Distinguish a sequence of geometric sums from its own sum

$u_n=1+2+\cdots+2^{n-1}=(2^n-1)/(2-1)=2^n-1$.

$S_n=\sum_{r=1}^n(2^r-1)=2(2^n-1)/(2-1)-n=2^{n+1}-n-2$. The first few sequence terms are $1,3,7,15$, whose partial sums are a different sequence.