In mathematics, the **Thue–Morse sequence** or **Prouhet–Thue–Morse sequence** or **parity sequence**^{ [1] } is the binary sequence (an infinite sequence of 0s and 1s) obtained by starting with 0 and successively appending the Boolean complement of the sequence obtained thus far. The first few steps of this procedure yield the strings 0 then 01, 0110, 01101001, 0110100110010110, and so on, which are prefixes of the Thue–Morse sequence. The full sequence begins:

- Definition
- Direct definition
- Fast sequence generation
- Recurrence relation
- L-system
- Characterization using bitwise negation
- Infinite product
- Properties
- In combinatorial game theory
- The Prouhet–Tarry–Escott problem
- Fractals and turtle graphics
- Equitable sequencing
- Hash collisions
- History
- See also
- Notes
- References
- Further reading
- External links

- 01101001100101101001011001101001....
^{ [1] }

The sequence is named after Axel Thue and Marston Morse.

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

To compute the *n*th element *t _{n}*, write the number

This method leads to a fast method for computing the Thue–Morse sequence: start with *t*_{0} = 0, and then, for each *n*, find the highest-order bit in the binary representation of *n* that is different from the same bit in the representation of *n* − 1. If this bit is at an even index, *t _{n}* differs from

In pseudo-code form:

`defgenerate_sequence(seq_length:int):"""Thue–Morse sequence."""value=0forn=0toseq_length-1by1:# Note: assumes an even number of bits in the word size, and two's complement arithmetic so that when n == 0, x is odd (e.g. 31 or 63)x=index_of_highest_one_bit(n^(n-1))if((x&1)==0):# bit index is even, so toggle valuevalue=1-valueyieldvalue`

The resulting algorithm takes constant time to generate each sequence element, using only a logarithmic number of bits (constant number of words) of memory.^{ [3] }

The Thue–Morse sequence is the sequence *t _{n}* satisfying the recurrence relation

for all non-negative integers *n*.^{ [2] }

The Thue–Morse sequence is a morphic word:^{ [4] } it is the output of the following Lindenmayer system:

Variables | 0, 1 |
---|---|

Constants | None |

Start | 0 |

Rules | (0 → 01), (1 → 10) |

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 2^{n} elements have been specified, forming a string *s*, then the next 2^{n} elements must form the bitwise negation of *s*. Now we have defined the first 2^{n+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.

So

*T*_{0}= 0.*T*_{1}= 01.*T*_{2}= 0110.*T*_{3}= 01101001.*T*_{4}= 0110100110010110.*T*_{5}= 01101001100101101001011001101001.*T*_{6}= 0110100110010110100101100110100110010110011010010110100110010110.- And so on.

The sequence can also be defined by:

where *t*_{j} is the *j*th element if we start at *j* = 0.

The Thue–Morse sequence contains many *squares*: instances of the string , where denotes the string , , , or , where for some and is the bitwise negation of .^{ [5] } For instance, if , then . The square appears in starting at the 16th bit. Since all squares in are obtained by repeating one of these 4 strings, they all have length or for some . contains no *cubes*: instances of . There are also no *overlapping squares*: instances of or .^{ [6] }^{ [7] } The critical exponent of is 2.^{ [8] }

The Thue–Morse sequence is a uniformly recurrent word: given any finite string *X* in the sequence, there is some length *n _{X}* (often much longer than the length of

The sequence *T*_{2n} is a palindrome for any *n*. Furthermore, let *q*_{n} be a word obtained by counting the ones between consecutive zeros in *T*_{2n} . For instance, *q*_{1} = 2 and *q*_{2} = 2102012. Since *T _{n}* does not contain

The **Thue–Morse morphism ***μ* is defined on alphabet {0,1} by the substitution map *μ*(0) = 01, *μ*(1) = 10: every 0 in a sequence is replaced with 01 and every 1 with 10.^{ [12] } If *T* is the Thue–Morse sequence, then *μ*(*T*) is also *T*. Thus, *T* is a fixed point of *μ*. The morphism *μ* is a prolongable morphism on the free monoid {0,1}^{∗} with *T* as fixed point: *T* is essentially the *only* fixed point of *μ*; 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 generalized to the concept of an automatic sequence.

