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Ab initio calculations have been performed to unravel the origin of therecently found superlattice peaks in the trilayer nickelate La$_4$Ni$_3$O$_8$.These peaks arise from static charge ordering of Ni$^{2+}$/ Ni$^{1+}$ stripesoriented at 45$^{\circ}$ to the Ni-O bonds. An insulating state originates froma combination of structural distortions and magnetic order, with the gap beingformed solely within the d$_{x^2-y^2}$ manifold of states. When doped,electrons or holes would go into these states, in a similar fashion to whatoccurs in the cuprates. Analogous calculations suggest that checkerboard chargeorder should occur in the bilayer nickelate La$_3$Ni$_2$O$_6$. These resultsreveal a close connection between La$_4$Ni$_3$O$_8$ and La$_3$Ni$_2$O$_6$ withLa$_{2-x}$Sr$_x$NiO$_4$ for x=1/3 and x=1/2, respectively.

Charge ordering in Ni$^{1+}$/Ni$^{2+}$ nickelates: La$_4$Ni$_3$O$_8$ and La$_3$Ni$_2$O$_6$