  Novel fermionic quasiparticles with integer pseudospins in some energy bands,
such as pseudospin-1 triple-point fermions, recently attract increasing
interest since they are beyond the conventional spin-$1/2$ Dirac and Weyl
counterparts. In this paper, we propose a class of pseudospin-1 fermioic
excitations emerging in topological metal bands, dubbed double-triple-point
(DTP) fermions. We first present a general three-band continuum model with
$C_4$ symmetry in three dimensions, which has three types of threefold
degenerate points in the bands classified by their topological charges
$C=\pm4,\pm2,0$, respectively. They are dubbed DTPs as spin-1 generalization of
double-Weyl points. We then construct two-dimensional and three-dimensional
tight-binding lattice models of topological metal bands with exotic DTP
fermions near the DTPs. In two dimensions, the band gaps close at a trivial DTP
with zero Berry phase, which occurs at the transition between the normal and
topological insulator phases. In three dimensions, the topological properties
of three different DTP fermions in lattice systems are further investigated,
and the effects of breaking $C_4$ symmetry are also studied, which generally
leads to splitting each quadratic DTP into two linear triple points and gives
topological phase diagrams. Using ultracold fermionic atoms in optical
lattices, the proposed models can be realized and the topological properties of
the DTP fermions can be detected.
