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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  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 | Sloane | 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, ... |
SEE ALSO: Fractional Part, Wythoff
Array, Wythoff's Game
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. http://mathworld.wolfram.com/BeattySequence.html
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