Hello, as you may know, only the quadratic irrationals are predictable, (Lagrange theorem), the others being the exponential and some variants as well as some points with the Bessel function. I am afraid there is nothing else new in that area since approx. 1850. But... there is a pattern if we look at the generalized continued fraction of Pi, (or 1/Pi) . As far as I know : we know nothing from the CFE of the cube root of 2 either. There is a pattern in the CFE of Pi but it is with the generalized expansion of the continued fraction. It is more general but we lose the convergence property. And that's all that is known for this subject, nothing better has been found since Euler and the 1850s thereafter. Euler's traditional transformation between continuous fractions has not advanced an inch since. I have a program that expands a real number into 1000 different developments, I put all these transformations in my big table of 17.3 billion constants and it has not given anything until now. It is mainly based on variants of the greedy algorithm. Unless someone finds better than what Euler found: we're stuck. In other words , I have no idea where to look beyond : Pi = [3, 7, 15, 1, 292, 1, 1, 1, 2, 1, 3, 1, 14, 2, ...] ?? Best regards, Simon Plouffe