Sturmian word: Difference between revisions
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imported>CosmiaNebula (Sturmian word) |
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The Sturmian word can be computed thus as an algorithm:
* If x > 1, then it is the inverse of the Sturmian word for (1/x). So we have reduced to the case of <math>0 < x
* Iterate over <math>floor(1x), floor(2x), floor(3x), \dots </math>
* 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.
The problem:
* Given a positive rational number <math>\frac mn</math>, specified by two positive integers <math>m, n</math>, output its entire Sturmian word.
* Given a quadratic real number <math>\frac{\sqrt{a} + m}{n}</math>, specified by three positive integers <math>a, m, n </math>, where <math>a</math> is not a perfect square, output the first <math>k</math> letters of its Sturmian word when given a positive number <math>k</math>.
Stretch goal: calculate the Sturmian word for other kinds of definable real numbers, such as cubic roots.
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Revision as of 22:44, 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 x > 1, then it is the inverse of the Sturmian word for (1/x). 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.