论文标题
口吃的康威序列仍然是康威序列
Stuttering Conway Sequences Are Still Conway Sequences
论文作者
论文摘要
通过读取当前值的数字，将相同的数字分组在一起：从1开始，序列读取：1，11，11，21，121，121，111221，312211等来迭代获得外观序列（OEIS A005150）。从任何数字$ d \ neq 1 $开始，都提供了Conway的序列：$ D $，$ 1D $，$ 111D $，$ 311D $，$ 13211D $等（OEIS A006715）。康威（Conway）推广了这些序列，并研究了它们的一些属性。在本文中，我们考虑了一个变体的“再次出现”，其中重复数字两次。我们证明，再次显示序列仅包含数字$ 1、2、4、6，D $，其中$ d $代表起始数字。这样的序列分解和连续长度的比率会收敛到康威的常数。实际上，这些属性是由外观序列和“经典”外观序列之间的通勤图引起的。类似的结果适用于“外观三次”序列。
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.
