 Solution

The primitive unit cell contains two carbon atoms and four edge-shared oxygen atoms, each contributing one half: $n=2+4/2=4$ atoms. Equivalently, one molecular repeat contributes its two carbon atoms but only half of each of its four shared oxygen atoms. There are therefore $3n=12$ phonon branches.

At the Brillouin zone centre, three rigid translations give three acoustic phonon branches; the remaining $3n-3$ are optical phonon branches:A free layer has two in-plane acoustic modes and an out-of-plane flexural acoustic mode. A two-dimensional lattice does not imply that its atoms can move only in two directions. Artificially suppressing all out-of-plane motion would instead define a different model with two acoustic and six optical branches.