  We analyze the electronic structure of the recently synthesized higher-order
nickelate Ruddlesden-Popper phases La$_{n+1}$Ni$_n$O$_{3n+1}$ ($n=4-6$) using
first-principles calculations. For all materials, our results show large
holelike Fermi surfaces with $d_{x^2-y^2}$ character that closely resemble
those of optimally hole-doped cuprates. For higher values of $n$, extra
non-cuprate-like bands of $d_{z^2}$ orbital character appear. These aspects
highlight that this Ruddlesden-Popper series can provide a means to modify the
electronic ground states of nickelates by tuning their dimensionality. With
their similarities and differences to the cuprates, this new family of
materials can potentially shed light on the physics of copper-based oxides.
