Explain the pattern and find the next five rows.

Hint: the symbols I chose have nothing to do with the pattern.

You're all very close, and I'll say that the next five rows have been correctly predicted by at least one person.  You've all noticed the little patterns, such as the pattern with prime numbered rows.  Think of a more unifying pattern that will cause the patterns you have observed.  It might be simpler than you expect.

I don't quite understand that algorithm, but it would seem to me that it doesn't successfully predict the rows already shown.

As far as real hints, I think that the columns are equally important as the rows.