The length $a(n)$ of the longest common subsequence of the nth Thue-Morse word and its bitwise complement is studied. An open problem suggested by Jean Berstel in 2006 is to find a formula for $a(n)$. In this paper we prove new lower bounds on $a(n)$ by explicitly constructing a common subsequence between the Thue-Morse words and their bitwise complement. We obtain the lower bound $a(n)=2^n(1-o(1))$, saying that when n grows large, the fraction of omitted symbols in the longest common subsequence of the nth Thue-Morse word and its bitwise complement goes to 0. We further generalize to any prefix of the Thue-Morse sequence, where we prove similar lower bounds.

长度 $\mathrm{一种}（ñ）$研究了第n个Thue-Morse单词的最长公共子序列及其按位补码。让·贝斯特尔（Jean Berstel）在2006年提出的一个开放问题是为$\mathrm{一种}（ñ）$。在本文中，我们证明了新的下界$\mathrm{一种}（ñ）$通过在Thue-Morse单词与其按位补语之间显式构造一个公共子序列。我们获得下界$\mathrm{一种}（ñ）=2^{ñ}（\mathrm{1个}-Ø（\mathrm{1个}））$，说当n变大时，第n个Thue-Morse单词的最长公共子序列及其按位补码中省略符号的分数变为0。我们进一步推广到Thue-Morse序列的任何前缀，在这里我们证明相似的下限。