Sturmian word
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You are encouraged to solve this task according to the task description, using any language you may know.
A Sturmian word is a binary sequence, finite or infinite, that makes up the cutting sequence for a positive real number x, as shown in the picture.
![](https://upload.wikimedia.org/wikipedia/commons/thumb/2/2d/Fibonacci_word_cutting_sequence.png/300px-Fibonacci_word_cutting_sequence.png)
The Sturmian word can be computed thus as an algorithm:
- If , then it is the inverse of the Sturmian word for . So we have reduced to the case of .
- Iterate over
- If is an integer, then the program terminates. Else, if , then the program outputs 0, else, it outputs 10.
The problem:
- Given a positive rational number , specified by two positive integers , output its entire Sturmian word.
- Given a quadratic real number , specified by integers , where is not a perfect square, output the first letters of Sturmian words when given a positive number .
(If the programming language can represent infinite data structures, then that works too.)
Stretch goal: calculate the Sturmian word for other kinds of definable real numbers, such as cubic roots.
The key difficulty is accurately calculating for large . Floating point arithmetic would lose precision. One can either do this simply by directly searching for some integer such that , or by more trickly methods, such as the continued fraction approach.
First calculate the continued fraction convergents to . Let be a convergent to , such that , then since the convergent sequence is the best rational approximant for denominators up to that point, we know for sure that, if we write out , the sequence would stride right across the gap . Thus, we can take the largest such that , and we would know for sure that .
In summary,
where is the first continued fraction approximant to with a denominator