A Balanced Three-term Generalization of Nicomachus' Identity
Seon-Hong Kim, Kenneth B. Stolarsky
TL;DR
The paper addresses generalizing Nicomachus' identity for sums of powers to a balanced three-term form that involves two triangular-number components. It develops a framework with two families of expressions $L_{m,o}(x),R_{m,o}(x),xP_{m,o}(x)$ and $L_{m,e}(x),R_{m,e}(x),xP_{m,e}(x)$, establishing a theorem that the difference between the left and right sides equals $xP(x)-x^2+x^3$, with all terms of degree 4 in $m$ and the $x=0$ case recovering the classical Nicomachus identity. The proof employs polynomial-first approaches and a coefficient-matching matrix to verify the identity, and also discusses determinant representations and related structural observations. A notable limiting case, the $\sqrt{11}$ identity, connects the generalized framework to squares of triangular numbers via the continued fraction of $(-1+\sqrt{11})/2$, illustrating a deep link between continued fractions, discriminants, and Somos-type integer phenomena within this generalization.
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.