 Binary order on the discrete cube

Identify a vertex $(x_1,\dots ,x_n)$ with the integer ${\sum }_{i=1}^n{2}^{i-1}{x}_i$. Binary order is increasing order of these integers. Its initial segment of size $a$ has $F(a)={\sum }_{j=0}^{a-1}{s}_2(j)$ induced edges, where $s_2(j)$ is the Hamming weight of the binary expansion of $j$.