With Mathematica I have:
f[w_] := 1/((1 - E^(-I w)/3)*(1 - E^(-I w)/4));
InverseFourierSequenceTransform[f[w], w, n]
(* Piecewise[{{4/3^n - 3/4^n, n >= 0}}, 0] *)
g[w] = f[w] // Apart
(*1 + 4/(-1 + 3 E^(I w)) - 3/(-1 + 4 E^(I w)) *)
InverseFourierSequenceTransform[g[w], w, n]
(* Piecewise[{{4/3^n - 3/4^n, n >= 0}}, 0] *)
Results are the same. You can check this code in here.(Only you must sign in)
