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{{Short description|Infinite binary sequence generated by repeated complementation and concatenation}}
[[Image:Morse-Thue sequence.gif|frame|right|This graphic demonstrates the repeating and complementary makeup of the Thue–Morse sequence.]]
In [[mathematics]], the '''Thue–Morse''' or '''Prouhet–Thue–Morse sequence''' is the [[binary sequence]] (an infinite sequence of 0s and 1s) that can be obtained by starting with 0 and successively appending the [[Boolean algebra|Boolean complement]] of the sequence obtained thus far.<ref name=A010060>{{Cite OEIS|A010060|Thue-Morse sequence}}</ref> It is sometimes called the '''fair share sequence''' because of its applications to [[fair division]] or '''parity sequence'''. The first few steps of this procedure yield the strings 0, 01, 0110, 01101001, 0110100110010110, and so on, which are the [[prefix (mathematics)|prefixes]] of the Thue–Morse sequence. The full sequence begins:

:01101001100101101001011001101001....<ref name=A010060 />

The sequence is named after [[Axel Thue]], [[Marston Morse]] and (in its extended form) [[Eugène Prouhet]].

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

=== Direct definition ===
[[File:Thue-Morse binary digit sum.svg|thumb|128px|When counting in binary, the digit sum modulo 2 is the Thue–Morse sequence]]

To compute the ''n''th element ''t<sub>n</sub>'', write the number ''n'' in [[Binary number|binary]].  If the [[Hamming weight|number of ones]] in this binary expansion is odd then ''t<sub>n</sub>''&nbsp;=&nbsp;1, if even then ''t<sub>n</sub>''&nbsp;=&nbsp;0.<ref name=AS15>{{harvtxt|Allouche|Shallit|2003|p=15}}</ref>  That is, ''t<sub>n</sub>'' is the [[parity bit|even parity bit]] for ''n''. [[John H. Conway]] ''et al''. deemed numbers ''n'' satisfying ''t<sub>n</sub>''&nbsp;=&nbsp;1 to be ''odious'' (intended to be similar to ''odd'') numbers, and numbers for which ''t<sub>n</sub>''&nbsp;=&nbsp;0 to be ''evil'' (similar to ''even'') numbers.

=== Fast sequence generation ===
This method leads to a fast method for computing the Thue–Morse sequence: start with {{math|1=''t''<sub>0</sub>&nbsp;=&nbsp;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 {{math|1=''n''&nbsp;&minus;&nbsp;1}}. If this bit is at an even index,  ''t<sub>n</sub>'' differs from {{math|1=''t''<sub>''n''&nbsp;&minus;&nbsp;1</sub>}}, and otherwise it is the same as {{math|1=''t''<sub>''n''&nbsp;&minus;&nbsp;1</sub>}}.

In [[Python (programming language)|Python]]:
<syntaxhighlight lang="python">
def generate_sequence(seq_length: int) -> int:
    """Thue–Morse sequence."""
    value: int = 1
    for n in range(seq_length):
        # Note: assumes that (-1).bit_length() gives 1
        x: int = (n ^ (n - 1)).bit_length() + 1
        if x & 1 == 0:
            # Bit index is even, so toggle value
            value = 1 - value
        yield value
</syntaxhighlight>
The resulting algorithm takes constant time to generate each sequence element, using only a [[L (complexity)|logarithmic number of bits]] (constant number of words) of memory.{{sfnp|Arndt|2011}}

=== Recurrence relation ===
The Thue–Morse sequence is the sequence ''t<sub>n</sub>'' satisfying the [[recurrence relation]]

:<math>\begin{align}
   t_0 &= 0,\\
   t_{2n} &= t_n,\\
   t_{2n+1} &= 1 - t_n,
 \end{align}</math>

for all non-negative integers ''n''.<ref name=AS15/>

=== L-system ===
[[File:Thue-Morse L-System.svg|thumb|256px|Thue–Morse sequence generated by an L-System]]

The Thue–Morse sequence is a [[morphic word]]:<ref>{{harvtxt|Lothaire|2011|p=11}}</ref> it is the output of the following [[L-system|Lindenmayer system]]:
{| class="wikitable"
|-
! Variables <td> 0, 1 </td>
|-
! Constants <td> None </td>
|-
! Start <td> 0 </td>
|-
! Rules <td> (0 → 01), (1 → 10) </td>
|}

