There are no de bruijn sequences of span n with complexity 2n ? 1 + n + 1

If s = (s₀, s₁,…, s_2ⁿ?1) is a binary de Bruijn sequence of span n, then it has been shown that the least length of a linear recursion that generates s, called the complexity of s and denoted by c(s), is bounded for n ? 3 by 2^{n ? 1} + n ? c(s) ? 2ⁿ ?1. A numerical study of the allowable values of c(s) for 3 ? n ? 6 found that all values in this range occurred except for 2^n?1 + n + 1. It is proven in this note that there are no de Bruijn sequences of complexity 2^n?1 + n + 1 for all n ? 3.