Heegaard splittings of the form H + nK

Suppose that a three-manifold M contains infinitely many distinct strongly irreducible Heegaard splittings H + nK, obtained by Haken summing the surface H with n copies of the surface K. We show that K is incompressible. All known examples, of manifolds containing infinitely many irreducible Heegaard splittings, are of this form. We also give new examples of such manifolds.