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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.

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1 Answer 1

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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.

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  • $\begingroup$ Thank you very much, this is very complete! $\endgroup$
    – EDX
    Aug 7, 2020 at 14:08

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