2 ”N” orbitals with a level offset, which roughly reproduces the conduction and valence bands closest to the Fermi level, as shown by the dashed lines in the middle panel of Fig.1. Here we take the nearest neighbor (M-N) hopping t = 1 . 2eV, the level offset Δ /t = 2 . 7, and the next nearest neighbor (M-M) hopping t $^{′}$/t = 0 . 35. The on-site interaction U/t = 6 is considered in the single band model, and the on-site interaction U/t = 6 is introduced on both M and N orbitals in the two orbital model $^{22}$. The band filling n is defined as the number of electrons/number of sites. For the single band model, n = 2 x , where x is the Li or Na content, since there are two M sites per unit cell, which results in bonding and antibonding bands, and the electrons are doped only in the bonding band. For the two band model, n = 1 corresponds to the non-doped case, and x and n is re- lated by n = 1+ x . We use fluctuation exchange (FLEX) method $^{23–25}$, which is kind of a self-consistent random phase approximation, to obtain the Green’s function. Then we solve the linearized Eliashberg equation, Here, G is the Green’s function matrix (with l, m, etc... labeling sites in a unit cell) obtained by FLEX. V is the spin singlet pairing interaction matrix given by V = 3 $_{2}$U $^{2}$χ$_{sp}$ − 1 $_{2}$U $^{2}$χ$_{ch}$ with the spin and charge suscepti- bility matrices χ$_{sp(ch)}$ ( q ) = χ$_{irr}$ ( q )[1 − (+) Uχ$_{irr}$ ( q )] − $^{1}$, where χ$_{irr}$ ( q ) is the irreducible susceptibility matrix χ$_{irr}$ ( q ) = − 1 N ∑ $_{k}$G ( k + q ) G ( k ) ( N is the number of k - point meshes). In the following, the maximum eigenvalue of the spin susceptibility matrix will be referred to as the spin susceptibility. We take up to 128 × 128 k -point meshes and up to 16384 Matsubara frequencies. The eigenvalue of the Eliashberg equation λ increases upon lowering the temperature and reaches unity at T = T$_{c}$ . model for the cuprates, where the oxygen 2 p$_{x}$ , 2 p$_{y}$ de- grees of freedom is integrated out from the original three band d - p model. By considering hopping integrals up to fourth nearest neighbors (Fig.1, upper panel), we can re- produce the single band that crosses the Fermi level in the first principles band calculations. 8,10,20,21 The band dispersion of the single band model with t = 0 . 65 (eV), t$_{2}$/t = − 0 . 03, t$_{3}$/t = 0 . 06, t$_{4}$/t = − 0 . 025 is shown by the green dash-dotted line in Fig.1. We first present calculation results for the single band model. In Fig.2, the eigenvalue of the Eliashberg equa- tion is shown as functions of temperature for spin singlet and triplet pairings for n = 0 . 2. It can be seen that the triplet pairing strongly dominates over singlet pair- ing. The gap of the spin triplet pairing has an f -wave form, where the nodes of the gap do not intersect the Fermi surface (the ridge of the Green’s function squared) as shown in Fig.3. On the other hand, the singlet pairing has a d -wave gap (not shown). The possibility of this type of spin triplet f -wave pairing was in fact proposed on a canonical triangular lattice in ref. $^{26}$. The triplet pairing dominates over singlet pairing because ferromag- netic spin fluctuations arise due to the dilute band filling To go a step further, we also consider a model on a single layer HC lattice consisting of alternating ”M” and FIG. 1: (color online) Upper : the single band model on a triangular lattice. Middle : the two band model on a hon- eycomb lattice. Lower : the first principles band calculation taken from ref. 8 and the band dispersion of the single band model (dash-dotted green) and the two band model(dashed red). λφ$_{lm}$ ( k ) = $_{−}$T N ∑ k ′ ∑ l ′ ,m ′ × V$_{lm}$ ( k − k $^{′}$) G$_{ll}$ ′ ( k $^{′}$) G$_{mm}$ ′ ( − k $^{′}$) φ$_{l}$ ′ m ′ ( k $^{′}$) . III. SINGLE BAND MODEL: CALCULATION RESULTS