  We propose a novel storage scheme for three-nucleon (3N) interaction matrix
elements relevant for the normal-ordered two-body approximation used
extensively in ab initio calculations of atomic nuclei. This scheme reduces the
required memory by approximately two orders of magnitude, which allows the
generation of 3N interaction matrix elements with the standard truncation of
$E_{3\max}=28$, well beyond the previous limit of 18. We demonstrate that this
is sufficient to obtain ground-state energies in $^{132}$Sn converged to within
a few MeV with respect to the $E_{3\max}$ truncation. In addition, we study the
asymptotic convergence behavior and perform extrapolations to the un-truncated
limit. Finally, we investigate the impact of truncations made when evolving
free-space 3N interactions with the similarity renormalization group. We find
that the contribution of blocks with angular momentum $J_{\rm rel}>9/2$ is
dominated by a basis-truncation artifact which vanishes in the large-space
limit, so these computationally expensive components can be neglected. For the
two sets of nuclear interactions employed in this work, the resulting binding
energy of $^{132}$Sn agrees with the experimental value within theoretical
uncertainties. This work enables converged ab initio calculations of heavy
nuclei.
