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A triangulation of S^2 has non-negative curvature if every vertex has six or fewer triangles adjacent to it. Thurston showed that non-negative curvature triangulations correspond to lattice points in a moduli space of flat cone metrics on S^2. In joint work with Peter Smillie, we use an arithmetic technique of Siegel to count such lattice points. The appropriately weighted number of triangulations with 2n triangles is an explicit constant times the ninth divisor power sum of n.