Very odd sequences.

Inglis, Nicholas F. J. and Wiseman, Julian D. A. (1995) Very odd sequences. Journal of Combinatorial Theory, Series A, 71 (1). pp. 89-96. ISSN 0097-3165

Abstract

Suppose that n ε and a = (a0, …, an − 1) is a sequence of length n with ai ε {0, 1}. For 0 k n − 1, let We call the sequence avery odd if Ak is odd for 0 k n − 1. We prove that there are very odd sequences of length n> 1 if and only if the order of 2 is odd in the multiplicative group of integers modulo 2n − 1.