Exponential Lower Bounds for the Numbers of Skolem-Type Sequences

G.K. Bennett1, M.J. Grannell1, T.S. Griggs1
1Department of Pure Mathematics The Open University Walton Hail Milton Keynes MK7 6AA UNITED KINGDOM

Abstract

It was shown by Abrham that the number of pure Skolem sequences of order \(n\), \(n \equiv 0\) or \(1 \pmod{4}\), and the number of extended Skolem sequences of order \(n\), are both bounded below by \(2^{\left\lfloor \frac{n}{3} \right\rfloor}\). These results are extended to give similar lower bounds for the numbers of hooked Skolem sequences, split Skolem sequences, and split-hooked Skolem sequences.