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.