We discuss an interesting sequence defined recursively; namely, sequence A105774 from the On-Line Encyclopedia of Integer Sequences, and study some of its properties. Our main tools are Fibonacci representation, finite automata, and the Walnut theorem-prover. We also prove two new results about synchronized sequences.

翻译：本文讨论了一个由递归定义的趣味数列，即在线整数序列百科全书中的序列A105774，并研究了其若干性质。我们的主要工具包括斐波那契表示、有限状态自动机以及Walnut定理证明器。此外，我们还证明了关于同步序列的两个新结论。