Conway’s constant

Conway’s constant $\lambda \approx 1.303577296$ gives the asymptotic rate of growth in the length between $a_i$ and $a_{i+1}$ in most look and say sequences. That is, given a function $d(n)$ that gives us the number of digits of $n$ in base 10, then

$$\underset{i\to \mathrm{\infty }}{lim}\frac{a_{i+1}}{a_i}=\lambda .$$

For example, starting with $n=1$ and skipping ahead to $a_7$, we observe

$i$	$a_i$	$\frac{a_{i+1}}{a_i}$
7	13112221	1.333333333…
8	1113213211	1.25
9	31131211131221	1.4
10	13211311123113112211	1.428571428…
11	11131221133112132113212221	1.3
12	3113112221232112111312211312113211	1.307692307…

Conway’s constant is the largest zero of this degree 71 polynomial:

$x^{71}-x^{69}-2x^{68}-x^{67}+2x^{66}+2x^{65}+x^{64}-x^{63}-x^{62}-x^{61}-x^{60}-x^{59}$ $+2x^{58}+5x^{57}+3x^{56}-2x^{55}-10x^{54}-3x^{53}-2x^{52}+6x^{51}+6x^{50}+x^{49}+9x^{48}$ $-3x^{47}-7x^{46}-8x^{45}-8x^{44}+10x^{43}+6x^{42}+8x^{41}-5x^{40}-12x^{39}+7x^{38}-7x^{37}$ $+7x^{36}+x^{35}-3x^{34}+10x^{33}+x^{32}-6x^{31}-2x^{30}-10x^{29}-3x^{28}+2x^{27}+9x^{26}$ $-3x^{25}+14x^{24}-8x^{23}-7x^{21}+9x^{20}+3x^{19}-4x^{18}-10x^{17}-7x^{16}+12x^{15}$ $+7x^{14}+2x^{13}-12x^{12}-4x^{11}-2x^{10}+5x^9+x^7-7x^6+7x^5-4x^4+12x^3-6x^2+3x-6$

Title	Conway’s constant
Canonical name	ConwaysConstant
Date of creation	2013-03-22 18:02:36
Last modified on	2013-03-22 18:02:36
Owner	PrimeFan (13766)
Last modified by	PrimeFan (13766)
Numerical id	4
Author	PrimeFan (13766)
Entry type	Definition
Classification	msc 11A63