Is there a simple increasing sequence $\{ x_n \}$ of positive integers such that $x_n \mbox{ mod } b$ is the $n$-th digit of some known mathematical irrational constant in some integer base $b$? Perhaps a sequence simple enough that it is possible to prove that the digits in question are evenly distributed?
Of course the sequence $\lfloor b^n\pi \rfloor$ fits the bill for the number $\pi$ but it is intractable. A sequence such as $y_n = \lfloor b^n\pi \rfloor -b \phi_n$ where $\phi_n$ is an integer, such that $\{ y_n \}$ is smooth enough, for instance a very smoothly growing sequence of perfect squares?
I have spent years trying to solve this mystery using dynamical systems, ergodic stochastic processes and their equilibrium distributions (see here) but to no avail so far. I have made great progress no doubt, but I feel like I am traveling through space trying to reach the closest star besides the sun, and I am just barely outside the solar system.