The *generating series* of *T* over the binary field is the formal power series

This power series is algebraic over the field of rational functions, satisfying the equation^{ [13] }

The set of *evil numbers* (numbers with ) forms a subspace of the nonnegative integers under nim-addition (bitwise exclusive or). For the game of Kayles, evil nim-values occur for few (finitely many) positions in the game, with all remaining positions having odious nim-values.

The Prouhet–Tarry–Escott problem can be defined as: given a positive integer *N* and a non-negative integer *k*, partition the set *S* = { 0, 1, ..., *N*-1 } into two disjoint subsets *S*_{0} and *S*_{1} that have equal sums of powers up to k, that is:

- for all integers
*i*from 1 to*k*.

This has a solution if *N* is a multiple of 2^{k+1}, given by:

*S*_{0}consists of the integers*n*in*S*for which*t*= 0,_{n}*S*_{1}consists of the integers*n*in*S*for which*t*= 1._{n}

For example, for *N* = 8 and *k* = 2,

- 0 + 3 + 5 + 6 = 1 + 2 + 4 + 7,
- 0
^{2}+ 3^{2}+ 5^{2}+ 6^{2}= 1^{2}+ 2^{2}+ 4^{2}+ 7^{2}.

The condition requiring that *N* be a multiple of 2^{k+1} is not strictly necessary: there are some further cases for which a solution exists. However, it guarantees a stronger property: if the condition is satisfied, then the set of *k*th powers of any set of *N* numbers in arithmetic progression can be partitioned into two sets with equal sums. This follows directly from the expansion given by the binomial theorem applied to the binomial representing the *n*th element of an arithmetic progression.

For generalizations of the Thue–Morse sequence and the Prouhet–Tarry–Escott problem to partitions into more than two parts, see Bolker, Offner, Richman and Zara, "The Prouhet–Tarry–Escott problem and generalized Thue–Morse sequences".^{ [14] }

Using turtle graphics, a curve can be generated if an automaton is programmed with a sequence. When 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 by an angle of π/3 radians (60°)

The resulting curve converges to the Koch curve, a fractal curve of infinite length containing a finite area. This illustrates the fractal nature of the Thue–Morse Sequence.^{ [15] }

It is also possible to draw the curve precisely using the following instructions:^{ [16] }

- If
*t*(*n*) = 0, rotate by an angle of π radians (180°), - If
*t*(*n*) = 1, move ahead by one unit, then rotate by an angle of π/3 radians.

In their book on the problem of fair division, Steven Brams and Alan Taylor invoked the Thue–Morse sequence but did not identify it as such. When allocating a contested pile of items between two parties who agree on the items' relative values, Brams and Taylor suggested a method they called *balanced alternation*, or *taking turns taking turns taking turns . . . *, as a way to circumvent the favoritism inherent when one party chooses before the other. An example showed how a divorcing couple might reach a fair settlement in the distribution of jointly-owned items. The parties would take turns to be the first chooser at different points in the selection process: Ann chooses one item, then Ben does, then Ben chooses one item, then Ann does.^{ [17] }

Lionel Levine and Katherine E. Stange, in their discussion of how to fairly apportion a shared meal such as an Ethiopian dinner, proposed the Thue–Morse sequence as a way to reduce the advantage of moving first. They suggested that “it would be interesting to quantify the intuition that the Thue–Morse order tends to produce a fair outcome.”^{ [18] }

Robert Richman addressed this problem, but he too did not identify the Thue–Morse sequence as such at the time of publication.^{ [19] } He presented the sequences *T _{n}* as step functions on the interval [0,1] and described their relationship to the Walsh and Rademacher functions. He showed that the

