Sturmian word: Difference between revisions
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The Sturmian word can be computed thus as an algorithm: |
The Sturmian word can be computed thus as an algorithm: |
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* If x |
* If <math>x>1</math>, then it is the inverse of the Sturmian word for <math>1/x</math>. So we have reduced to the case of <math>0 < x \leq 1</math>. |
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* Iterate over <math>floor(1x), floor(2x), floor(3x), \dots </math> |
* Iterate over <math>floor(1x), floor(2x), floor(3x), \dots </math> |
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* If <math>kx </math> is an integer, then the program terminates. Else, if <math>floor((k-1)x) = floor(kx)</math>, then the program outputs 0, else, it outputs 10. |
* If <math>kx </math> is an integer, then the program terminates. Else, if <math>floor((k-1)x) = floor(kx)</math>, then the program outputs 0, else, it outputs 10. |
Revision as of 22:45, 31 January 2024
![Task](http://static.miraheze.org/rosettacodewiki/thumb/b/ba/Rcode-button-task-crushed.png/64px-Rcode-button-task-crushed.png)
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 three positive integers , where is not a perfect square, output the first letters of its Sturmian word when given a positive number .
Stretch goal: calculate the Sturmian word for other kinds of definable real numbers, such as cubic roots.