Extended Langford sequences with small defects (Q1268727)

A \(k\)-extended Langford sequence of defect \(d\) and length \(m\) is a sequence \(s_1,\dots,s_{2m+1}\) in which \(s_k=\varepsilon\), where \(\varepsilon\) is the null symbol and each other member of the sequence comes from the set \(S=\{d,d+1,\dots,d+m-1\}\). Each \(j\in S\) occurs exactly twice in the sequence and the two occurences are separated by exactly \(j-1\) symbols. In this paper necessary conditions for the existence of such a sequence are given and it is shown that when \(d=2,3\) these conditions are sufficient, too.