PlanetMath: Conway's constant

(more info)

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Conway's constant

(Definition)

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

$\displaystyle \lim_{i \to \infty} \frac{a_{i + 1}}{a_i} = \lambda.$

For example, starting with and skipping ahead to , we observe

		$\frac{a_{i + 1}}{a_i}$
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:

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