Great, Andre! I edited your LaTeX so the list doesn't shoot off the page.

Now everyone can see how curious this sequence is! It becomes even more interesting if one leaves out the first term and writes the rest like this:

$$\frac{1}{1},$$

$$\frac{1}{2}, \frac{2}{1},$$

$$\frac{1}{3}, \frac{3}{2}, \frac{2}{3}, \frac{3}{1},$$

$$\frac{1}{4}, \frac{4}{3}, \frac{3}{5}, \frac{5}{2}, \frac{2}{5}, \frac{5}{3}, \frac{3}{4}, \frac{4}{1},$$

$$\frac{1}{5}, \frac{5}{4}, \frac{4}{7}, \frac{7}{3}, \frac{3}{8}, \frac{8}{5}, \frac{5}{7}, \frac{7}{2}, \frac{2}{7}, \frac{7}{5}, \frac{5}{8}, \frac{8}{3}, \frac{3}{7}, \frac{7}{4}, \frac{4}{5}, \frac{5}{1},$$

$$\frac{1}{6}, \frac{6}{5}, \frac{5}{9}, \frac{9}{4}, \frac{4}{11}, \frac{11}{7}, \frac{7}{10}, \frac{10}{3}, \frac{3}{11}, \frac{11}{8}, \frac{8}{13}, \frac{13}{5}, \frac{5}{12}, \frac{12}{7}, \frac{7}{9}, \frac{9}{2}, \frac{2}{9}, \frac{9}{7}, \frac{7}{12}, \frac{12}{5}, \frac{5}{13}, \frac{13}{8}, \frac{8}{11}, \frac{11}{3}, \frac{3}{10}, \frac{10}{7}, \frac{7}{11}, \frac{11}{4}, \frac{4}{9}, \frac{9}{5}, \frac{5}{6}, \frac{6}{1},$$

and so on.