  A class of explicit pseudo two-step Runge-Kutta-Nystr\"{o}m (GEPTRKN) methods
for solving second-order initial value problems $y'' = f(t,y,y')$, $y(t_0) =
y_0$, $y'(t_0)=y'_0$ has been studied. This new class of methods can be
considered a generalized version of the class of classical explicit pseudo
two-step Runge-Kutta-Nystr\"{o}m methods. %The new methods will be denoted by
GEPTRKN methods. We proved that an $s$-stage GEPTRKN method has step order of
accuracy $p=s$ and stage order of accuracy $r=s$ for any set of distinct
collocation parameters $(c_i)_{i=1}^s$. Super-convergence for order of accuracy
of these methods can be obtained if the collocation parameters $(c_i)_{i=1}^s$
satisfy some orthogonality conditions. We proved that an $s$-stage GEPTRKN
method can attain order of accuracy $p=s+2$. Numerical experiments have shown
that the new methods work better than classical methods for solving non-stiff
problems even on sequential computing environments. By their structures, the
new methods will be much more efficient when implemented on parallel computers.
