$$U_{687}(x, y) = \frac{\sqrt{2 x \left(\sqrt{4 y + 1} + 1\right) + 1}}{2} + \frac{1}{2}$$

$$T_{687}(n, m, k) = \begin{cases}1&\text{if n=0, m=0} ,\ \\\frac{k {\binom{k - n - 1}{n - 1}}}{n}&\text{if m=0, n>0} ,\ \\\frac{\left(-1\right)^{m - 1} k {\binom{k - n - 1}{n - 1}} {\binom{2 m - n - 1}{m - 1}}}{m}&\text{if m>0} \end{cases} $$