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…

 

Source: Wikipedia

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}

 

Let me know if you’re enjoying these Math Bites? M x

Advertisements

2 comments

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s