  A cosmological model with a gravitational Lagrangian $L_g(R)\propto R+A R^n$
is set up to account for the presently observed re-acceleration of the
universe. The evolution equation for the scale factor $a$ of the universe is
analyzed in detail for the two parameters $n=2$ and $n=4/3$, which were
preferred by previous studies of the early universe. The initial conditions are
specified at a red-shift parameter $z\approx 0$. The fit to the observable data
fixes the free parameter $A$. The analysis shows that the model with $n=2$
agrees better with present data. Then, if we set $w(q)=-1$ at $z=0$,
corresponding to the deceleration parameter $q\approx -1/2$, we find that at
$z\approx 0.5$, $w(q)$ has evolved to $w\approx -0.72$, corresponding to
$q\approx 0$. At $z\approx 1$ we find $w\approx 0$ corresponding to $q\approx
1/2$. These results are compared with the flat Friedmann model with cold matter
and Lambda-term (LCDM model) for the same initial conditions at $z\approx 0$.
The other choice of the model with $n=4/3$ allows for big crunch. However this
possibility is eliminated by the fit of $A$ to the present data.
