  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 $q \cdot a > b$. 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 $\Phi(q)^{n} C_q $ for some constant $C_q$ which depends on
$q$ but not on $n$. We demonstrate that $\Phi(1)$ is the golden ratio,
$\Phi(2)$ is equal to the tribonacci constant, $\Phi(k)$ is $(k+1)$-bonacci
constant. Furthermore, we prove that the function $\Phi(q)$ 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.
