What is the function $E(x)$?

I think it is the floor, $E(x) = lfloor xrfloor$. If $x$ is an integer, then you find the expression for the sum of integers, and if $x$ is not an integer, you add the missing part of the rectangle. In French, the notation $E(x)$ is used for the floor since it is called "partie entière".

$E(x)$ the floor or the greatest integer function : $E(x)=[x]$, $x=[x]+q$, where $[x]$ is integer $n$ and $q=x-[x]$ is the fractional part. $$S=int_{0}^{x} [t] dt=int_{0}^{1} 0 dt+ int_{1}^{2} 1 dt+int_{2}^{3} 2 dt+.......+int_{n-1}^{n} (n-1) dt+int_{n}^{n+q} n dx$$ $$S=1+2+3+4+...+(n-1)+nq=n(n-1)+nq=n(n-1)/2+nq$$ $$ implies S=[x]([x]-1)/2+[x](x-[x]).$$