4 problems

Let $a_n$ denote the number of Motzkin paths of length $n$ satisfying the rule specified in the source. For $n\geq 1$, let A005802 and A216947 denote the cited OEIS sequences, with…

Terminal-pattern divisibility conjecture. The following implications hold:

Possible-roots conjecture. For $M$, the rational roots of $w_l(M)$ lie in

Let $d$ be a positive integer, and let the admissible unit steps be … where $\{\epsilon_1,\dots,\epsilon_{d+1}\}$ is the standard basis of $\mathbb{R}^{d+1}$. Consider paths from t…