$$U_{690}(x, y) = \frac{\sqrt{\frac{8 \sqrt{3} x \sin{\left(\frac{\operatorname{asin}{\left(\frac{3 \sqrt{3} \sqrt{y}}{2} \right)}}{3} \right)}}{3 \sqrt{y}} + 1}}{2} + \frac{1}{2}$$

$$T_{690}(n, m, k) = \begin{cases}1&\text{if m+n=0 } ,\ \\\frac{k {\binom{k - n - 1}{n - 1}} {\binom{3 m + n - 1}{m}}}{2 m + n} \end{cases} $$