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In this study, we address the open problem of determining the general term of the sequence \(4, 9, 15, 60, 106, \ldots \) , which counts the number of solutions to the Diophantine equation: \( \sum _{k=1}^N F_{n_k} = 2^a, \qquad N=1,2,3,\ldots , \) where \(F_n\) represents the Fibonacci sequence. This sequence, now registered as OEIS A356928 (2022), was introduced by the author. We describe exact computation via dynamic programming, provide computed data for small N, discuss fitted recurrences and their limitations, and outline routes toward a rigorous asymptotic.

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General Theory for Diophantine Equations in Fibonacci Sums: An Open Problem

  • 会议论文

  • 发表日期：

    2026年

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  • Pagdame Tiebekabe

摘要

In this study, we address the open problem of determining the general term of the sequence \(4, 9, 15, 60, 106, \ldots \) , which counts the number of solutions to the Diophantine equation: \( \sum _{k=1}^N F_{n_k} = 2^a, \qquad N=1,2,3,\ldots , \) where \(F_n\) represents the Fibonacci sequence. This sequence, now registered as OEIS A356928 (2022), was introduced by the author. We describe exact computation via dynamic programming, provide computed data for small N, discuss fitted recurrences and their limitations, and outline routes toward a rigorous asymptotic.