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arXiv · 2511.15133

A Balanced Three-term Generalization of Nicomachus' Identity

Seon-Hong Kim·Kenneth B. Stolarsky
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Abstract

We present a generalization of the classical Nicomachus' identity for the sum of the first $n$ cubes. Unlike previous generalizations, it has three rather than two terms, and involves not just one, but two distinct triangular numbers, and each term is of degree $4$ in $\lfloor n/2 \rfloor$. The asymptotic behavior for large $n$ leads to continued fractions with remarkable (but conjectural) properties. Moreover, we give a way of looking at squares of triangular numbers that involves the square root of $11$ and show it is a limiting case of a non-obvious identity involving truncations of the continued fraction expansion of that square root. The details involve a nonlinear recurrence that (with appropriate initial conditions) unexpectedly produces only integers, a ``Somos-type'' phenomenon.

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Seon-Hong Kim, Kenneth B. Stolarsky. 2025-11-19. A Balanced Three-term Generalization of Nicomachus' Identity. https://arxiv.org/abs/2511.15133

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