ and this paper of Pemantle and Wilf for general bounded non-decreasing integer sequences. But I haven't seen the specific question before.