On the binomial transforms of Ap\'ery-like sequences

In the proof of the irrationality of ζ(3) and ζ(2), Ap\'ery defined two integer sequences through $3$-term recurrences, which are known as the famous Ap\'ery numbers. Zagier, Almkvist--Zudilin and Cooper successively introduced the other $13$ sporadic sequences through variants of Ap\'ery's $3$-term recurrences. All of the $15$ sporadic sequences are called Ap\'ery-like sequences. Motivated by Gessel's congruences mod $24$ for the Ap\'ery numbers, we investigate the congruences in the form uₙ≡ αⁿ Nα~(α∈ Z,Nα∈ N⁺) for all of the $15$ Ap\'ery-like sequences ₙ≥ 0. Let Nα be the largest positive integer such that uₙ≡ αⁿ Nα for all non-negative integers n. We determine the values of maxα|α ∈ Z\ for all of the $15$ Ap\'ery-like sequences ₙ≥ 0.The binomial transforms of Ap\'ery-like sequences provide us a unified approach to this type of congruences for Ap\'ery-like sequences.