Beatty sequence

In mathematics, a Beatty sequence (or homogeneous Beatty sequence) is the sequence of integers found by taking the floor of the positive multiples of a positive irrational number. Beatty sequences are named after Samuel Beatty, who wrote about them in 1926.

Rayleigh's theorem, named after Lord Rayleigh, states that the complement of a Beatty sequence, consisting of the positive integers that are not in the sequence, is itself a Beatty sequence generated by a different irrational number.

Beatty sequences can also be used to generate Sturmian words.

A positive irrational number ${\displaystyle r}$ generates the Beatty sequence

If ${\displaystyle r>1\,,}$ then ${\displaystyle s=r/(r-1)}$ is also a positive irrational number. These two numbers naturally satisfy the equation ${\displaystyle 1/r+1/s=1}$. The two Beatty sequences they generate,

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