Jorg Arndt's example generating Pascal's triangle and a Fibonacci-like sequence was considerably cooler... another I invented quite a while back was 1000 / (500 + sqrt(249990)) = 1.00001000020000500014000420013200429014300... where the sequence 1,2,5,14,42,... is the Catalan numbers (which easily give you the central binomial coefficients). How did I do that? Well, again, one way is using generating functions. But the continued fraction of quadratic irrationals is ultimately periodic and once you know its period you can easily solve for the surd, so again, this is easy to construct from the desired sequence without any intelligence.