This might be an example of what SBG is looking for. The infinite product prod(1 - z^n) shows up in the theory of partitions. It's also a theta function. When you multiply it out, it turns out that most of the terms z^k have a coefficient of 0. When k is a pentagonal number j(3j+-1)/2, the coefficient of z^k is +-1, depending on whether j is even or odd. The power series begins 1 - z - z^2 + z^5 + z^7 - z^12 - z^15 + z^22 + z^26 - ... If you evaluate this at z = 1/2, and write the value in binary, you get a predictable pattern. It's not periodic, but it is predictable; and it's relatively cheap to compute single bits without reference to what's come before. This number is clearly irrational, and most likely transcendental, but I don't know if transcendence is proved. The non-zero portions may be thin enough that you can show it's not a quadratic irrational. Rich