Hooked k-extended Skolem sequences

Abstract

A hooked k-extended Skolem sequence of order n is a sequence s₁s₂…s_2n+2 in which s_k = s_2n+1 = ε (ε is the null symbol) and each j ϵ {1, 2, …, n} occurs exactly twice, the two occurrences separated by exactly j − 1 symbols. It is proved that, with the exception of (k, n) = (2, 1), such a sequence exists if and only if n ≡ 0, 1 (mod 4) for k even, and n ≡ 2, 3 (mod 4) for k odd. This result is then used to give an alternative proof of the existence of bicyclic Steiner triple systems.