Chain addition cycles

For each seed s = (s(1), s(2), ..., s(n)) of elements s(i) chosen from the ring Z(m) of integers modulo m, the infinite sequence S = S(m, s) = (s(k): k is an element of N) satisfying s(n+k) = s(k) + s(k+1) (addition in Z(m)) for every positive integer k is the (m, n) chain,addition sequence generated by the seed s. We investigate the maximal period, L-n(m), of chain addition cycles with seed length n (modulo m). The general problem is reduced to finding L-n(p(k)) for primes p and it is shown that if L-n(p(2)) not equal L-n(p), then L-n(p(k)) = p(k-1) L-n(p) for positive integers k. Further, conditions guaranteeing that L-n(p(2)) not equal L-n(p) are given. (C) 2002 Elsevier Science Inc. All rights reserved.