Van der Corput sequence

A van der Corput sequence is an example of the simplest one-dimensional low-discrepancy sequence over the unit interval; it was first described in 1935 by the Dutch mathematician J. G. van der Corput. It is constructed by reversing the base-n representation of the sequence of natural numbers (1, 2, 3, …).

The $b$ -ary representation of the positive integer $n\geq 1$ is

n~=~\sum _{k=0}^{L-1}d_{k}(n)b^{k}~=~d_{0}(n)b^{0}+\cdots +d_{L-1}(n)b^{L-1},

where $b$ is the base in which the number $n$ is represented, and $0\leq d_{k}(n)<b;$ that is, the $k$ -th digit in the $b$ -ary expansion of $n.$ The $n$ -th number in the van der Corput sequence is

g_{b}(n)~=~\sum _{k=0}^{L-1}d_{k}(n)b^{-k-1}~=~d_{0}(n)b^{-1}+\cdots +d_{L-1}(n)b^{-L}.

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