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 questionis: 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.

**Answer**

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.

**Attribution***Source : Link , Question Author : EDX , Answer Author : Sharky Kesa*