=== Characterization using bitwise negation ===
The Thue–Morse sequence in the form given above, as a sequence of [[bit]]s, can be defined [[recursion|recursively]] using the operation of [[bitwise negation]].
So, the first element is 0.
Then once the first 2<sup>''n''</sup> elements have been specified, forming a string ''s'', then the next 2<sup>''n''</sup> elements must form the bitwise negation of ''s''.
Now we have defined the first 2<sup>''n''+1</sup> 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''<sub>0</sub> = 0.
* ''T''<sub>1</sub> = 01.
* ''T''<sub>2</sub> = 0110.
* ''T''<sub>3</sub> = 01101001.
* ''T''<sub>4</sub> = 0110100110010110.
* ''T''<sub>5</sub> = 01101001100101101001011001101001.
* ''T''<sub>6</sub> = 0110100110010110100101100110100110010110011010010110100110010110.
* And so on.

In [[Python (programming language)|Python]]:
<syntaxhighlight lang="python">
def thue_morse_bits(n):
	"""Return an int containing the first 2**n bits of the Thue-Morse sequence, low-order bit 1st."""
	bits = 0
    for i in range(n):
        bits |= ((1 << (1 << i)) - 1 - bits) << (1 << i)
    return bits
</syntaxhighlight>

Which can then be converted to a (reversed) string as follows: 
<syntaxhighlight lang="python">
n = 7
print(f"{thue_morse_bits(n):0{1<<n}b}")
</syntaxhighlight>

=== Infinite product ===
The sequence can also be defined by:
: <math> \prod_{i=0}^{\infty} \left(1 - x^{2^i}\right) = \sum_{j=0}^{\infty} (-1)^{t_j} x^j,</math>
where ''t''<sub>''j''</sub> is the ''j''th element if we start at ''j'' = 0.

== Properties ==
The Thue–Morse sequence contains many ''squares'': instances of the string <math>XX</math>, where <math>X</math> denotes the string <math>A</math>, <math>\overline{A}</math>, <math>A \overline{A}A</math>, or <math>\overline{A}A \overline{A}</math>, where <math>A=T_{k}</math> for some <math>k\ge 0</math> and <math>\overline{A}</math> is the bitwise negation of <math>A</math>.{{sfnp|Brlek|1989}} For instance, if <math>k=0</math>, then <math>A=T_{0}=0 </math>. The square <math>A \overline{A}AA \overline{A}A = 010010</math> appears in <math>T</math> starting at the 16th bit. Since all squares in <math>T</math> are obtained by repeating one of these 4 strings, they all have length <math>2^n</math> or <math>3\cdot2^n</math> for some <math>n\ge 0</math>. <math>T</math> contains no ''cubes'': instances of <math>XXX</math>. There are also no ''overlapping squares'': instances of <math>0X0X0</math> or <math>1X1X1</math>.<ref name=ACOW113>{{harvtxt|Lothaire|2011|p=113}}</ref><ref name=PF103>{{harvtxt|Pytheas Fogg|2002|p=103}}</ref> The [[Critical exponent of a word|critical exponent]] of <math>T</math> is 2.{{sfnp|Krieger|2006}}

The Thue–Morse sequence is a [[uniformly recurrent word]]: given any finite string ''X'' in the sequence, there is some length ''n<sub>X</sub>'' (often much longer than the length of ''X'') such that ''X'' appears in ''every'' block of length ''n<sub>X</sub>''.<ref name=ACOW30>{{harvtxt|Lothaire|2011|p=30}}</ref>{{sfnp|Berthé|Rigo|2010}} Notably, the Thue–Morse sequence is uniformly recurrent ''without'' being either [[periodic sequence|periodic]] or eventually periodic (i.e., periodic after some initial nonperiodic segment).<ref name=ACOW31>{{harvtxt|Lothaire|2011|p=31}}</ref>

The sequence ''T''<sub>2''n''</sub> is a [[Palindromic number|palindrome]] for any ''n''. Furthermore, let ''q''<sub>''n''</sub> be a word obtained by counting the ones between consecutive zeros in ''T''<sub>2''n''</sub> . For instance, ''q''<sub>1</sub> = 2 and ''q''<sub>2</sub> = 2102012. Since ''T<sub>n</sub>'' does not contain ''overlapping squares'', the words ''q<sub>n</sub>'' are palindromic [[squarefree word]]s.

