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.
Definition
A positive irrational number 
generates the Beatty sequence
If 
then 
is also a positive irrational number. They naturally satisfy
and the sequences
form a pair of complementary Beatty sequences.
A more general non-homogeneous Beatty sequence takes the form
where 
is a real number. For 
, the complementary non-homogeneous Beatty sequences can be found by making 
so that
form a pair of complementary Beatty sequences.
Examples
For r = the golden mean, we have s = r + 1. In this case, the sequence 
, known as the lower Wythoff sequence, is
Podcasts:
