AI 中文总结
该研究针对标准基向量生成的广义斯坦豪斯三角形，证明其截断系数分布具有周期性，推导了1的数量的递推关系、有理生成函数及渐近增长率，明确了其1密度为零，给出经典情形的显式公式，统一描述了其周期性与计数结构。
AI 中文摘要
广义斯坦豪斯s-三角形由二进制序列通过将每s个连续元素的块替换为其模2和得到。我们研究由标准基向量生成的此类三角形，通过二项式系数表达其元素，证明连续行的截断系数分布具有纯周期性，并确定其精确周期。该周期性导出了一个2的幂次导数恒等式及分解为相同有限块的性质，由此得到1的数量的精确递推关系，证明在每个剩余类上，该权重是长度的仿射函数。我们还推导了有理生成函数，确定了渐近增长率，证明这些标准三角形的1密度为零。针对经典情形的前两个标准向量和块权重，得到了显式公式。这些结果为标准基向量生成的广义斯坦豪斯三角形的周期性与计数结构提供了统一描述。
英文摘要
A generalized Steinhaus $s$-triangle is obtained from a binary sequence by repeatedly replacing each block of $s$ consecutive entries by its sum modulo $2$. We study the triangles generated by canonical basis vectors. Expressing their entries through bi$^{s}$nomial coefficients, we prove that the truncated coefficient profiles of the successive rows are purely periodic and determine their exact period. This periodicity yields a power-of-two derivative identity and a decomposition into identical finite blocks. We consequently obtain an exact recurrence for the number of ones and show that, on every residue class, this weight is an affine function of the length. We also derive a rational generating function, determine the asymptotic growth rate, and prove that these canonical triangles have zero density of ones. Explicit formulas are obtained for the first two canonical vectors and for the block weight in the classical case. These results provide a unified description of the periodic and enumerative structure of generalized Steinhaus triangles generated by canonical basis vectors.
Comments15 pages
