On Symmetric and Antisymmetric Balanced Binary Sequences

Let X = (x1, . . . , xn) be a finite binary sequence of length n, i.e., xi = ±1 for all i. The derived sequence of X is the binary sequence ∂X = (x1x2, . . . , xn−1xn) of length n− 1, and the derived triangle of X is the collection ∆X of all derived sequences ∂X for 0 ≤ i ≤ n−1. We say that X is balanced if its derived triangle ∆X contains as many +1’s as −1’s. This concept was introduced by Steinhaus in 1963. It is known that balanced binary sequences occur in every length n ≡ 0 or 3 mod 4, and in none other. In this paper, we solve the problem of determining all possible lengths of symmetric and of antisymmetric balanced binary sequences. We prove that (1) there exists a symmetric balanced binary sequence of length n if and only if n ≡ 0, 3 or 7 mod 8, and (2) there exists an antisymmetric balanced binary sequence of length n if and only if n ≡ 4 mod 8.