Beatty Sequence

The Beatty sequence is a spectrum sequence with an irrational base. In other words, the Beatty sequence corresponding to an irrational number  is given by , , , ..., where  is the floor function. If  and  are positive irrational numbers such that

then the Beatty sequences , , ... and , , ... together contain all the positive integers without repetition.

The sequences for particular values of  and  are given in the following table (Sprague 1963; Wells 1986, pp. 35 and 40), where  is the golden ratio.

parameter	OEIS	sequence
	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, ...

REFERENCES:

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.