On the binomial transforms of Apéry-like sequences

In the proof of the irrationality of $\zeta (3)$ and $\zeta (2)$, Apéry defined two integer sequences through $3$-term recurrences, which are known as the famous Apéry numbers. Zagier, Almkvist–Zudilin and Cooper successively introduced the other $13$ sporadic sequences through variants of Apéry’s $3$-term recurrences. All of the $15$ sporadic sequences are called Apéry-like sequences. Motivated by Gessel’s congruences mod $24$ for the Apéry numbers, we investigate the congruences in the form $u_n\equiv {\alpha }^n(\mathrm{m}\mathrm{o}\mathrm{d}{N}_{\alpha })(\alpha \in \mathbb{Z},N_{\alpha }\in {\mathbb{N}}^{+})$ for all of the $15$ Apéry-like sequences $\{u_n{\}}_{n\ge 0}$. Let $N_{\alpha }$ be the largest positive integer such that $u_n\equiv {\alpha }^n(\mathrm{m}\mathrm{o}\mathrm{d}{N}_{\alpha })$ for all non-negative integers $n$. We determine the values of $\max \{N_{\alpha }\mathrm{\mid }\alpha \in \mathbb{Z}\}$ for all of the $15$ Apéry-like sequences $\{u_n{\}}_{n\ge 0}$.The binomial transforms of Apéry-like sequences provide us a unified approach to this type of congruences for Apéry-like sequences.