F_{k+2}^{2} - F_{k+1}^{2} = F_{k} \cdot F_{k+3} \\
F_{k+1}^{2} + F_{k-1}^{2} = F_{k} \cdot (F_{k+2} + F_{k+1}) \\
F_{k}^{2} + F_{k-1}^{2} + F_{k-1}^{2} = F_{k} \cdot (F_{k} + F_{k-1} + F_{k} + F_{k} + F_{k-1}) \\
F_{k}^{2} + 2F_{k-1}^{2}  = F_{k} \cdot (3F_{k} + 2F_{k-1} ) \\