Is there a closed form expression for $G(n,k,m)=\sum_{i=0}^{n}F_{ik+m}$ or at least for $G(n,k,1)$?
I could find \begin{align} G(n,k,0)&=\frac{F_{nk+k}-(-1)^kF_{nk}-F_k}{L_k-(-1)^k-1}\\\\ &=G(n,k,1)-G(n,k,-1) \end{align} (with $L$ as the Lucas number), but no expression for $m\in\mathbb Z$.
