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

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$