Joshua Cooper and Aaron Dutle showed why the Thue–Morse order provides a fair outcome for discrete events.^{ [21] } They considered the fairest way to stage a Galois duel, in which each of the shooters has equally poor shooting skills. Cooper and Dutle postulated that each dueler would demand a chance to fire as soon as the other's *a priori* probability of winning exceeded their own. They proved that, as the duelers’ hitting probability approaches zero, the firing sequence converges to the Thue–Morse sequence. In so doing, they demonstrated that the Thue–Morse order produces a fair outcome not only for sequences *T _{n}* of length

Thus the mathematics supports using the Thue–Morse sequence instead of alternating turns when the goal is fairness but earlier turns differ monotonically from later turns in some meaningful quality, whether that quality varies continuously^{ [19] } or discretely.^{ [21] }

Sports competitions form an important class of equitable sequencing problems, because strict alternation often gives an unfair advantage to one team. Ignacio Palacios-Huerta proposed changing the sequential order to Thue–Morse to improve the * ex post * fairness of various tournament competitions, such as the kicking sequence of a penalty shoot-out in soccer.^{ [22] } He did a set of field experiments with pro players and found that the team kicking first won 60% of games using ABAB (or *T*_{1}), 54% using ABBA (or *T*_{2}), and 51% using full Thue–Morse (or *T*_{n}). As a result, ABBA is undergoing extensive trials in FIFA (European and World Championships) and English Federation professional soccer (EFL Cup).^{ [23] } An ABBA serving pattern has also been found to improve the fairness of tennis tie-breaks.^{ [24] } In competitive rowing, *T*_{2} is the only arrangement of port- and starboard-rowing crew members that eliminates transverse forces (and hence sideways wiggle) on a four-membered coxless racing boat, while *T*_{3} is one of only four rigs to avoid wiggle on an eight-membered boat.^{ [25] }

Fairness is especially important in player drafts. Many professional sports leagues attempt to achieve competitive parity by giving earlier selections in each round to weaker teams. By contrast, fantasy football leagues have no pre-existing imbalance to correct, so they often use a “snake” draft (forward, backward, etc.; or *T*_{1}).^{ [26] } Ian Allan argued that a “third-round reversal” (forward, backward, backward, forward, etc.; or *T*_{2}) would be even more fair.^{ [27] } Richman suggested that the fairest way for “captain A” and “captain B” to choose sides for a pick-up game of basketball mirrors *T*_{3}: captain A has the first, fourth, sixth, and seventh choices, while captain B has the second, third, fifth, and eighth choices.^{ [19] }

The initial 2^{k} bits of the Thue–Morse sequence are mapped to 0 by a wide class of polynomial hash functions modulo a power of two, which can lead to hash collisions.^{ [28] }

The Thue–Morse sequence was first studied by Eugène Prouhet in 1851,^{ [29] } 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 the threefold repetition rule aimed at preventing infinitely protracted games by declaring repetition of moves a draw. At the time, consecutive identical board states were required to trigger the rule; the rule was later amended to the same board position reoccurring three times at any point, as the sequence shows that the consecutive criterion can be evaded forever.

- Dejean's theorem
- Fabius function
- Gray code
^{ [30] }^{ [31] }^{ [32] } - Komornik–Loreti constant
- Prouhet–Thue–Morse constant

