Thue-Morse sequence
The Thue-Morse sequence begins:
- 01101001100101101001011001101001...
| Table of contents |
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2 Some properties 3 History 4 See also 5 External links |
Definition
There are several equivalent ways of defining the Thue-Morse sequence.
Recurrence relation
The Thue-Morse sequence is the sequence 
satisfying 
and
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:
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 nX (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 nX 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.
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. Since Thue published in Norwegian, his work was ignored at first; 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.
See also
- Prouhet-Thue-Morse constant
- Sturmian word, of which the Thue-Morse sequence is a standard example
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
- Sequence A010060 from Sloane's On-Line Encyclopedia of Integer Sequences
- Sequence A001285 from Sloane's OEIS for the sequence over (1,2)
