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arXiv · 2609.31548

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

Subhasis Panda·Aditya Kumar Dash·Utkal Keshari Dutta
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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 $n\geq2$, \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 $\varepsilon_n=1$ if $n\equiv1\pmod{12}$ and $\varepsilon_n=0$ otherwise. This result extends the corresponding reciprocal-sum result for Fibonacci numbers due to Hwang, Park and Song to balancing numbers.

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Subhasis Panda, Aditya Kumar Dash, Utkal Keshari Dutta. 2026-09-25. On the infinite sum of reciprocals of the fourth powers of balancing numbers. https://arxiv.org/abs/2609.31548

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