# Maths Bite: Conway’s Constant

### Look-and-Say Sequences

A Look-and-Say sequence was first introduced and analysed by John Conway. An example of such series is:

1, 11, 21, 1211, 111221, 312211, 13112221, 1113213211….

To generate the next number in the sequence from the previous term, read off the digits of the previous number, counting the number of digits in groups of the same digit. For example:

• 1 is read as ‘one 1’ = 11
• 11 is read as ‘two 1s’ = 21
• 21 is read as ‘one 2, one 1’ = 1211

If we start with any digit x from 0 to 9, then x will remain the last digit of the sequence. When x does not equal 1, the sequence is as follows:

x, 1x, 111x, 311x, 13211x, 111312211x, 31131122211x…

The Conway sequence, named by Vardi in 1991, is a look-and-say sequence with the starting digit 3.

### Growth in Length and Conway’s Constant

The terms of the sequence eventually grow in length about 30% per generation. If Ln denotes the number of digits in the n-th term of the sequence, the limit of the ratio

$\frac{L_{n + 1}}{L_n}$

is Conway’s constant:

$\lim_{n \to \infty}\frac{L_{n+1}}{L_{n}} = \lambda$

where λ = 1.303577269034…

Conway’s constant is the unique positive real root of the following polynomial:

\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}

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