Triples (x,y,r) with r|xy and (r|x or r|y): a Divisor-Function Identity and an O(n2/3) Algorithm for the Sequence A268756

Let a(n) be the number of triples of positive integers (x,y,r) such that xy <= n, r divides xy, and r divides x or y (OEIS A268756). The original entry in the OEIS provides only an enumeration program of cost O(n^2). We establish an exact combinatorial identity a(n) = 2*D_3(n) - T(n), where D_3(n) is the summatory function of d_3(k) (OEIS A061201) and T(n) is the sum of d(gcd(x,y)) over xy <= n (OEIS A268732). Equivalently, the Dirichlet generating series of the increments is proved to be zeta(s)^2 * (2*zeta(s) - zeta(2s)). Using these structural properties, we derive a precise second-order asymptotic expansion a(n) = n * (ln^2 n + c_1 * ln n + c_0) + O(n^(1/2+epsilon)), where the constants c_1 = -0.18164... and c_0 = 2.59373... are explicitly evaluated using Euler's constant, the first Stieltjes constant gamma_1, and the derivative zeta'(2). Furthermore, we develop an exact sublinear algorithm that computes a(n) using a prefix table of the divisor function D(x) and a modified Dirichlet hyperbola method in O(n^(2/3) log n) arithmetic operations and O(n^(2/3)) memory. The algorithm matches the known terms and has been validated against independent sieves up to n=10^7. In Python 3.12, the value a(10^12) = 761,048,161,394,384 is returned in 3.4 s, and a(10^13) is computed in 22.6 s.