Drawing inspiration from a recent paper of Heuberger, Krenn, and Lipnik, we define the class of strongly k-recursive sequences. We show that every k-automatic sequence is strongly $k$-recursive, therefore k-recursive, and discuss that the converse is not true. We also show that the class of strongly k-recursive sequences is a proper subclass of the class of k-regular sequences, and we present some explicit examples. We then extend the proof techniques to answer the same question for the class of k-recursive sequences.

翻译：受Heuberger、Krenn和Lipnik近期论文的启发，我们定义了强k递归序列类。我们证明每个k自动序列都是强k递归的，因此也是k递归的，并讨论了其逆命题不成立。我们还表明强k递归序列类是k正则序列类的一个真子类，并给出了一些显式示例。随后，我们扩展了证明技术，以解答k递归序列类的相同问题。