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 $\left\lfloor{\theta}\right\rfloor$, $\left\lfloor{2\theta}\right\rfloor$, $\left\lfloor{3\theta}\right\rfloor$, ..., where $\left\lfloor{x}\right\rfloor$ is the Floor Function. If $\alpha$ and $\beta$ are Positive Irrational Numbers such that

$${1\over\alpha}+{1\over\beta}=1,$$

then the Beatty sequences $\left\lfloor{\alpha}\right\rfloor$, $\left\lfloor{2\alpha}\right\rfloor$, ... and $\left\lfloor{\beta}\right\rfloor$, $\left\lfloor{2\beta}\right\rfloor$, ... together contain all the Positive Integers without repetition.

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.