  We compute the perturbative expansion for the effective interaction $W$ of
the half-filled 2-dimensional Hubbard model. We derive extensions of standard
RPA resummations that include arbitrarily high order contributions in the
$W_{\uparrow\uparrow}$ and $W_{\uparrow\downarrow}$ basis. Using algorithmic
tools we explore the static $Q$-dependent interaction as well as the same-time
quantity both in momentum- and real-space. We emphasize the absence of
screening in the Hubbard interaction where we find an enhanced repulsive local
$W_{\uparrow\downarrow}$ with a non-zero attractive $W_{\uparrow\uparrow}$.
Finally, starting from only a locally repulsive bare interaction find an
emergent non-local nearest-neighbor attraction for low temperatures at
sufficiently large values of $U/t$ which may be key to understanding pairing
processes in the model.