The '''Thue–Morse [[monoid morphism|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.<ref name=BLRS70>{{harvtxt|Berstel|Lauve|Reutenauer|Saliola|2009|p=70}}</ref> If ''T'' is the Thue–Morse sequence, then ''μ''(''T'') is also ''T''. Thus, ''T'' is a [[fixed point (mathematics)|fixed point]] of ''μ''. The morphism ''μ'' is a [[prolongable morphism]] on the [[free monoid]] {0,1}<sup>&lowast;</sup> 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]]

:<math>t(Z) = \sum_{n=0}^\infty T(n) Z^n \ . </math>

This [[power series]] is algebraic over the field of rational functions, satisfying the equation<ref name=BLRS80>{{harvtxt|Berstel|Lauve|Reutenauer|Saliola|2009|p=80}}</ref>

:<math>Z + (1+Z)^2 t + (1+Z)^3 t^2 = 0 </math>

=== In combinatorial game theory ===
The set of ''evil numbers'' (numbers <math>n</math> with <math>t_n=0</math>) forms a subspace of the nonnegative integers under [[nim-addition]] ([[bitwise operation|bitwise]] [[exclusive or]]).  For the game of [[Kayles]], evil [[nim-value]]s occur for few (finitely many) positions in the game, with all remaining positions having odious nim-values.

=== The Prouhet–Tarry–Escott problem === <!-- This section is linked to from [[Prouhet–Tarry–Escott problem]]. -->
The [[Prouhet–Tarry–Escott problem]] can be defined as: given a positive integer ''N'' and a non-negative integer ''k'', [[Partition of a set|partition]] the set ''S'' = { 0, 1, ..., ''N''-1 } into two [[Disjoint sets|disjoint]] subsets ''S''<sub>0</sub> and ''S''<sub>1</sub> that have equal sums of powers up to k, that is:
:<math> \sum_{x \in S_0} x^i = \sum_{x \in S_1} x^i</math> for all integers ''i'' from 1 to ''k''.

This has a solution if ''N'' is a multiple of 2<sup>''k''+1</sup>, given by:
* ''S''<sub>0</sub> consists of the integers ''n'' in ''S'' for which ''t<sub>n</sub>'' = 0,
* ''S''<sub>1</sub> consists of the integers ''n'' in ''S'' for which ''t<sub>n</sub>'' = 1.

For example, for ''N'' = 8 and ''k'' = 2,
:{{nowrap|1= 0 + 3 + 5 + 6 = 1 + 2 + 4 + 7,}}
:{{nowrap|1= 0<sup>2</sup> + 3<sup>2</sup> + 5<sup>2</sup> + 6<sup>2</sup> = 1<sup>2</sup> + 2<sup>2</sup> + 4<sup>2</sup> + 7<sup>2</sup>.}}

The condition requiring that ''N'' be a multiple of 2<sup>''k''+1</sup> 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".{{sfnp|Bolker|Offner|Richman|Zara|2016}}

=== Fractals and turtle graphics ===
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 [[radian]]s (60°)

The resulting curve converges to the [[Koch snowflake|Koch curve]], a [[fractal curve]] of
infinite length containing a finite area. This illustrates the fractal nature of the Thue–Morse Sequence.{{sfnp|Ma|Holdener|2005}}

It is also possible to draw the curve precisely using the following instructions:<ref>{{Cite web|url=http://blog.zacharyabel.com/2012/01/thue-morse-navigating-turtles/|title=Thue-Morse Navigating Turtles|last=Abel|first=Zachary|date=January 23, 2012|website=Three-Cornered Things}}</ref>

*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.

