周期平衡二元三角形
Periodic balanced binary triangles
摘要：
大小为$n$的二进制三角形是由0和1组成的三角形，具有$n$行，使用与标准杨辉三角形形模$2$相同的局部规则构建。一个二元三角形被称为平衡的，如果组成这个三角形的0和1的个数之间的绝对差至多是$1$。本文证明了对所有正整数$n$，存在大小为$n$的平衡二元三角形.这是通过考虑周期平衡二元三角形来实现的，周期平衡二元三角形是平衡二元三角形，其中每行、列或对角线是周期序列。
英文摘要：
A binary triangle of size $n$ is a triangle of zeroes and ones, with $n$ rows, built with the same local rule as the standard Pascal triangle modulo $2$. A binary triangle is said to be balanced if the absolute difference between the numbers of zeroes and ones that constitute this triangle is at most $1$. In this paper, the existence of balanced binary triangles of size $n$, for all positive integers $n$, is shown. This is achieved by considering periodic balanced binary triangles, that are balanced binary triangles where each row, column or diagonal is a periodic sequence.
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。
