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Beatty Sequence


The Beatty sequence is a spectrum sequence with an irrational base. In other words, the Beatty sequence corresponding to an irrational number theta is given by |_theta_|, |_2theta_|, |_3theta_|, ..., where |_x_| is the floor function. If alpha and beta are positive irrational numbers such that

 1/alpha+1/beta=1,

then the Beatty sequences |_alpha_|, |_2alpha_|, ... and |_beta_|, |_2beta_|, ... together contain all the positive integers without repetition.

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

parameterOEISsequence
alpha=sqrt(2)A0019511, 2, 4, 5, 7, 8, 9, 11, 12, ...
beta=2+sqrt(2)A0019523, 6, 10, 13, 17, 20, 23, 27, 30, ...
alpha=sqrt(3)A0228381, 3, 5, 6, 8, 10, 12, 13, 15, 17, ...
beta=1/2(3+sqrt(3))A0544062, 4, 7, 9, 11, 14, 16, 18, 21, 23, 26, ...
alpha=eA0228432, 5, 8, 10, 13, 16, 19, 21, 24, 27, 29, ...
beta=e/(e-1)A0543851, 3, 4, 6, 7, 9, 11, 12, 14, 15, 17, 18, ...
alpha=piA0228443, 6, 9, 12, 15, 18, 21, 25, 28, 31, 34, ...
beta=pi/(pi-1)A0543861, 2, 4, 5, 7, 8, 10, 11, 13, 14, 16, 17, 19, ...
alpha=phiA0002011, 3, 4, 6, 8, 9, 11, 12, 14, 16, 17, 19, 21, ...
beta=phi^2A0019502, 5, 7, 10, 13, 15, 18, 20, 23, 26, 28, 31, 34, ...

See also

Fractional Part, Wythoff Array, Wythoff's Game

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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.

Referenced on Wolfram|Alpha

Beatty Sequence

Cite this as:

Weisstein, Eric W. "Beatty Sequence." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/BeattySequence.html

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