===Equitable sequencing===
In their book on the problem of [[fair division]], [[Steven Brams]] and [[Alan D. Taylor|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.{{sfnp|Brams|Taylor|1999}}

[[Lionel Levine]] and [[Katherine E. Stange]], in their discussion of how to fairly apportion a shared meal such as an [[Ethiopian cuisine|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.”{{sfnp|Levine|Stange|2012}}

Robert Richman addressed this problem, but he too did not identify the Thue–Morse sequence as such at the time of publication.<ref name="richman">{{harvtxt|Richman|2001}}</ref>   He presented the sequences [[#Characterization using bitwise negation|''T<sub>n</sub>'']] as [[step function]]s on the interval [0,1] and described their relationship to the [[Walsh function|Walsh]] and [[Hans Rademacher|Rademacher]] functions.  He showed that the ''n''th [[derivative]] can be expressed in terms of  ''T<sub>n</sub>''.  As a consequence, the step function arising from  ''T<sub>n</sub>'' is [[Orthogonality|orthogonal]] to [[polynomial]]s of [[Degree of a polynomial|order]] ''n''&nbsp;&minus;&nbsp;1.  A consequence of this result is that a resource whose value is expressed as a [[Monotonic function|monotonically]] decreasing [[continuous function]] is most fairly allocated using a sequence that converges to Thue–Morse as the function becomes [[Flat function|flatter]].  An example showed how to pour cups of [[Drip brew|coffee]] of equal strength from a carafe with a [[Nonlinear system|nonlinear]] [[concentration]] [[gradient]], prompting a whimsical article in the popular press.{{sfnp|Abrahams|2010}}

Joshua Cooper and Aaron Dutle showed why the Thue–Morse order provides a fair outcome for discrete events.<ref name="cooper">{{harvtxt|Cooper|Dutle|2013}}</ref>  They considered the fairest way to stage a [[Évariste Galois|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|''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 [[#Characterization using bitwise negation|''T<sub>n</sub>'']] of length ''2<sup>n</sup>'', but for sequences of any 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<ref name="richman" /> or discretely.<ref name="cooper" />

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 (association football)#Procedure|penalty shoot-out]] in soccer.{{sfnp|Palacios-Huerta|2012}} 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''<sub>1</sub>), 54% using ABBA (or ''T''<sub>2</sub>), and 51% using full Thue–Morse (or ''T''<sub>n</sub>).  As a result, ABBA is undergoing [[Penalty shoot-out (association football)#Advantage to team kicking first?|extensive trials]] in [[FIFA#FIFA competitions|FIFA (European and World Championships)]] and English Federation professional soccer ([[EFL Cup]]).{{sfnp|Palacios-Huerta|2014}} An ABBA serving pattern has also been found to improve the fairness of [[Tennis scoring system#Scoring a tiebreak game|tennis tie-breaks]].{{sfnp|Cohen-Zada|Krumer|Shapir|2018}} In [[Rowing (sport)|competitive rowing]], ''T''<sub>2</sub> is the only arrangement of [[Port and starboard|port- and starboard-rowing]] crew members that eliminates transverse forces (and hence sideways wiggle) on a four-membered coxless racing boat, while ''T''<sub>3</sub> is one of only four [[Boat rigging|rigs]]  to avoid wiggle on an eight-membered boat.{{sfnp|Barrow|2010}}

Fairness is especially important in [[Draft (sports)|player drafts]]. Many professional sports leagues attempt to achieve [[Parity (sports)|competitive parity]] by giving earlier selections in each round to weaker teams. By contrast, [[Fantasy football (American)|fantasy football leagues]] have no pre-existing imbalance to correct, so they often use a “snake” draft (forward, backward, etc.; or ''T''<sub>1</sub>).<ref>{{cite web |url=http://www.nfl.com/fantasyfootball/help/drafttypes |archive-url=https://web.archive.org/web/20181012053900/http://www.nfl.com/fantasyfootball/help/drafttypes |archive-date=October 12, 2018 |title=Fantasy Draft Types |work=[[National Football League|NFL.com]]}}</ref> Ian Allan argued that a “third-round reversal” (forward, backward, backward, forward, etc.; or ''T''<sub>2</sub>) would be even more fair.<ref>{{cite web |first=Ian |last=Allan |url=https://www.fantasyindex.com/2014/07/16/fantasy-news/third-round-reversal-drafts |title=Third-Round Reversal Drafts |date=July 16, 2014 |work=Fantasy Index |access-date=September 1, 2020}}</ref> Richman suggested that the fairest way for “captain A” and “captain B” to choose sides for a [[Streetball|pick-up game of basketball]] mirrors  ''T''<sub>3</sub>:  captain A has the first, fourth, sixth, and seventh choices, while captain B has the second, third, fifth, and eighth choices.<ref name="richman" />

