Very odd sequences

Suppose that n ϵ N 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.