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| A '''Ducci sequence''' is a sequence of [[n-tuple|''n''-tuples]] of [[integer]]s. Given an ''n''-tuple of integers <math>(a_1,a_2,...,a_n)</math>, the next ''n''-tuple in the sequence is formed by taking the absolute differences of neighbouring integers:
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| :<math>(a_1,a_2,...,a_n) \rightarrow (|a_1-a_2|, |a_2-a_3|, ..., |a_n-a_1|)\, .</math>
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| Another way of describing this is as follows. Arrange ''n'' integers in a circle and make a new circle by taking the difference between neighbours, ignoring any minus signs; then repeat the operation. Ducci sequences are named after Enrico Ducci, the Italian mathematician credited with their discovery.
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| Ducci sequences are also known as the '''Ducci map''' or the '''n-number game'''. Open problems in the study of these maps still remain.<ref name="Chamberland&Thomas2004">{{cite journal
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| | last1 = Chamberland
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| | first1 = Marc
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| | last2 = Thomas
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| | first2 = Diana M.
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| | title = The N-Number Ducci Game
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| | journal = Journal of Difference Equations and Applications
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| | volume = 10
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| | issue = 3
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| | pages = 33–36
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| | publisher = [[Taylor & Francis]]
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| | location = London
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| | year = 2004
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| | url = http://www.math.grinnell.edu/~chamberl/papers/ducci_survey.pdf
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| | accessdate = 2009-01-26}}</ref>
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| ==Properties==
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| From the second ''n''-tuple onwards, it is clear that every integer in each ''n''-tuple in a Ducci sequence is greater than or equal to 0 and is less than or equal to the difference between the maximum and mimimum members of the first ''n''-tuple. As there are only a finite number of possible ''n''-tuples with these constraints, the sequence of n-tuples must sooner or later repeat itself. Every Ducci sequence therefore eventually becomes [[Periodic function|periodic]].
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| If ''n'' is a power of 2 every Ducci sequence eventually reaches the ''n''-tuple (0,0,...,0) in a finite number of steps.<ref name="Chamberland&Thomas2004" />
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| <ref name="Chamberland2003">{{cite journal
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| | doi = 10.1080/1023619021000041424
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| | last = Chamberland
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| | first = Marc
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| | title = Unbounded Ducci sequences
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| | journal = Journal of Difference Equations and Applications
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| | volume = 9
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| | issue = 10
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| | pages = 887–895
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| | publisher = [[Taylor & Francis]]
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| | location = London
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| | year = 2003
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| | url = http://www.math.grinnell.edu/~chamberl/papers/ducci_unbounded.pdf
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| | accessdate = 2009-01-26}}</ref>
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| <ref name="Andriychenko&Chamberland2000">{{cite journal
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| | last1 = Andriychenko
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| | first1 = Oleksiy
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| | last2 = Chamberland
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| | first2 = Marc
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| | title = Iterated Strings and Cellular Automata
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| | journal = [[The Mathematical Intelligencer]]
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| | volume = 22
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| | issue = 4
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| | pages = 33–36
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| | publisher = [[Springer Verlag]]
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| | location = New York, NY
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| | year = 2000
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| | doi = 10.1007/BF03026764}}</ref>
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| If ''n'' is ''not'' a power of two, a Ducci sequence will either eventually reach an ''n''-tuple of zeros or will settle into a periodic loop of 'binary' ''n''-tuples; that is, ''n''-tuples which contain only two different digits.
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| An obvious generalisation of Ducci sequences is to allow the members of the ''n''-tuples to be any real numbers rather than just integers. The properties presented here do not always hold for these generalisations. For example, a Ducci sequence starting with the ''n''-tuple (1, ''q'', ''q''<sup>2</sup>, ''q''<sup>3</sup>) where ''q'' is the (irrational) positive root of the cubic <math>x^3 - x^2 - x - 1 = 0</math> does not reach (0,0,0,0) in a finite number of steps, although in the limit it converges to (0,0,0,0).<ref name="Brockman">{{cite journal
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| | last1 = Brockman
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| | first1 = Greg
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| | title = Asymptotic behaviour of certain Ducci sequences
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| | journal = [[Fibonacci Quarterly]]
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| | url = http://www.cut-the-knot.org/Curriculum/Algebra/GregBrockman/GregBrockmanDucciSequencesPaper.pdf
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| | year = 2007}}
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| </ref>
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| ==Examples==
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| Ducci sequences may be arbitrarily long before they reach a tuple of zeros or a periodic loop. The 4-tuple sequence starting with (0, 653, 1854, 4063) takes 24 iterations to reach the zeros tuple.
