THUE-MORSE SEQUENCE

In mathematics and its applications, the 'Thue-Morse sequence', or 'Prouhet-Thue-Morse sequence', is a certain binary sequence whose initial segments alternate (in a certain sense).
The Thue-Morse sequence begins:
: 01101001100101101001011001101001...
However, sometimes other symbols are used besides 0 and 1, such as 1 and 2, or 1 and 0 (in the opposite order), or left and right, up and down, etc.
Thus one may speak of the Thue-Morse sequence ''on'' a given ordered pair.

Contents

Definition

Direct definition

Recurrence relation

L-system

Characterization using bitwise negation

Infinite product

Some properties

The Prouhet-Tarry-Escott Problem

Fractals and Turtle graphics

History

See also

External links

Definition

There are several equivalent ways of defining the Thue-Morse sequence.

Direct definition

To compute the $n$th element $t\_n$, write the number $n$ in binary. If the number of ones in this binary expansion is odd then $t\_n=1$, if even then $t\_n=0$. For this reason John H. Conway et al. call numbers $n$ satisfying $t\_n=1$ '''od''ious' numbers and numbers for which $t\_n=0$ '''ev''il' numbers.

Recurrence relation

The Thue-Morse sequence is the sequence $t\_n$ satisfying $t\_0\; =\; 0$ and
:$t\_\{2n\}\; =\; t\_n$
:$t\_\{2n+1\}\; =\; 1-t\_n$
for all positive integers n.

L-system

The Thue-Morse sequence is the output of the following Lindenmayer system:
'variables' 0 1
'constants' none
'start' 0
'rules' (0 → 01), (1 → 10)

Characterization using bitwise negation

The Thue-Morse sequence in the form given above, as a sequence of bits, can be defined recursively using the operation of bitwise negation.
So, the first element is 0.
Then once the first 2ⁿ elements have been specified, forming a string s, then the next 2ⁿ elements must form the bitwise negation of s.
Now we have defined the first 2ⁿ⁺¹ elements, and we recurse.
Spelling out the first few steps in detail:

★ We start with 0.

★ The bitwise negation of 0 is 1.

★ Combining these, the first 2 elements are 01.

★ The bitwise negation of 01 is 10.

★ Combining these, the first 4 elements are 0110.

★ The bitwise negation of 0110 is 1001.

★ Combining these, the first 8 elements are 01101001.

★ And so on.

Infinite product

The sequence can also be defined by:
: $prod\_\{i=0\}^\{\}\; (1\; -\; x^\{2^\{i\}\})\; =\; sum\_\{j=0\}^\{infty\}\; (-1)^\{t\_j\}\; x^\{j\}\; mbox\{,\}\; !$
where t_j is the jth element if we start at j = 0.

Some properties

Because each new block in the Thue-Morse sequence is defined by forming the bitwise negation of the beginning, and this is repeated at the beginning of the next block, the Thue-Morse sequence is filled with ''squares'': consecutive strings that are repeated.
That is, there are many instances of XX, where X is some string.
However, there are no ''cubes'': instances of XXX.
There are also no ''overlapping squares'': instances of 0X0X0 or 1X1X1.
The statement above that the Thue-Morse sequence is "filled with squares" can be made precise:
It is a recurrent sequence, meaning that given any finite string X in the sequence, there is some length n_X (often much longer than the length of X) such that X appears in ''every'' block of length n.
The easiest way to make a recurrent sequence is to form a periodic sequence, one where the sequence repeats entirely after a given number m of steps.
Then n_X can be set to any multiple of m that is larger than twice the length of X.
But the Morse sequence is recurrent ''without'' being periodic, nor even eventually periodic (meaning periodic after some nonperiodic initial segment).
One can define a function f from the set of binary sequences to itself by replacing every 0 in a sequence with 01 and every 1 with 10.
Then if T is the Thue-Morse sequence, then f(T) is T again; that is, T is a fixed point of f.
In fact, T is essentially the ''only'' fixed point of f; the only other fixed point is the bitwise negation of T, which is simply the Thue-Morse sequence on (1,0) instead of on (0,1).
This property may be generated to the concept of an automatic sequence.
=== In combinatorial game theory ===
The set of ''evil numbers'' (numbers $n$ with $t\_n=0$) forms a subspace of the nonnegative integers under nim-addition (bitwise exclusive or). For the game of Kayles, the evil numbers form the sparse space—the subspace of nim-values which occur for few (finitely many) positions in the game—and the odious numbers are the common coset.

The Prouhet-Tarry-Escott Problem

The Prouhet-Tarry-Escott Problem can be defined as: partition the set of all integers from 0 to N into two disjunct subsets S₀ and S₁, such that
$sum\_\{n\; in\; S0\}\; n^k\; =\; sum\_\{n\; in\; S1\}\; n^k$
This equation is true if

★ N is a power of 2,

★ and S₀ contains all n (nt_n=0, S₁ contains all n (nt_n=1.

★ and k=0...ld(N+1)

Fractals and Turtle graphics

A Turtle Graphics is the curve that is generated if an automaton is programmed with a sequence.
If the Thue-Morse Sequence members are used in order to select program states:

★ If t(n) = 0, move ahead by one unit,

★ If t(n) = 1, rotate counterclockwise by an angle of π/3,
the resulting curve converges to the Koch snowflake, a fractal curve of
infinite length containing a finite area.
This illustrates the fractal nature of the Thue-Morse Sequence.

History

The Thue-Morse sequence was first studied by P. Prouhet in 1851, who applied it to number theory.
However, Prouhet did not mention the sequence explicitly; this was left to Axel Thue in 1906, who used it to found the study of combinatorics on words.
The sequence was only brought to worldwide attention with the work of Marston Morse in 1921, when he applied it to differential geometry.
The sequence has been discovered independently many times, not always by professional research mathematicians; for example, Max Euwe, a chess grandmaster and mathematics teacher, discovered it in 1929 in an application to chess: by using its cube-free property (see above), he showed how to circumvent a rule aimed at preventing infinitely protracted games by declaring repetition of moves a draw.

See also

★ Prouhet-Thue-Morse constant

External links

★ The Ubiquitous Prouhet-Thue-Morse Sequence. Allouche, J.-P.; Shallit, J. O. Many applications and some history

★ MathWorld: Thue-Morse Sequence. Some other applications

★ Thue-Morse Sequence

★ Thue-Morse Sequence over (1,2)

★ When Thue-Morse meets Koch A paper showing an astonishing similarity between the Thue-Morse Sequence and the Koch snowflake

★ Reducing the influence of DC offset drift in analog IPs using the Thue-Morse Sequence A technical application of the Thue-Morse Sequence