  In this paper, we aim to perform a systematical investigation on the field
equations and Noether potentials for the higher-order gravity theories endowed
with Lagrangians depending on the metric and the Riemann curvature tensor,
together with $i$th ($i=1,2,\cdot\cdot\cdot$) powers of the
Beltrami-d'Alembertian operator $\Box$ acting on the latter. We start with a
detailed derivation of the field equations and the Noether potential
corresponding to the Lagrangian $\sqrt{-g}L_R(R,\Box R,\cdot\cdot\cdot,\Box^m
R)$ through the direct variation of the Lagrangian and a method based upon the
conserved current. Next the parallel analysis is extended to a more generic
Lagrangian $\sqrt{-g}L_{\text{Ric}}(g^{\mu\nu}, R_{\mu\nu},\Box R_{\mu\nu},
\cdot\cdot\cdot,\Box^m R_{\mu\nu})$, as well as to the generalization of the
Lagrangian $\sqrt{-g}L_{\text{Ric}}$, which depends on the metric $g^{\mu\nu}$,
the Riemann tensor $R_{\mu\nu\rho\sigma}$ and $\Box^i R_{\mu\nu\rho\sigma}$s.
Finally, all the results associated to the three types of Lagrangians are
extended to the Lagrangian relying on an arbitrary tensor and the variables via
$\Box^i$ acting on such a tensor. In particular, we take into consideration of
equations of motion and Noether potentials for nonlocal gravity models. For
Lagrangians involving the variables $\Box^i R$, $\Box^i R_{\mu\nu}$ and $\Box^i
R_{\mu\nu\rho\sigma}$, our investigation provides their concrete Noether
potentials and the field equations without the derivative of the Lagrangian
density with respect to the metric. Besides, the Iyer-Wald potentials
associated to these Lagrangians are also presented.
