# Ratio of length of consecutive terms in Look and Say sequence

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.

• 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. Jun 2, 2021 at 12:56
• Jun 2, 2021 at 13:13