The Koi patterns that you encounter in the pond are connected to the pattern of Koi that you’re swimming around.
So for the first pattern (Sutato), you will only encounter other Sutato Koi for the first 4 Sutato patterns you have in your collection. For every encounter after that, there’s a chance that you’ll encounter the next pattern up.
As you continue to collect more and more Sutato, the chance of encountering the next highest level increases (which can make collecting those last few in a set difficult).
So if you’re stuck, wondering why you can’t see any of the next level of Koi, then you need to collect a few more of your current pattern before a new pattern will be unlocked. This also means that you can’t really skip collecting a pattern (say if you didn’t like the pattern) since you’ll need to get at least 4 before the next pattern will become available.