Very odd sequences
About 20 years ago, J. Pelikán made the following conjecture. Let \(a_0, a_1, \dots, a_{n-1}\) be a binary sequence of length \(n\), and let \(A_k= \sum_{i=0}^{n-k-1} a_i a_{i+k}\), \(0\leq k\leq n\); then, for \(n\geq 5\), some \(A_k\) is even. The present authors show that this conjecture is false, determining precisely those values of \(n\) for which it fails. However, it seems that this result was obtained earlier by \textit{F. J. MacWilliams} and \textit{A. M. Odlyzko} in a 1977 paper which appeared in the same journal [J. Comb. Theory, Ser. A 22, 110-114 (1977; Zbl 0345.12010)].
scientific article; zbMATH DE number 774646
scientific article