=== Hash collisions ===
The initial {{math|2<sup>''k''</sup>}} bits of the Thue–Morse sequence are mapped to 0 by a wide class of polynomial [[hash function]]s modulo a [[power of two]], which can lead to [[hash collision]]s.<ref>{{cite journal |last1=Pachocki |first1=Jakub |last2=Radoszewski |first2=Jakub |title=Where to Use and How not to Use Polynomial String Hashing |journal=Olympiads in Informatics |date=2013 |volume=7 |pages=90–100 |url=https://ioinformatics.org/journal/INFOL119.pdf}}</ref>

===Riemann zeta function===
Certain linear combinations of Dirichlet series whose coefficients are terms of the Thue–Morse sequence give rise to identities involving the Riemann Zeta function (Tóth, 2022 <ref>
{{cite journal|author1-link=Tóth|last1=Tóth|first1=László|title=Linear Combinations of Dirichlet Series Associated with the Thue-Morse Sequence|journal=Integers|volume=22|year=2022|issue=article 98|arxiv=2211.13570 }}
</ref>). For instance:
:<math>  \begin{align}
    \sum_{n\geq1} \frac{5 t_{n-1} + 3 t_n}{n^2} &= 4 \zeta(2) = \frac{2 \pi^2}{3}, \\
	\sum_{n\geq1} \frac{9 t_{n-1} + 7 t_n}{n^3} &= 8 \zeta(3),\end{align}</math>
where <math>(t_n)_{n\geq0}</math> is the <math>n^{\rm th}</math> term of the Thue–Morse sequence. In fact, for all <math>s</math> with real part greater than <math>1</math>, we have
:<math>  (2^s+1) \sum_{n\geq1} \frac{t_{n-1}}{n^s} + (2^s-1) \sum_{n\geq1} \frac{t_{n}}{n^s} = 2^s \zeta(s).</math>

== History ==
The Thue–Morse sequence was first studied by {{ill|Eugène Prouhet|fr}} in 1851,<ref>[https://cs.uwaterloo.ca/~shallit/Papers/ubiq15.pdf The ubiquitous Prouhet-Thue-Morse sequence by Jean-Paul Allouche and Jeffrey Shallit]</ref> 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 [[Multiple discovery|discovered independently]] many times, not always by professional research mathematicians; for example, [[Max Euwe]], a [[Grandmaster (chess)|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.

==See also==
* [[Dejean's theorem]]
* [[Fabius function]]
* [[Square-free word#First difference of the Thue–Morse sequence|First difference of the Thue–Morse sequence]]
* [[Gray code]]<ref name="Fredricksen_1992"/><ref name="Erickson_2018"/>
* [[Komornik–Loreti constant]]
* [[Prouhet–Thue–Morse constant]]

==Notes==
{{reflist|refs=
<ref name="Fredricksen_1992">{{cite journal |author-first=Harold |author-last=Fredricksen |title=Gray codes and the Thue-Morse-Hedlund sequence |journal=Journal of Combinatorial Mathematics and Combinatorial Computing  |volume=11 |date=1992 |pages=3–11 |location=Naval Postgraduate School, Department of Mathematics, Monterey, California, USA}}</ref>
<ref name="Erickson_2018">{{cite web |author-first=John |author-last=Erickson |date=2018-10-30 |title=On the Asymptotic Relative Change for Sequences of Permutations |url=https://www.researchgate.net/project/On-the-Asymptotic-Relative-Change-for-Sequences-of-Permutations |access-date=2021-01-31 }} [https://www.academia.edu/37676013/On_the_Asymptotic_Relative_Change_for_Sequences_of_Permutations]</ref>

