Look and Say Sequence

I came across the so-called "look-and-say sequence" popularised by the mathematician John Conway while watching this YouTube video:

There is a good explanation of the sequence on Wikipedia, part of which I've copied below:

In mathematics, the look-and-say sequence is the sequence of integers beginning as follows: 

1, 11, 21, 1211, 111221, 312211, 13112221, 1113213211, ... 

This is sequence A005150 in the OEIS.

To generate a member of the sequence from the previous member, read off the digits of the previous member, counting the number of digits in groups of the same digit. For example: 

1 is read off as "one 1" or 11.
11 is read off as "two 1s" or 21.
21 is read off as "one 2, then one 1" or 1211.
1211 is read off as "one 1, one 2, then two 1s" or 111221.
111221 is read off as "three 1s, two 2s, then one 1" or 312211.

What Conway found was that the ratio of the lengths of successive terms tended toward a constant value of \(\lambda\) = 1.303577269034... or to express it more precisely:
$$\lim_{n \to \infty} \frac{L_{n+1}}{L_n} = \lambda$$ where \(L_n\) denotes the number of digits in the n-th term of the sequence. Moreover, Conway showed that \(\lambda\) is an algebraic number and in fact the positive solution of a polynomial of degree 71 described in this Wikipedia article.

Of course, if allowing complex number solutions, there will be 71 solutions and when plotted on the Argand diagram they look like this (notice \(\lambda\) on the far right of the positive x-axis):