A non-periodic sequence of s and s, with no 3 consecutive identical blocks of any length.

Ways of constructing it:

**1.** For each nonnegative integer , let if the number of ‘s in the binary representation of is even, and if this number is odd. (Conway calls these *evil* and *odious* numbers, respectively)

**2.** At each step, append a copy of the sequence so far, with s and s interchanged.

i.e. Start with . Append to that. .

Append to that .

Append to that .

Append to that . etc.

**3.** Start with . Apply and .

If is in binary notation, then , with the same number of binary s as , and , containing one more . So:

i.e. the sequence is determined by its odd-numbered terms, and each of those is different to the previous (even) term.

Interpreted as a binary fraction, , the *Morse-Thue constant*, it’s intimately related to period-doubling bifurcation and the Mandelbrot set.

The **Tarry-Escott problem**

Can the set be partitioned into two disjoint subsets and such that

The Thue-Morse sequence determines such a partitioning.

Let be the set of all with ,

and be the set of with .

e.g. Take .

The first 16 terms of the sequence are , so

, and .

And sure enough…

,

,

,

.

This works for any polynomial of degree .

If if a polynomial of degree not exceeding , then

**Further reading**

Savchev, Andreescu – Mathematical Miniatures (2002)

Manfred Schroeder – Fractals, Chaos, Power Laws