Integral representations for Mersenne and Horadam-Fermat numbers

In this note we first derive integral representations for Mersenne numbers $M_{kn}$ and Horadam-Fermat numbers $\mathcal{F}_{kn}$, then we use those to provide integral representations for Mersenne numbers $M_{kn+r}$ and Horadam-Fermat numbers $\mathcal{F}_{kn+r}$, where $n\in\mathbb{Z}_{>0}=\{1,2,3,\ldots\}$ is a non-negative integer, $k\in\mathbb{Z}_{>0}=$ $\{1,2,3,\ldots\}$ is an arbitrary but fixed positive integer, while $r\in\mathbb{Z}_{\geqslant0}$ is an arbitrary but fixed non-negative integer.