Lattice Points in the Astroid: An Exact Integer Test and a Convex Hull Walk for the Sequence A216877

Let a(n) be the number of integer points (x, y) with |x|^(2/3) + |y|^(2/3) <= n^(2/3), corresponding to the OEIS sequence A216877. The original entry contains only a brute-force program of cost O(n^2). This paper provides an exact integer criterion for membership, proving that for integers x, y >= 1 the inequality holds if and only if n^2 - x^2 - y^2 >= 0 and (n^2 - x^2 - y^2)^3 >= 27 * x^2 * y^2 * n^2. Using this exact test and algebraic symmetry, the total count a(n) is reduced to a single monotone sum S over the boundary. We compute S by a walk along the lower convex hull of the lattice points above the curve, where every decision is an exact integer comparison. The Stern-Brocot descent step is solved in closed form by using the tangent point of the astroid. The algorithm requires approximately 1.04 * n^(2/3) iterations, verified numerically up to n = 10^10. In Python 3.12, the value a(10^9) = 1,178,097,245,096,271,057 is computed in 5.4 s, and a(10^10) takes 29 s, presenting a significant speedup over standard O(n) column-counting approaches (8.0 s for n = 10^6).