  Introduced in 2012, by Zhang, Zhao, and Nijhoff, the trilinear Boussinesq
equation is the natural form of the equation for the $\tau$-function of the
lattice Boussinesq system. In this paper we study various aspects of this
equation: its highly nontrivial derivation from the bilinear lattice AKP
equation under dimensional reduction, a quadrilinear dual lattice equation,
conservation laws, and periodic reductions leading to higher-dimensional
integrable maps and their Laurent property. Furthermore, we consider a higher
Gel'fand-Dikii lattice system, its periodic reductions and Laurent property. As
a special application, from both a trilinear Boussinesq recurrence as well as a
higher Gel'fand-Dikii system of three bilinear recurrences, we establish
Somos-like integer sequences.
