以10為基數的范德科皮特序列前n項(n從0至999)的圖示
范德科皮特序列(英語:van der Corput sequence)是定義在單位區間上的一維低差異序列,由荷蘭數學家約翰內斯·范德科皮特於1935年提出。將以基數b表示的自然數列反轉後便可得到范德科皮特序列。
使用基數b可將自然數n表示為
![{\displaystyle n=\sum _{k=0}^{L-1}d_{k}(n)b^{k},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/24d41846a7f859c8e454b9d1f3d2b80e2a8f531f)
其中第k位為dk(n),滿足0 ≤ dk(n) < b。
由此,可以得到范德科皮特序列的第n位:
![{\displaystyle g_{b}(n)=\sum _{k=0}^{L-1}d_{k}(n)b^{-k-1}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/24663095108966037060fc804baf2f8658bb24f0)
例如,以10為基數的范德科皮特序列的前幾項為
![{\displaystyle \left\{{\tfrac {1}{10}},{\tfrac {2}{10}},{\tfrac {3}{10}},{\tfrac {4}{10}},{\tfrac {5}{10}},{\tfrac {6}{10}},{\tfrac {7}{10}},{\tfrac {8}{10}},{\tfrac {9}{10}},{\tfrac {1}{100}},{\tfrac {11}{100}},{\tfrac {21}{100}},{\tfrac {31}{100}},{\tfrac {41}{100}},{\tfrac {51}{100}},{\tfrac {61}{100}},{\tfrac {71}{100}},{\tfrac {81}{100}},{\tfrac {91}{100}},{\tfrac {2}{100}},{\tfrac {12}{100}},{\tfrac {22}{100}},{\tfrac {32}{100}},\ldots \right\},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2cbc3b8897cf240466ba218e77c32d9e3649067d)
而以2為基數的范德科皮特序列的前幾項則為
![{\displaystyle \left\{{\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 \right\}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ca8fdc78e5aec5a35f778b3437b038d7084ecf1b)
參考文獻[編輯]
- van der Corput, J.G., Verteilungsfunktionen. I. Mitt., Proc. Akad. Wet. Amsterdam, 1935, 38: 813–821, Zbl 0012.34705 (德語)
- Kuipers, L.; Niederreiter, H., Uniform distribution of sequences, Dover Publications: 129,158, 2005 [1974], ISBN 0-486-45019-8, Zbl 0281.10001