}}

==References==
{{refbegin}}
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* {{cite book |last1=Allouche |first1=Jean-Paul |last2=Shallit |first2=Jeffrey |author2-link=Jeffrey Shallit |isbn=978-0-521-82332-6 |publisher=[[Cambridge University Press]] |title=Automatic Sequences: Theory, Applications, Generalizations |date=2003 |zbl=1086.11015}}
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* {{cite journal |last1=Bolker |first1=Ethan |last2=Offner |first2=Carl |last3=Richman |first3=Robert |last4=Zara |first4=Catalin |title=The Prouhet–Tarry–Escott problem and generalized Thue–Morse sequences |journal=Journal of Combinatorics |date=2016 |volume=7 |issue=1|pages=117–133 |doi=10.4310/JOC.2016.v7.n1.a5 |arxiv=1304.6756|s2cid=118040795 }}}
* {{cite book |title=The Win-Win Solution: Guaranteeing Fair Shares to Everybody |last1=Brams |first1=Steven J. |last2=Taylor |first2=Alan D. |pages=[https://archive.org/details/winwinsolutiongu00stev/page/36 36–44] |isbn=978-0-393-04729-5 |publisher=W. W. Norton & Co., Inc. |date=1999 |url-access=registration |url=https://archive.org/details/winwinsolutiongu00stev/page/36}}
*{{cite journal |last=Brlek |first=Srećko |title=Enumeration of Factors in the Thue-Morse Word |journal=[[Discrete Applied Mathematics]] |date=1989 |volume=24 |issue=1–3 |pages=83–96 |doi=10.1016/0166-218x(92)90274-e|doi-access= }}
*{{cite journal |last1=Cohen-Zada |first1=Danny |last2=Krumer |first2=Alex |last3=Shapir |first3=Offer Moshe |title=Testing the effect of serve order in tennis tiebreak |journal=[[Journal of Economic Behavior and Organization]] |date=2018 |volume=146 |pages=106–115 |doi=10.1016/j.jebo.2017.12.012 |s2cid=89610106 |url=https://www.sciencedirect.com/science/article/pii/S0167268117303530|url-access=subscription }}
*{{cite journal |last1=Cooper |first1=Joshua |last2=Dutle |first2=Aaron |title=Greedy Galois Games |journal=[[American Mathematical Monthly]] |date=2013 |volume=120 |issue=5 |pages=441–451 |url=http://www.math.sc.edu/~cooper/ThueMorseDueling.pdf |doi=10.4169/amer.math.monthly.120.05.441 |arxiv=1110.1137|s2cid=1291901 }}
* {{cite book |title=Developments in Language Theory: Proceedings 10th International Conference, DLT 2006, Santa Barbara, California, USA, June 26-29, 2006 |volume=4036 |series=Lecture Notes in Computer Science |editor1-first=Oscar H. |editor1-last=Ibarra |editor2-first=Zhe |editor2-last=Dang |publisher=[[Springer-Verlag]] |date=2006 |isbn=978-3-540-35428-4 |first=Dalia |last=Krieger |chapter=On critical exponents in fixed points of non-erasing morphisms |pages=280–291 |zbl=1227.68074}}
* {{cite journal |last1=Levine |first1=Lionel |last2=Stange |first2=Katherine E. |title=How to Make the Most of a Shared Meal: Plan the Last Bite First |journal=[[American Mathematical Monthly]] |date=2012 |volume=119 |issue=7 |pages=550–565 |url=http://math.colorado.edu/~kstange/papers/EthiopianDinner.pdf |doi=10.4169/amer.math.monthly.119.07.550 |arxiv=1104.0961|s2cid=14537479 }}
* {{cite book |last=Lothaire |first=M. |author-link=M. Lothaire |title=Algebraic combinatorics on words |others=With preface by Jean Berstel and Dominique Perrin |edition=Reprint of the 2002 hardback |series=Encyclopedia of Mathematics and Its Applications |volume=90 |publisher=Cambridge University Press |date=2011 |isbn=978-0-521-18071-9 |zbl=1221.68183}}
* {{cite journal |last1=Ma |first1=Jun |last2=Holdener |first2=Judy |author2-link=Judy A. Holdener |doi=10.1142/S0218348X05002908 |issue=3 |journal=[[Fractals (journal)|Fractals]] |mr=2166279 |pages=191–206 |title=When Thue-Morse meets Koch |url=http://www2.kenyon.edu/people/holdenerj/StudentResearch/WhenThueMorsemeetsKochJan222005.pdf |volume=13 |date=2005}}
* {{cite journal |last=Palacios-Huerta |first=Ignacio |title=Tournaments, fairness and the Prouhet–Thue–Morse sequence |journal=Economic Inquiry |date=2012 |volume=50 |issue=3 |pages=848–849 |url=http://www.palacios-huerta.com/docs/EI-Tournaments_and_PTM_sequence.pdf |doi=10.1111/j.1465-7295.2011.00435.x|s2cid=54036493 }}
* {{cite book |title=Beautiful Game Theory |last=Palacios-Huerta |first=Ignacio |isbn=978-0691144023 |publisher=Princeton University Press |date=2014 }}
* {{cite book |author-last=Pytheas Fogg |author-first=N. |editor-last1=Berthé |editor-first1=Valérie |editor-last2=Ferenczi |editor-first2=Sébastien |editor-last3=Mauduit |editor-first3=Christian |editor-last4=Siegel |editor-first4=A. |title=Substitutions in dynamics, arithmetics and combinatorics |series=Lecture Notes in Mathematics |volume=1794 |location=Berlin, Germany |publisher=[[Springer-Verlag]] |date=2002 |isbn=978-3-540-44141-0 |zbl=1014.11015}}
*{{cite journal |last=Richman |first=Robert |title=Recursive Binary Sequences of Differences |journal=[[Complex Systems (journal)|Complex Systems]] |date=2001 |volume=13 |issue=4 |pages=381–392 |url=http://www.complex-systems.com/pdf/13-4-3.pdf}}
{{refend}}

