On the Sum of the Sixth Powers of Fibonacci Numbers

Kunle Adegoke; Olawanle Layeni

paper

On the Sum of the Sixth Powers of Fibonacci Numbers

arXiv:2401.10221

Abstract

Let $(G_{k})_{k \in Z}$ 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.