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| A '''van der Corput sequence''' is a [[low-discrepancy sequence]] over the [[unit interval]] first published in 1935 by the [[Netherlands|Dutch]] mathematician [[Johannes van der Corput|J. G. van der Corput]]. It is constructed by reversing the [[base (exponentiation)|base ''n'' representation]] of the sequence of [[natural number]]s (1, 2, 3, …). For example, the [[decimal]] van der Corput sequence begins:
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| :0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 0.01, 0.11, 0.21, 0.31, 0.41, 0.51, 0.61, 0.71, 0.81, 0.91, 0.02, 0.12, 0.22, 0.32, …
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| whereas the [[binary numeral system|binary]] van der Corput sequence can be written as:
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| :0.1<sub>2</sub>, 0.01<sub>2</sub>, 0.11<sub>2</sub>, 0.001<sub>2</sub>, 0.101<sub>2</sub>, 0.011<sub>2</sub>, 0.111<sub>2</sub>, 0.0001<sub>2</sub>, 0.1001<sub>2</sub>, 0.0101<sub>2</sub>, 0.1101<sub>2</sub>, 0.0011<sub>2</sub>, 0.1011<sub>2</sub>, 0.0111<sub>2</sub>, 0.1111<sub>2</sub>, …
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| or, equivalently, as:
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| :<math>\tfrac{1}{2}, \tfrac{1}{4}, \tfrac{3}{4}, \tfrac{1}{8}, \tfrac{5}{8}, \tfrac{3}{8}, \tfrac{7}{8}, \tfrac{1}{16}, \tfrac{9}{16}, \tfrac{5}{16}, \tfrac{13}{16}, \tfrac{3}{16}, \tfrac{11}{16}, \tfrac{7}{16}, \tfrac{15}{16}, \ldots</math>
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| The elements of the van der Corput sequence (in any base) form a [[dense set]] in the unit interval: for any real number in [0, 1] there exists a [[subsequence]] of the van der Corput sequence that [[limit of a sequence|converges]] towards that number. They are also [[equidistributed]] over the unit interval.
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| ==See also==
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| * [[Bit-reversal permutation]]
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| * [[Constructions of low-discrepancy sequences]]
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| * [[Halton sequence]], a natural generalization of the van der Corput sequence to higher dimensions
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| ==References==
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| * J. G. van der Corput, ''Verteilungsfunktionen''. Proc. Ned. Akad. v. Wet., 38:813–821, 1935
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| * {{citation | last=Kuipers | first=L. | last2= Niederreiter | first2=H. | title = Uniform distribution of sequences | publisher=[[Dover Publications]] | date=2005 | isbn=0-486-45019-8 | page=129,158 }}
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| ==External links==
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| * [http://mathworld.wolfram.com/vanderCorputSequence.html Van der Corput sequence] at [[MathWorld]]
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| [[Category:Quasirandomness]]
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| [[Category:Diophantine approximation]]
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| [[Category:Sequences and series]]
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