The three-sphere, S^3, has trivial cohomology except for H^0(S^3) and H^3(S^3) which are both isomorphic to the integers. Thus the even and odd cohomologies are both isomorphic to the integers. Because the three-sphere is of dimension three, which is less than five, the third Steenrod square is trivial on its cohomology and so the first nontrivial differential is just d_5 = H. The later differentials increase the degree of a cohomology class by more than three and so are again trivial; thus the twisted K-theory is just the cohomology of the operator d_3 which acts on a class by cupping it with the 3-class H.
