Experimental results show that, when the order $n$ is odd, there are de Bruijn sequences such that the corresponding complement sequence and the reverse sequence are the same. In this paper, we propose one efficient method to generate such de Bruijn sequences. This solves an open problem asked by Fredricksen forty years ago for showing the existence of such de Bruijn sequences when the odd order $n >1$. Moreover, we refine a characterization of de Bruijn sequences with the same complement and reverse sequences and study the number of these de Bruijn sequences, as well as the distribution of de Bruijn sequences of the maximum linear complexity.

翻译：实验结果表明，当阶数$n$为奇数时，存在其对应补序列与逆序列相同的德布鲁因序列。本文提出一种高效生成此类德布鲁因序列的方法，从而解决了Fredricksen在四十年前提出的关于证明奇数阶$n>1$时此类序列存在性的公开问题。此外，我们完善了具有相同补序列与逆序列的德布鲁因序列的特征刻画，研究了此类序列的数量分布，并分析了最大线性复杂度的德布鲁因序列的分布特性。