The Existence of Two New Classes of Near-Skolem Type Sequences

Let m,n be two positive integers, m ≤ n. An m-near-Skolem sequence of order n and defect m is a sequence m − Sn = (s1, s2, · · · , s2n−2) of 2n − 2 non-negative integers such that the following conditions hold: 1. for each k ∈ {1, 2, · · ·,m−1,m+1, · · · , n} there exist exactly two elements si, sj ∈ m − Sn such that si = sj = k 2. if si = sj = k, then |i − j | = k For example, (4, 1, 1, 2, 4, 2) is an example of a 3-near-Skolem sequence of order 4. In this article we prove that the necessary conditions are also suffi-cient for the existence of two new classes of near-Skolem type sequences; one with a hook two places from the last entry of the sequence and one with a hook two places and one place from the last entry. The sequence