Sturmian word

An infinite binary word is Sturmian if it has exactly $n+1$ distinct contiguous subwords of length $n$ for every positive integer $n$.

For an irrational number $0<\alpha <1$, the characteristic Sturmian word of slope $\alpha$ is the binary sequenceEach term is zero or one because the arguments differ by less than one. This fixes an intercept convention: it is the lower mechanical word with intercept $\alpha$.

For $\alpha =(3-\sqrt{5})/2$, the beginning is $0100101001\dots$, the Fibonacci word. Like every Sturmian word, it has exactly $n+1$ distinct factors of length $n$. Its relation to continued fractions makes its recursive block structure reflect the partial quotients of its slope.