Structure and Growth of R-Bonacci Words

A binary word is called q-decreasing, for $q>0$, if inside this word each of length-maximal (in the local sense) occurrences of a factor of the form 0ᵃ1ᵇ, $a>0$, 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 for any real positive q the number of q-decreasing words of length n grows as Cq · Φ(q)ⁿ for some constant Cq which depends on q but not on n. From previous works, it is already known that Φ(1) is the golden ratio, Φ(2) is equal to the tribonacci constant, Φ(k) is $(k+1)$-bonacci constant. We prove that the function Φ(q) is strictly increasing, discontinuous at every positive rational point, and exhibits a fractal structure related to the Stern-Brocot tree and Minkowski's question mark function.