What you see here is the sequence of moves to solve the Towers of Hanoi puzzle. The sequence goes up to 2 to the power of the number of rings in your puzzle. The picture here is the sequence for 8 discs, so 256 moves. “If you plot log_2 (binary log), this shape (in red) would be under that log plot, except for the powers of 2 which would be exactly on the log plot!” Simon has noticed.
Simon’s earlier video about the math behind the Towers of Hanoi puzzle: