The continued fraction: $${\cal R}_\eta(a,b) =\,\frac{{\bf \it a}}{\displaystyle \eta+\frac{\bf \it b^2}{\displaystyle \eta +\frac{4{\bf \it a}^2}{\displaystyle \eta+\frac{9 {\bf \it b}^2}{\displaystyle \eta+{}_{\ddots}}}}}$$
enjoys attractive algebraic properties such as a striking arithmetic-geometric mean relation and elegant links with elliptic-function theory. The fraction presents a computational challenge, which we could not resist.