Beatty 序列 -- 来自 Wolfram MathWorld

Beatty 序列

Beatty 序列是一个谱序列，其基数为无理数。换句话说，对应于无理数的 Beatty 序列由 , , , ..., 给出，其中是向下取整函数。如果和是正无理数，使得

那么 Beatty 序列 , , ... 和 , , ... 一起包含所有正整数，且不重复。

下表给出了和的特定值的序列 (Sprague 1963; Wells 1986, pp. 35 and 40)，其中是黄金比例。

参数	OEIS	序列
	A001951	1, 2, 4, 5, 7, 8, 9, 11, 12, ...
	A001952	3, 6, 10, 13, 17, 20, 23, 27, 30, ...
	A022838	1, 3, 5, 6, 8, 10, 12, 13, 15, 17, ...
	A054406	2, 4, 7, 9, 11, 14, 16, 18, 21, 23, 26, ...
	A022843	2, 5, 8, 10, 13, 16, 19, 21, 24, 27, 29, ...
	A054385	1, 3, 4, 6, 7, 9, 11, 12, 14, 15, 17, 18, ...
	A022844	3, 6, 9, 12, 15, 18, 21, 25, 28, 31, 34, ...
	A054386	1, 2, 4, 5, 7, 8, 10, 11, 13, 14, 16, 17, 19, ...
	A000201	1, 3, 4, 6, 8, 9, 11, 12, 14, 16, 17, 19, 21, ...
	A001950	2, 5, 7, 10, 13, 15, 18, 20, 23, 26, 28, 31, 34, ...

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分数部分, Wythoff 数组, Wythoff 游戏

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参考文献

Gardner, M. Penrose Tiles and Trapdoor Ciphers... and the Return of Dr. Matrix, reissue ed. New York: W. H. Freeman, p. 21, 1989.Graham, R. L.; Lin, S.; and Lin, C.-S. "Spectra of Numbers." Math. Mag. 51, 174-176, 1978.Guy, R. K. Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, p. 227, 1994.Sloane, N. J. A. A Handbook of Integer Sequences. Boston, MA: Academic Press, pp. 29-30, 1973.Sloane, N. J. A. and Plouffe, S. The Encyclopedia of Integer Sequences. San Diego, CA: Academic Press, p. 18, 1995.Sprague, R. Recreations in Mathematics: Some Novel Puzzles. London: Blackie and Sons, 1963.Sloane, N. J. A. Sequences A000201/M2322, A001950/M1332, A001951/M0955, A001952/M2534, A022838, A022843, A022844, A054406, A054385, and A054386 in "The On-Line Encyclopedia of Integer Sequences."Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, p. 35, 1986.

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Beatty 序列

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Weisstein, Eric W. “Beatty 序列。” 来自 MathWorld--Wolfram Web 资源。 https://mathworld.net.cn/BeattySequence.html

Beatty 序列

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