  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.

Dynamical Systems;Formal Languages and Automata Theory;Combinatorics