For any integer k>0, what do you think H(k,k) represents?

k=2    H(2,2) = H(2-1,min(1,2-1)) + H(2-2,min(2,2-2)) =
              = H(1,1) + H(0,0) =
              = 1 + 1 =
              = 2

k=3    H(3,3) = H(3-1,min(1,3-1)) + H(3-2,min(2,3-2)) +
              + H(3-3,min(3,3-3) =
              = H(2,1) + H(1,1) + H(0,0) =
              = 1 + 1 + 1 =
              = 3

Without going further (and without proof) it seems to me that H(k,k) = k.