Let $(a_n)_{n=0}^{\infty}$ be a sequence with $a_0 \in \mathbb{Z}$ and $a_n \in \mathbb{N}$ for every $n \geq 1$. Let $(q_n)_{n=-1}^{\infty}$ be the denominator sequence of the convergents of the infinite simple continued fraction $[a_0; a_1, a_2, \dots]$, defined by