Contrariwise, to me you seem to be the confused party. I am contrasting the compression given by lzw or zip as discussed in the article, which would be unable

Contrariwise, to me you seem to be the confused party. I am contrasting the compression given by lzw or zip as discussed in the article, which would be unable to significantly compress the bit sequence of pi, with the "compression technique" of simply calculating the same sequence (in order!) with a short program. A lookup table is irrelevant, since we're talking about printing out the first 10 billion pi digits in order by a simple algorithm for computing them, not about anything involving calculating their frequency or probability. (In any case, since those digits pass statistical randomness tests, the frequency distribution wouldn't tell you much, which is why zip-style compression fails on pi.) The point I'm making is that running a very short program can print a 10 billion digit sequence which stats-based compression methods would fail to significantly compress. The digits of pi happen to be a good example of this, but there are likely plenty of others. Regards the second part of your comment, the notion of a "deterministic pattern" is intriguing, but what does it mean? If you mean "not random" then in the algorithmic information world that equates to "can be printed by a program shorter than itself". Obviously any (including random) sequence can be printed by some program, but the presence of regularity (or "patterns" if you like) in a sequence allows shorter encodings, either by creating a dictionary of recurring subsequences, as in lzw or zip style compression, or by some other means as in the calculation of pi's digits. I suggest you read up on AIT or Kolmogorov complexity at the wikipedia links I posted to get an idea of what my comment was actually about :)