A Unique Square in a Fixed-Offset Family of OEIS A392677: A Computer-Assisted Proof via an Elliptic Curve

Let S2(n) = sum_{i=1}^n i^2(-1)^{ceil(sqrt(i))}, which is OEIS A392677. We prove that for every integer k >= 18, |S2(k(k+1)-18)| is a perfect square if and only if k = 32; at that point S2(1038) = 4317^2. The square condition reduces exactly to the quartic 16y^2 = 17q^4 - 1256q^2 + 29799 and then maps to the elliptic curve Y^2 = X^3 - 1256X^2 + 506583X. PARI/GP establishes rank 1, while Magma certifies the full Mordell-Weil generator and the complete integral-point list. The square-coordinate filter leaves only q = 65. This is a computer-assisted theorem for the single fixed-offset family n = k(k+1)-18; it does not classify all square values of A392677.