半完全等差数列、超立方体的除法和Pell常数
Semicomplete Arithmetic Sequences, Division of Hypercubes, and the Pell Constant
摘要：
在本文中，我们产生了一些延续我们以前的工作分区成分数。具体来说，我们研究严格增加整数序列$\{n_j\}$，使有分区的所有整数小于地板的$G$，其中$G=\frac{n_1}{j} + \frac{n_2}{j} + \cdots + \frac{n_{j-1}}{j}+\frac{n_j}{j}$，和所有的被加数是不同的条款从$\frac{n_1}{j} + \frac{n_2}{j} + \cdots + \frac{n_{j-1}}{j}+\frac{n_j}{j}$。我们将这种序列称为\enquote{semicomplete}。我们发现只有三个半完全等差数列。我们还研究了序列，给出了最大数量的片，$M$维超立方体可以切割成使用$N-1$超平面。我们发现这些在一维、二维、三维和四维中是半完全的。顺便说一下，我们使用我们的生成函数之一来产生似乎是佩尔常数的新恒等式，这个数与负佩尔方程的解的密度密切相关。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
