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.

翻译：若一个二进制词中，所有形如$0^a1^b$（$a>0$）的最大长度因子均满足$q \cdot a > b$（$q>0$），则该词称为$q$-递减词。我们将$q$-递减词与直线$y=qx$切割序列的特定前缀建立了双射对应。证明表明，长度为$n$的$q$-递减词数量以$\Phi(q)^{n} C_q$的形式增长，其中常数$C_q$仅依赖于$q$而与$n$无关。我们证明$\Phi(1)$为黄金比例，$\Phi(2)$等于三波那契常数，而$\Phi(k)$为$(k+1)$-波那契常数。进一步证明，函数$\Phi(q)$严格递增，在每个正有理点处不连续，并呈现出与Stern--Brocot树及Minkowski问号函数相关的分形结构。