Semicomplete Arithmetic Sequences, Division of Hypercubes, and the Pell Constant

Abstract

In this paper we produce a few continuations of our previous work on partitions into fractions. Specifically, we study strictly increasing integer sequences $\{n_j\}$ such that there are partitions for all integers less than the floor of $G$, where $G=\frac{n_1}{j} + \frac{n_2}{j} + \cdots + \frac{n_{j-1}}{j}+\frac{n_j}{j}$, and all summands are distinct terms drawn from $\frac{n_1}{j} + \frac{n_2}{j} + \cdots + \frac{n_{j-1}}{j}+\frac{n_j}{j}$. We call such sequences \enquote{semicomplete}. We find that there are only three semicomplete arithmetic sequences. We also study sequences that give the maximum number of pieces that an $M$ dimensional hypercube can be cut into using $N-1$ hyperplanes. We find that these are semicomplete in one, two, three, and four dimensions. As an aside, we use one of our generating functions to produce what appears to be a new identity for the Pell constant, a number which is closely connected to the density of solutions to the negative Pell equation.