TITLE: Why is the Conway 'Look and Say' sequences constant defined by this polynom?
QUESTION [6 upvotes]: In his work on 'Look and Say' sequences,for instance beginning with $1$.
$$1//
11//
21//
1211//
111221//
312212$$
If $L_n$ is the length of the $n-th$ sequences, then it follows from Conway work that :
$$\lim_{n\to\infty} \
\frac{L_{n+1}}{L_n} =\lambda=1.303577269034... $$
where $\lambda$ is the unique real, stricly positive root of
\begin{align}
  x^{71} - x^{69}   - 2x^{68}  - x^{67}   + 2x^{66}  + 2x^{65}  + x^{64}   - x^{63} \\
- x^{62}  - x^{61}   - x^{60}   - x^{59}   + 2x^{58}  + 5x^{57}  + 3x^{56}  - 2x^{55}  - 10x^{54} \\
- 3x^{53}- 2x^{52}  + 6x^{51}  + 6x^{50}  + x^{49}   + 9x^{48}  - 3x^{47}  - 7x^{46}  - 8x^{45}  \\
- 8x^{44} + 10x^{43} + 6x^{42}  + 8x^{41}  - 5x^{40}  - 12x^{39} + 7x^{38}  - 7x^{37}  + 7x^{36}  \\
+ x^{35}  - 3x^{34}  + 10x^{33} + x^{32}   - 6x^{31}  - 2x^{30}  - 10x^{29} - 3x^{28}  + 2x^{27}  \\
+ 9x^{26} - 3x^{25}  + 14x^{24} - 8x^{23}   - 7x^{21} + 9x^{20}  -3x^{19} - 4x^{18}  \\
- 10x^{17} - 7x^{16} + 12x^{15} + 7x^{14}  + 2x^{13}  - 12x^{12} - 4x^{11}  - 2x^{10}  + 5x^9     \\
+ x^7      - 7x^6    + 7x^5     - 4x^4     + 12x^3    - 6x^2     + 3x       - 6
\end{align}
My question is: why that polynom? How did Conway manage to get it? Is it an approximation of the experimental values of $\lambda$ he got?
If there exists any paper, I would appreciate to read it. Thanks for your help.

REPLY [4 votes]: Have a look here for the derivation: http://www.njohnston.ca/2010/10/a-derivation-of-conways-degree-71-look-and-say-polynomial/
The gist of it is every term after the 8th term can be constructed from some of 92 strings. Then its a matter of counting how the sequence length increases and then computing this limit.