Editing Look-and-say sequence (section)

=== Growth in length ===
The terms eventually grow in length by about 30% per generation. In particular, if ''L''<sub>''n''</sub> denotes the number of digits of the ''n''-th member of the sequence, then the [[Limit (mathematics)|limit]] of the ratio <math>\frac{L_{n + 1}}{L_n}</math> exists and is given by
<math display="block">\lim_{n \to \infty} \frac{L_{n+1}}{L_{n}} = \lambda</math>

where &lambda; = 1.303577269034... {{OEIS|id=A014715}} is an [[algebraic number]] of degree 71.<ref name="Martin2006" /> This fact was proven by Conway, and the constant &lambda; is known as '''Conway's constant'''. The same result also holds for every variant of the sequence starting with any seed other than 22.

==== Conway's constant as a polynomial root ====
Conway's constant is the unique positive [[real root]] of the following [[polynomial]] {{OEIS|id=A137275}}:
<math display="block">\begin{matrix}
             &           &\qquad            &           &\qquad            &           &\qquad            &            & +1x^{71}  &           \\
   -1x^{69}  & -2x^{68}  &   -1x^{67}  & +2x^{66}  &  +2x^{65}  &  +1x^{64} &  -1x^{63}  &  -1x^{62}  & -1x^{61}  & -1x^{60}  \\
   -1x^{59}  & +2x^{58}  &  +5x^{57}  & +3x^{56}  & -2x^{55}  & -10x^{54} &  -3x^{53}  &  -2x^{52}  & +6x^{51}  & +6x^{50}  \\
   +1x^{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}  & +1x^{35}  &  -3x^{34} & +10x^{33}  &  +1x^{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}   &           & +1x^{7}   &  -7x^{6}  &  +7x^{5}   &  -4x^{4}  & +12x^{3}   &  -6x^{2}   &  +3x^{1}  & -6x^{0}  \\
\end{matrix}
</math>

This polynomial was correctly given in Conway's original ''Eureka'' article,<ref name="Conway-original-article" />
but in the reprinted version in the book edited by Cover and Gopinath<ref name="Conway-original-article" /> the term <math>x^{35}</math> was incorrectly printed with a minus sign in front.<ref>
{{Cite book
  | last = Vardi
  | first = Ilan
  | title = Computational Recreations in Mathematica
  | publisher = [[Addison-Wesley]]
  | year = 1991
  | isbn = 0-201-52989-0
}}
</ref>