  We prove that a sequence $h$ of non-negative integers is the Hilbert function
of some Artinian Gorenstein algebra with the strong Lefschetz property if and
only if it is an SI-sequence. This generalizes the result by T. Harima which
characterizes the Hilbert functions of Artinian Gorenstein algebras with the
weak Lefschetz property. We also provide classes of Artinian Gorenstein
algebras obtained from the ideal of points in $\mathbb{P}^n$ such that some of
their higher Hessians have non-vanishing determinants. Consequently, we provide
families of such algebras satisfying the SLP.
