Continued fractions for some transcendental numbers

We consider series of the form $$\frac{p}{q}+\sum _{j=2}^{\mathrm{\infty }}\frac{1}{x_j},$$ where $x_1=q$ and the integer sequence $(x_n)$ satisfies a certain non-autonomous recurrence of second order, which entails that $x_n\mathrm{\mid }{x}_{n+1}$ for $n\ge 1$. It is shown that the terms of the sequence, and multiples of the ratios of successive terms, appear interlaced in the continued fraction expansion of the sum of the series, which is a transcendental number.