Pelikán's conjecture and cyclotomic cosets

The following conjecture was recently made by J. Pelikán. Let a₀ ,…, a_n be an (n + 1)-tuple of 0's and 1's; let A_k = ?_i=0^n?ka_ia_i+k for k = 0,…, n. Then if n ? 4 some A_k is even.This paper shows that Pelikán's conjecture is false for infinitely many values of n. On the other hand it is also shown that the conjecture is true for most values of n, and a characterization is given of those values of n for which it fails.