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| <math>
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| (0, 653, 1854, 4063) \rightarrow
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| (653, 1201, 2209, 4063) \rightarrow
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| (548, 1008, 1854, 3410) \rightarrow
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| </math>
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| <math>
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| \cdots \rightarrow
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| (0, 0, 128, 128) \rightarrow
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| (0, 128, 0, 128) \rightarrow
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| (128, 128, 128, 128) \rightarrow
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| (0, 0, 0, 0)
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| </math>
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| This 5-tuple sequence enters a period 15 binary 'loop' after 7 iterations.
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| <math> | |
| \begin{matrix}
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| 1 5 7 9 9 \rightarrow &
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| 4 2 2 0 8 \rightarrow &
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| 2 0 2 8 4 \rightarrow &
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| 2 2 6 4 2 \rightarrow &
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| 0 4 2 2 0 \rightarrow &
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| 4 2 0 2 0 \rightarrow \\
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| 2 2 2 2 4 \rightarrow &
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| 0 0 0 2 2 \rightarrow &
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| 0 0 2 0 2 \rightarrow &
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| 0 2 2 2 2 \rightarrow &
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| 2 0 0 0 2 \rightarrow &
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| 2 0 0 2 0 \rightarrow \\
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| 2 0 2 2 2 \rightarrow &
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| 2 2 0 0 0 \rightarrow &
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| 0 2 0 0 2 \rightarrow &
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| 2 2 0 2 2 \rightarrow &
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| 0 2 2 0 0 \rightarrow &
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| 2 0 2 0 0 \rightarrow \\
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| 2 2 2 0 2 \rightarrow &
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| 0 0 2 2 0 \rightarrow &
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| 0 2 0 2 0 \rightarrow &
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| 2 2 2 2 0 \rightarrow &
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| 0 0 0 2 2 \rightarrow &
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| \cdots \quad \quad \\
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| \end{matrix}
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| </math> | |
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| The following 6-tuple sequence shows that sequences of tuples whose length is not a power of two may still reach a tuple of zeros:
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| <math>
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| \begin{matrix}
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| 1 2 1 2 1 0 \rightarrow &
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| 1 1 1 1 1 1 \rightarrow &
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| 0 0 0 0 0 0 \\
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| \end{matrix}
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| </math>
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| ==Modulo two form==
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| When the Ducci sequences enter binary loops, it is possible to treat the sequence in modulo two. That is:<ref name="Breauer">Florian Breuer, "Ducci sequences in higher dimensions" in INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 7 (2007) [http://www.integers-ejcnt.org/h24/h24.pdf]</ref>
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| :<math>(|a_1-a_2|, |a_2-a_3|, ..., |a_n-a_1|)\ = (a_1+a_2, a_2+a_3, ..., a_n + a_1)\ mod 2</math>
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| This forms the basis for proving when the sequence vanish to all zeros.
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| ==Cellular automata==
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| [[File:CA rule 102.png|140px|right]]
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| The linear map in modulo 2 can further be identified as the cellular automata denoted as '''rule 102''' in [[Wolfram code]] and related to [[rule 90]] through the Martin-Odlyzko-Wolfram map.<ref>S Lettieri, JG Stevens, DM Thomas, "Characteristic and minimal polynomials of linear cellular automata" in Rocky Mountain J. Math, 2006.</ref><ref>M Misiurewicz, JG Stevens, DM Thomas, "Iterations of linear maps over finite fields", Linear Algebra and Its Applications, 2006</ref> Rule 102 reproduces the [[Sierpinski triangle]].<ref>Weisstein, Eric W. "Rule 102." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/Rule102.html</ref>
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| ==Other related topics==
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| The Ducci map is an example of a [[difference equation]], a category that also include [[non-linear dynamics]], [[chaos theory]] and [[numerical analysis]]. Similarities to [[cyclotomic polynomials]] have also been pointed out.<ref>F. Breuer et al. 'Ducci-sequences and cyclotomic polynomials' in [[Finite Fields and Their Applications]] 13 (2007) 293–304</ref> While there are no practical applications of the Ducci map at present, its connection to the highly applied field of difference equations led <ref name="Brockman"/> to conjecture that a form of the Ducci map may also find application in the future.
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| ==References==
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| {{reflist}}
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| ==External links==
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| * [http://www.math.jussieu.fr/theses/2002/breuer/homepage/research.html Ducci Sequence]
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| * [http://www.cut-the-knot.org/SimpleGames/IntIter.shtml Integer Iterations on a Circle] at [[Cut-the-Knot]]
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| {{DEFAULTSORT:Ducci Sequence}}
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| [[Category:Sequences and series]]
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| [[Category:Number theory]]
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Greetings! I am Marvella and I feel comfy when people use the full name. South Dakota is her beginning location but she requirements to move because of her family members. To collect coins is what her family and her enjoy. For many years I've been working as a payroll clerk.
Feel free to visit my blog :: www.gaysphere.net