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, ... |
The complementary sequences with parameters 
and 
are the Wythoff sequences.
Ashburn (2026b) determined when a positive integer
has exactly one or exactly two representations as the sum of one term from each of
these two sequences.
For the array with rows 
, 
, let 
count the rows containing 
. Ashburn (2026a) proved that, for every positive
integer 
,
the set of 
satisfying 
has positive natural density.
For 
,
the rows 
and 
are disjoint iff 
is odd and divides 
. The array and row counts are OEIS A128440
and OEIS A358359, respectively. Ashburn (2026ab)
reports Claude assistance with computations and states that he checked the proofs
and computations.
."
5 Oct 2026a.