On amicable sequences and orthogonal designs

In this paper we give a general theorem which can be used to multiply the length of amicable sequences keeping the amicability property and the type of the sequences. As a consequence we have that if there exist two, four or eight amicable sequences of length m and type (al, a2), (al, a2, a3, a4) or (al, a2, ... , a8) then there exist amicable sequences of length ℓ ≡ 0 (mod m) and of the same type. We also present a theorem that produces a set of 2v amicable sequences from a set of v (not necessary amicable) sequences and a construction method for amicable sequences of type (al, al, a2, a2, ... , av, av) from v pairs of disjoint (0, ±1) amicable sequences. Using these results we can obtain many infinite classes of orthogonal designs. Actually, if there exists an orthogonal design of order n and of type (al, a2,... , av), which is constructed from sequences, then there exists an infinite family of orthogonal designs of the same type which is constructed from appropriate sequences.