Andy Latto: But without the "silly" generating functions, do we have any explanation of why if we take the continued fraction expansion of 1.004009016025036049064081100, and truncate before the first large entry, we end up with a decimal that continues with

121144169196225256289324361... Surely this is not coincidence! Continued fractions give us an efficient way to find the smallest-denominatored fraction that starts with the first ten three-digit squares, but unless I'm missing something, they give no insight as to why this expansion should continue with the next 10 3-digit squares, and the generating function approach does. --well, true, but this should be obvious :) Of course the only integer sequences one is going to be able to generate and keep going for a long time are those with rational functions of x as generating functions. But if you have any particular such sequence and want to know the P/Q, the easiest fastest way to find P and Q is using the continued fraction. So no: they*do* give an insight as to why this expansion should continue with the next 10 3-digit squares, which is: via the continued fraction, you have FOUND the generating function of x, just evaluated at x=1000 (in the present case).