Beatty sequence

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Suppose r and s are positive irrational numbers satisfying

$\frac1r + \frac1s = 1.$

Then the sequences $( \lfloor nr \rfloor)$ and $( \lfloor ns \rfloor)$ are a pair of complementary Beatty sequences, and each is a Beatty sequence.

1 Beatty's Theorem
2 Example
3 Relation with Sturmian sequences
4 History
5 References
6 External links

[edit] Beatty's Theorem

A pair of complementary Beatty sequences partition the set of positive integers.

Proof: We must show that every positive integer lies in one and only one of the two sequences and shall do so by considering the positions occupied by all the fractions $j / r$ and $r / s$ when they are jointly listed in nondecreasing order.

To see that no two of the numbers can occupy the same position (as a single number), suppose to the contrary that $j / r = k / s$ for some $j$ and $k$ . Then $r / s = j / k$ , a rational number, but also, $r / s = r (1 - 1 / r) = r - 1,$ not a rational number. Therefore no two of the numbers occupy the same position.

For any $j / r$ , there are $j$ numbers $i/r \le j/r$ and $\lfloor js/r \rfloor$ numbers $k/s \le j/r$ , so that the position of $j / r$ in the list is $j + \lfloor js/r \rfloor$ . The equation $1 / r + 1 / s = 1$ implies

$j + \lfloor js/r \rfloor = j + \lfloor j(s - 1) \rfloor = \lfloor js \rfloor$ .

Likewise, the position of $k / s$ in the list is $\lfloor kr \rfloor$ .

Conclusion: every positive integer (that is, every position in the list) is of the form $\lfloor nr \rfloor$ or of the form $\lfloor ns \rfloor$ , but not both.

[edit] Example

For $r =$ the golden mean, we have $s = r + 1.$ In this case, the sequence $( \lfloor nr \rfloor)$ , known as the lower Wythoff sequence, is

1, 3, 4, 6, 8, 9, 11, 12, 14, 16, 17, 19, 21, 22, 24, 25, 27, 29, ... (sequence A000201 in OEIS).

and the sequence $( \lfloor ns \rfloor)$ , the upper Wythoff sequence, is

2, 5, 7, 10, 13, 15, 18, 20, 23, 26, 28, 31, 34, 36, 39, 41, 44, 47, ... (sequence A001950 in OEIS).

[edit] Relation with Sturmian sequences

The first difference

$\lfloor (n+1)r\rfloor-\lfloor nr\rfloor$

of the Beatty sequence associated to the irrational number $r$ is a characteristic Sturmian word over the alphabet $\{\lfloor r\rfloor,\lfloor r\rfloor+1\}$ .

[edit] History

Beatty sequences solve a problem posed in the American Mathematical Monthly by Samuel Beatty in 1926. It is probably one of the most often cited problems ever posed in the Monthly.

[edit] References

Beatty, Samuel (1926). "Problem 3173". American Mathematical Monthly 33 (3): 159. Solution, vol. 34 (1927), p. 159.

Stolarsky, Kenneth (1976). "Beatty sequences, continued fractions, and certain shift operators". Canadian Mathematical Bulletin 19 (4): 473–482. MR 0444558. Includes many references.