For a general Martens-special chain of cycles $Γ$ of type $k$ we prove that the gonality is equal to $k+2$. Although $\dim (W^1_{k+2} (Γ))=k$ we prove that $w^1_{k+2}(Γ)=0$. We also compute the gonality sequence of $Γ$ and we prove it is divisorial complete. We prove that a general Martens-special discrete chain of cycles $G$ of type $k$ has the same gonality sequence.