condensed matter - Dispersion relation in tight binding model with even indices only

Given a tight binding model with Hamiltonian

$H= \sum_{i(even)}t[c_{i+1}^\dagger c_i+h.c]$

containing even indices only, how can we find out the dispersion relation?

Attempt:

My guess is that the dispersion relation is just $\frac{-2tcos(ka)}{2}$ i.e. half the regular tight binding model. I could not find the source for dispersion relation with two different hopping amplitudes. If I had, I would just replace one with zero.