==Further reading==
* {{cite book |last=Bugeaud |first=Yann |title=Distribution modulo one and Diophantine approximation |series=Cambridge Tracts in Mathematics |volume=193 |location=Cambridge |publisher=[[Cambridge University Press]] |date=2012 |isbn=978-0-521-11169-0 |zbl=1260.11001}}
* {{cite book |last=Lothaire |first=M. |author-link=M. Lothaire |title=Applied combinatorics on words |others=A collective work by Jean Berstel, Dominique Perrin, Maxime Crochemore, Eric Laporte, Mehryar Mohri, Nadia Pisanti, Marie-France Sagot, [[Gesine Reinert]], [[Sophie Schbath]], Michael Waterman, Philippe Jacquet, [[Wojciech Szpankowski]], Dominique Poulalhon, Gilles Schaeffer, Roman Kolpakov, Gregory Koucherov, Jean-Paul Allouche and Valérie Berthé |series=Encyclopedia of Mathematics and Its Applications |volume=105 |location=Cambridge |publisher=[[Cambridge University Press]] |date=2005 |isbn=978-0-521-84802-2 |zbl=1133.68067 |url-access=registration |url=https://archive.org/details/appliedcombinato0000loth}}

== External links ==
{{Commons category}}
* {{springer|title=Thue-Morse sequence|id=p/t120090}}
* {{MathWorld|urlname=Thue-MorseSequence|title=Thue-Morse Sequence|ref=none}}
* Allouche, J.-P.; Shallit, J. O. [http://www.cs.uwaterloo.ca/~shallit/Papers/ubiq15.pdf The Ubiquitous Prouhet-Thue-Morse Sequence]. (contains many applications and some history)
* Thue–Morse Sequence over (1,2) {{OEIS|id=A001285}}
* {{OEIS el|1=A000069|2=Odious numbers: numbers with an odd number of 1's in their binary expansion}}
* {{OEIS el|1=A001969|2=Evil numbers: numbers with an even number of 1's in their binary expansion}}
* [https://web.archive.org/web/20090912104941/http://www.geocities.com/jan.schat/ThueMorse.PDF Reducing the influence of DC offset drift in analog IPs using the Thue-Morse Sequence]. A technical application of the Thue–Morse Sequence
* [http://reglos.de/musinum MusiNum - The Music in the Numbers]. Freeware to generate self-similar music based on the Thue–Morse Sequence and related number sequences.
* {{cite web|last1=Parker|first1=Matt|author-link1=Matt Parker|title=The Fairest Sharing Sequence Ever|url=https://www.youtube.com/watch?v=prh72BLNjIk|publisher=standupmaths|access-date=20 January 2016|format=video |ref=none}}

{{DEFAULTSORT:Thue-Morse Sequence}}
[[Category:Binary sequences]]
[[Category:Fixed points (mathematics)]]
[[Category:Parity (mathematics)]]