Small and Large Weights in Steinhaus Triangles

Let $\{0=w_0<w_1<w_2<\ldots<w_m\}$ be the set of weights of binary Steinhaus triangles of size $n$, and let $W_i$ be the set of sequences in $\mathbb{F}_2^n$ that generate triangles of weight $w_i$. We obtain the values of $w_i$ and the corresponding sets $W_i$ for $i\in\{1,2,3,m\}$, and partial results about $w_{m-1}$ and $W_{m-1}$.