On the Sum of the Sixth Powers of Fibonacci Numbers

Let $(G_k{)}_{k\in \mathbb{Z}}$ be any sequence obeying the recurrence relation of the Fibonacci numbers. We derive formulas for ${\sum }_{j=1}^n{G}_{j+t}^6$ and ${\sum }_{j=1}^n(-1{)}^{j-1}{G}_{j+t}^5(G_{j+t-1}+G_{j+t+1})$, thereby extending the results of Ohtsuka and Nakamura who found simple formulas for ${\sum }_{j=1}^n{F}_j^6$ and ${\sum }_{j=1}^n{L}_j^6$, where $F_k$ and $L_k$ are the $k$th Fibonacci and Lucas numbers. We also evaluate ${\sum }_{j=1}^n{G}_{j+t}^3{G}_{j+t+1}^3$ and ${\sum }_{j=1}^n{G}_{j+t-1}^2{G}_{j+t}{G}_{j+t+1}{G}_{j+t+2}^2$, of which the results of Treeby are particular cases.