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We consider all hopping amplitude for all sites the same, $t_{12}=t$. Here, $g_{11}$ is the bare Green function for an uncoupled site, ie.,   \begin{equation}  g_{11}=\frac{1}{\omega-\epsilon_0}  \end{equation} Under these considerations we have  \begin{equation}  G_{21}=g_{21} + g_{22} t_{21} G_{11} + g_{22} t_{23} G_{31},  \end{equation}  and similar equations for $G_{31}$, etc. This sequence of Green functions for $G_{n1}$ generates an infinite number of equations. Now, it comes the trick. Consider that site $2$ is in reality another semi-infinite chain coupled to site $1$. Then, the equation for $G_{21}$ becomes simply  \begin{equation}  G_{12}=g_{ch} t_{21} G_{11}  \end{equation}  By replacing this expression for $G_{12}$ in $G_{11}$ we can close the Dyson equation and to obtain explicitly $G_{11}$. Therefore  \begin{equation}  G_{11}=g_{11}+ g_{11}|t|^2 g_{ch} G_{11}  \end{equation}