Knights are 24/13 times faster than the king

On an infinite chess board, how much faster can the knight reach a square when compared to the king, in average? More generally, for coprime b>a ∈ Z≥ 1 such that $a+b$ is odd, define the $(a,b)$-knight and the king as \[ Na,b= \{(a,b), (b,a), (-a,b), (-b,a), (-b,-a), (-a,-b), (a,-b), (b, -a)\}, \] \[ K=\{(1,0), (1,1), (0,1), (-1,1), (-1,0), (-1,-1), (0,-1), (1,-1)\} ⊆ Z^2, \] respectively. One way to formulate this question is by asking for the average ratio, for p∈ Z² in a box, between min∈ Z≥ 1 ~|~ p∈ hN\ and min∈ Z≥ 1 ~|~ p∈ hK\, where hA = ₁+⋯+aₕ ~|~ a₁,…, aₕ ∈ A\ is the h-fold sumset of A. We show that this ratio equals 2(a+b)b²/(a²+3b²).