The following problem has already been discussed in mathoverflow, for example here
It is essentially an elliptic curve of rank $1$, a generator of the Mordell-Weil group being the point $Q = (\frac{4}{11},-\frac{1}{11},1)$. (I'm taking $(-1,1,0)$ as the identity.) Since this point is on the non identity component of the curve, looking at his multiples we find at some point a solution $nQ$ with positive coordinate. This happens for the first time for $n=9$. Here are the first $8$ multiples of $Q$. (In red the part of the curve with positive coordinates)
And here the first $9$ multiples
It's quite strange that the coordinates of $9Q$ are so bigger, especially since if we look at $10Q$, $11Q$, etc, the coordinates are quite small.
If we change the generator something similar happens, again the first positive solution is much bigger then the previous ones, although the difference is less evident. (For $Q=(\frac{11}{9},-\frac{5}{9},1)$ it happens for $n=13$.)
Is this just a coincidence?
