A (probably easy) question about exchanging the order of this double sum

Question

I would like to exchange the order of the following:

$\sum_{k=1}^{i-1}\sum_{r=1}^{2k}$ (stuff).

I feel like it should be easy, but so far I am only able to produce

$\sum_{r=1}^{2(i-1)}\sum_{k=??}^{??}$ (stuff).

In particular, I don't see how to reconcile the inequalities $1 \leq r\leq 2k\,$ and $1 \leq k \leq i-1$\, without introducing half integers into the sum.

Thanks,

Answer 1

The second sum should be $$\sum_{k=\lceil \frac r2\rceil}^{i-1}$$ $k$ is always at least $\frac r2$ and ranges up to $i-1$