  Three quantum particles with on-site repulsion and nearest-neighbour
attraction on a one-dimensional lattice are considered. The three-body
Schroedinger equation is reduced to a set of single-variable integral
equations. Energies of three-particle bound complexes (trions) are found from
self-consistency of the approximating matrix equation. In the case of spin-1/2
fermions, the ground state trion energy, the excited state energies, the trion
spectra and stability regions are obtained for total spins S = 1/2 and S = 3/2.
In the S = 1/2 sector, a narrow but finite parameter region is identified where
the ground state consists of a stable fermion pair and an unbound fermion. Also
presented is the reference case of spin-0 bosons.
