  We classify all the six derivative Lagrangians of gravity, whose traced field
equations are of second or third order, in arbitrary dimensions. In the former
case, the Lagrangian in dimensions greater than six, reduces to an arbitrary
linear combination of the six dimensional Euler density and the two linearly
independent cubic Weyl invariants. In five dimensions, besides the independent
cubic Weyl invariant, we obtain an interesting cubic combination, whose field
equations for static spherically symmetric spacetimes are of second order. In
the later case, in arbitrary dimensions we obtain two combinations, which in
dimension three, are equivalent to the complete contraction of two Cotton
tensors. Moreover, we also recover all the conformal anomalies in six
dimensions. Finally, we present some static, spherically symmetric solutions
for these Lagrangians.
