A Balanced Three-term Generalization of Nicomachus' Identity

TL;DR

This paper introduces a novel three-term generalization of Nicomachus' identity involving triangular numbers, degree-4 terms, and intriguing asymptotic and continued fraction properties, revealing new mathematical structures.

Contribution

It presents a new three-term generalization of Nicomachus' identity, involving two distinct triangular numbers and degree-4 terms, with connections to continued fractions and nonlinear recurrences.

Findings

Asymptotic behavior leads to special continued fractions.

Identifies a link between squares of triangular numbers and the square root of 11.

Recurrence relations produce integer sequences with Somos-type properties.

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.