On the Sum of the Sixth Powers of Fibonacci Numbers

Abstract

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.