Sturmian word

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.

Sturmian word
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Example Sturmian word when x = 0.618..., the golden ratio.

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 0 < x < 1.
• Iterate over $floor(1x),floor(2x),floor(3x),\dots$
• If $kx$ is an integer, then the program terminates. Else, if $floor((k-1)x)=floor(kx)$, then the program outputs 0, else, it outputs 10.