Indians find those sequences of non-negative real numbers $x_0, x_1,...$ mystical t hat satisfy $x_0 < 2021$, $x_{i+1} = \lfloor x_i \rfloor \{x_i\}$ for every $i \ge 0$, furthermore the sequence contains an integer different from $0$. How many sequences are mystical according to the Indians?
Problem
Source: 2021 Dürer Math Competition Finals Day 1 E4 E+2
Tags: floor function, Sequence, algebra