2025/11/19 by Kim, Seon-Hong, Stolarsky, Kenneth B.
Computer Science · Mathematics · #semigroups and automata theory #Analytic Number Theory Research #Advanced Mathematical Identities
paper · doi:10.48550/arxiv.2511.15133
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.