  We found that when the spinless model is off the half-filling regime ($\mu
\neq V$), the Helmholtz free energy (HFE) can be written as two
$\beta$-expansions: one expansion comes from the half-filling configuration and
another one that depends on the parameter $x = \mu - V$. We show numerically
that the chemical potential as a function of temperature satisfies a relation
similar to the one derived from the particle-hole symmetry of the fermionic
spinless model. We extend the $\beta$-expansion of the HFE of the
one-dimensional fermionic spinless Hubbard model up to order $\beta^8$.
