The Wikipedia article Look and Say sequence mentions:

"As $n$ tends to infinity, the ratio of length of consecutive terms $(\lambda)$ in the sequence approximately equals $1.303577269034\dots$"

It is clear that for a term of size $n$, the maximum possible size for next term is twice its size (when no two consecutive digits are equal) and the minimum possible is $2$ (when all digits are equal).

But how do I prove the fact above i.e $\lambda = 1.303577269034\dots$

PS: I am particularly interested in sequence starting with 1 although proof for general case will be very helpful.

  • 2
    $\begingroup$ Conway's original paper answers this. link.springer.com/content/pdf/10.1007/978-1-4612-4808-8_53.pdf. Basically he found 92 subsequences that don't interact with each other again and a 71 by 71 matrix which describes how many of those are in the derivative of each other. The asymptotic length is the power of the greatest eigenvalue, which is the $\lambda$ you have in your question. It is an algebraic number of degree 71. $\endgroup$
    – Derivative
    Jun 2, 2021 at 12:56
  • 1
    $\begingroup$ See also math.stackexchange.com/q/620905/620957 $\endgroup$
    – Milten
    Jun 2, 2021 at 13:13


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