It appears J.Shallit & I both came up with approx. same idea -- generating functions -- to explain Simon Plouffe's weird numbers. If x were something like 2^(-10000) and if the coeffs in the generating function grew like, say, 1.5^n, then this could explain nonrandomness out to about the (10^8)th bit, corresponding to about the (10^4)th series-coefficient. This is approximately right (i.e. that is about how far Plouffe's nonrandomness extended), so it looks to me like my/Shallit's explanations were correct. Now, thinking a bit more: can we now purposefully design even more dramatic examples than Plouffe had found? For that, there are several avenues to pursue. 1. If the recurrence (which I have not yet worked out) obeyed by the generating function coefficients happens to be a binary-friendly recurrence, e.g. itself involving coefficients which are, say, powers of 2, then this might be able to produce even-further-extending nonrandom patterns. 2. Another (related) possible avenue to look at, is double generating functions F(x,y). 3. If we can design the function F(x) to get series-coefficient growth which is only polynomial(n), not exponential, then we will get nonrandomness extending tremendously further than in Plouffe's numbers. The following kind of function of x: F(x,a,b) = (1-x^a)^(-b) where a,b, are positive integers, and sums and products of such functions, all have got series coeffs which automatically grow only polynomially with n and automatically are all-integer. Also, if b is half-integer such as b=5/2, then F(2*x,a,b) or F(4*x,a,b) have all-integer coefficients too, but that grow exponentially. The growth constants are however exactly powers of 2, which acts a lot like non-growth (i.e. growthconst=1) from the view of binary-friendliness. Therefore these F's will yield examples far more dramatically-long nonrandom than Plouffe's examples.