TOPICS

# 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, ...

Fractional Part, Wythoff Array, Wythoff's Game

## Explore with Wolfram|Alpha

More things to try:

## 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.

Beatty Sequence

## Cite this as:

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