Extended skolem sequences

Abstract

Abstract A k ‐extended Skolem sequence of order n is an integer sequence (s 1 , s 2 ,…, s 2n+1 ) in which s k = 0 and for each j ϵ {1,…,n}, there exists a unique i ϵ {1,…, 2n} such that s i = s i+j = j. We show that such a sequence exists if and only if either 1) k is odd and n ≡ 0 or 1 (mod 4) or (2) k is even and n ≡ 2 or 3 (mod 4). The same conditions are also shown to be necessary and sufficient for the existence of excess Skolem sequences. Finally, we use extended Skolem sequences to construct maximal cyclic partial triple systems. © 1995 John Wiley & Sons, Inc.