On the infinite sum of reciprocals of the fourth powers of balancing numbers

Abstract

In this note, we study the infinite reciprocal sum ${\sum }_{k=n}^{\infty }1/B_k^4$ involving the fourth powers of balancing numbers $B_n$. We show that, for every $n\ge 2$, \begin{equation*} \left\lfloor \left( \sum_{k=n}^{\infty}\frac{1}{B_k^4} \right)^{-1} \right\rfloor = B_n^4-B_{n-1}^4 -\left\lceil\frac{B_{2n-1}}{280}\right\rceil +\varepsilon_n, \end{equation*} where ${\epsilon }_n=1$ if $n\equiv 1(\mathrm{mod}12)$ and ${\epsilon }_n=0$ otherwise. This result extends the corresponding reciprocal-sum result for Fibonacci numbers due to Hwang, Park and Song to balancing numbers.

12 pages