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 | OEIS | 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, ... |