Generalized Steinhaus triangles generated by canonical basis vectors: periodicity and weight formulas

Abstract

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.