Beatty Sequence - Definition

Definition

A positive irrational number generates the Beatty sequence

If then is also a positive irrational number. They naturally satisfy

and the sequences

and

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

and

form a pair of complementary Beatty sequences. For other values of the procedure for finding complementary sequences is not as straightforward.

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