- 1 2 Sloane, N. J. A. (ed.). "SequenceA010060(Thue-Morse sequence)".
*The On-Line Encyclopedia of Integer Sequences*. OEIS Foundation. - 1 2 Allouche & Shallit (2003 , p. 15)
- ↑ Arndt (2011).
- ↑ Lothaire (2011 , p. 11)
- ↑ Brlek (1989).
- ↑ Lothaire (2011 , p. 113)
- ↑ Pytheas Fogg (2002 , p. 103)
- ↑ Krieger (2006).
- ↑ Lothaire (2011 , p. 30)
- ↑ Berthé & Rigo (2010).
- ↑ Lothaire (2011 , p. 31)
- ↑ Berstel et al. (2009 , p. 70)
- ↑ Berstel et al. (2009 , p. 80)
- ↑ Bolker et al. (2016).
- ↑ Ma & Holdener (2005).
- ↑ Abel, Zachary (January 23, 2012). "Thue-Morse Navigating Turtles".
*Three-Cornered Things*. - ↑ Brams & Taylor (1999).
- ↑ Levine & Stange (2012).
- 1 2 3 Richman (2001)
- ↑ Abrahams (2010).
- 1 2 Cooper & Dutle (2013)
- ↑ Palacios-Huerta (2012).
- ↑ Palacios-Huerta (2014).
- ↑ Cohen-Zada, Krumer & Shapir (2018).
- ↑ Barrow (2010).
- ↑ "Fantasy Draft Types".
*NFL.com*. Archived from the original on October 12, 2018. - ↑ Allan, Ian (July 16, 2014). "Third-Round Reversal Drafts".
*Fantasy Index*. Retrieved September 1, 2020. - ↑ Pachocki, Jakub; Radoszewski, Jakub (2013). "Where to Use and How not to Use Polynomial String Hashing" (PDF).
*Olympiads in Informatics*.**7**: 90–100. - ↑ The ubiquitous Prouhet-Thue-Morse sequence by Jean-Paul Allouche and Jeffrey Shallit
- ↑ Fredricksen, Harold (1992). "Gray codes and the Thue-Morse-Hedlund sequence".
*Journal of Combinatorial Mathematics and Combinatorial Computing (JCMCC)*. Naval Postgraduate School, Department of Mathematics, Monterey, California, USA.**11**: 3–11. - ↑ Erickson, John (2018-10-30). "On the Asymptotic Relative Change for Sequences of Permutations" . Retrieved 2021-01-31.
- ↑ Plousos, George (2020-06-21). "What is the relationship between the Gray code and the Thue–Morse sequence".
*Quora*. Archived from the original on 2020-12-17. Retrieved 2021-01-31.

In mathematics, **modular arithmetic** is a system of arithmetic for integers, where numbers "wrap around" when reaching a certain value, called the **modulus**. The modern approach to modular arithmetic was developed by Carl Friedrich Gauss in his book *Disquisitiones Arithmeticae*, published in 1801.

In mathematics and computing, **Fibonacci coding** is a universal code which encodes positive integers into binary code words. It is one example of representations of integers based on Fibonacci numbers. Each code word ends with "11" and contains no other instances of "11" before the end.

In abstract algebra, the **free monoid** on a set is the monoid whose elements are all the finite sequences of zero or more elements from that set, with string concatenation as the monoid operation and with the unique sequence of zero elements, often called the empty string and denoted by ε or λ, as the identity element. The free monoid on a set *A* is usually denoted *A*^{∗}. The **free semigroup** on *A* is the subsemigroup of *A*^{∗} containing all elements except the empty string. It is usually denoted *A*^{+}.

In mathematics, the **Prouhet–Thue–Morse constant**, named for Eugène Prouhet, Axel Thue, and Marston Morse, is the number—denoted by τ—whose binary expansion 0.01101001100101101001011001101001... is given by the Prouhet–Thue–Morse sequence. That is,

In mathematics, a **Sturmian word**, named after Jacques Charles François Sturm, is a certain kind of infinitely long sequence of characters. Such a sequence can be generated by considering a game of English billiards on a square table. The struck ball will successively hit the vertical and horizontal edges labelled 0 and 1 generating a sequence of letters. This sequence is a Sturmian word.

In number theory, an **odious number** is a positive integer that has an odd number of 1s in its binary expansion. Non-negative integers that are not odious are called evil numbers.

In combinatorics, a **squarefree word** is a word that does not contain any squares. A **square** is a word of the form XX, where X is not empty. Thus, a squarefree word can also be defined as a word that avoids the pattern XX.

In number theory, an **evil number** is a non-negative integer that has an even number of 1s in its binary expansion. These numbers give the positions of the zero values in the Thue–Morse sequence, and for this reason they have also been called the **Thue–Morse set**. Non-negative integers that are not evil are called odious numbers.

In mathematics and theoretical computer science, an **automatic sequence** (also called a ** k-automatic sequence** or a

In mathematics, the **Prouhet–Tarry–Escott problem** asks for two disjoint multisets *A* and *B* of *n* integers each, whose first *k* power sum symmetric polynomials are all equal. That is, the two multisets should satisfy the equations

**Combinatorics on words** is a fairly new field of mathematics, branching from combinatorics, which focuses on the study of words and formal languages. The subject looks at letters or symbols, and the sequences they form. **Combinatorics on words** affects various areas of mathematical study, including algebra and computer science. There have been a wide range of contributions to the field. Some of the first work was on square-free words by Axel Thue in the early 1900s. He and colleagues observed patterns within words and tried to explain them. As time went on, combinatorics on words became useful in the study of algorithms and coding. It led to developments in abstract algebra and answering open questions.

In the mathematical theory of non-standard positional numeral systems, the **Komornik–Loreti constant** is a mathematical constant that represents the smallest base *q* for which the number 1 has a unique representation, called its *q*-development. The constant is named after Vilmos Komornik and Paola Loreti, who defined it in 1998.

In computer science, the **complexity function** of a *word* or *string* is the function that counts the number of distinct *factors* of that string. More generally, the complexity function of a formal language counts the number of distinct words of given length.

In mathematics and computer science, a **morphic word** or **substitutive word** is an infinite sequence of symbols which is constructed from a particular class of endomorphism of a free monoid.

In mathematics, a **sesquipower** or **Zimin word** is a string over an alphabet with identical prefix and suffix. Sesquipowers are unavoidable patterns, in the sense that all sufficiently long strings contain one.

In mathematics and theoretical computer science, a pattern is an **unavoidable pattern** if it is unavoidable on any finite alphabet.

In mathematics, a **recurrent word** or **sequence** is an infinite word over a finite alphabet in which every factor occurs infinitely many times. An infinite word is recurrent if and only if it is a sesquipower.

In mathematics and computer science, the **critical exponent** of a finite or infinite sequence of symbols over a finite alphabet describes the largest number of times a contiguous subsequence can be repeated. For example, the critical exponent of "Mississippi" is 7/3, as it contains the string "ississi", which is of length 7 and period 3.

In mathematics, **Ostrowski numeration**, named after Alexander Ostrowski, is either of two related numeration systems based on continued fractions: a non-standard positional numeral system for integers and a non-integer representation of real numbers.

In mathematics and theoretical computer science, a ** k-regular sequence** is a sequence satisfying linear recurrence equations that reflect the base-

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Wikimedia Commons has media related to Thue-Morse sequence .

- "Thue-Morse sequence",
*Encyclopedia of Mathematics*, EMS Press, 2001 [1994] - Weisstein, Eric W. "Thue-Morse Sequence".
*MathWorld*. - Allouche, J.-P.; Shallit, J. O. The Ubiquitous Prouhet-Thue-Morse Sequence. (contains many applications and some history)
- Thue–Morse Sequence over (1,2) (sequence A001285 in the OEIS )
- OEIS sequenceA000069(Odious numbers: numbers with an odd number of 1's in their binary expansion)
- OEIS sequenceA001969(Evil numbers: numbers with an even number of 1's in their binary expansion)
- Reducing the influence of DC offset drift in analog IPs using the Thue-Morse Sequence. A technical application of the Thue–Morse Sequence
- MusiNum - The Music in the Numbers. Freeware to generate self-similar music based on the Thue–Morse Sequence and related number sequences.
- Parker, Matt. "The Fairest Sharing Sequence Ever" (video). standupmaths. Retrieved 20 January 2016.

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