  We introduce and investigate an effective five-band model for $t_{2g}$ and
$e_g$ electrons to describe doped cobalt oxides with Co$^{3+}$ and Co$^{4+}$
ions in two-dimensional CoO$_2$ triangular lattice layers, as in
Na$_{1-x}$CoO$_2$. The effective Hamiltonian includes anisotropic kinetic
energy (due to both direct Co-Co and indirect Co-O-Co hoppings), on-site
Coulomb interactions parameterized by intraorbital Hubbard repulsion $U$ and
full Hund's exchange tensor, crystal-field terms and Jahn-Teller static
distortions. We study it using correlated wave functions on $6\times 6$
clusters with periodic boundary conditions. The computations indicate low S=0
spin to high S=2 spin abrupt transition in the undoped systems when increasing
strength of the crystal field, while intermediate S=1 spins are not found.
Surprisingly, for the investigated realistic Hamiltonian parameters describing
low spin states in CoO$_2$ planes, doping generates high $S=\frac{5}{2}$ spins
at Co$^{4+}$ ions that are pairwise bound into singlets, seen here as pairs of
up and down spins. It is found that such singlet pairs self-organize at higher
doping into lines of spins with coexisting antiferromagnetic and ferromagnetic
bonds, forming stripe-like structures. The ground states are insulating within
the investigated range of doping because computed HOMO-LUMO gaps are never
small enough.
