A look-and-say sequence is obtained iteratively by reading off the digits of the current value, grouping identical digits together: starting with 1, the sequence reads: 1, 11, 21, 1211, 111221, 312211, etc. (OEIS A005150). Starting with any digit $d \neq 1$ gives Conway's sequence: $d$, $1d$, $111d$, $311d$, $13211d$, etc. (OEIS A006715). Conway popularised these sequences and studied some of their properties. In this paper we consider a variant subbed "look-and-say again" where digits are repeated twice. We prove that the look-and-say again sequence contains only the digits $1, 2, 4, 6, d$, where $d$ represents the starting digit. Such sequences decompose and the ratio of successive lengths converges to Conway's constant. In fact, these properties result from a commuting diagram between look-and-say again sequences and "classical" look-and-say sequences. Similar results apply to the "look-and-say three times" sequence.

中文翻译：

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口吃的康威序列仍然是康威序列

通过读取当前值的数字，将相同的数字组合在一起，迭代地获得look-and-say序列：从1开始，序列读取：1、11、21、1211、111221、312211等（OEIS A005150 ）。以任何数字 $d \neq 1$ 开头给出康威的序列：$d$、$1d$、$111d$、$311d$、$13211d$ 等（OEIS A006715）。Conway 推广了这些序列并研究了它们的一些特性。在本文中，我们考虑了一个变体，其中数字重复两次。我们证明再看一遍序列只包含数字 $1, 2, 4, 6, d$，其中 $d$ 代表起始数字。这样的序列分解并且连续长度的比率收敛到康威常数。事实上，这些性质是由look-and-say again序列和“