Structure and growth of ℝ-bonacci words

A binary word is called q-decreasing, for q>0, if every of its length maximal factors of the form 0^a1^b, a>0, satisfies q · a > b. We bijectively link q-decreasing words with certain prefixes of the cutting sequence of the line y=qx. We show that the number of q-decreasing words of length n grows as Φ(q)^n C_q for some constant C_q which depends on q but not on n. We demonstrate that Φ(1) is the golden ratio, Φ(2) is equal to the tribonacci constant, Φ(k) is (k+1)-bonacci constant. Furthermore, we prove that the function Φ(q) is strictly increasing, discontinuous at every positive rational point, exhibits a fractal structure related to the Stern–Brocot tree and Minkowski's question mark function.