1. If , then there exists such that ,
2. If , then, for every , there is exactly one such that .
(As and range through , the array , called the associative array of , ranges through all of .) An example of a fractal sequence is 1, 1, 1, 1, 2, 1, 2, 1, 3, 2, 1, 3, 2, 1, 3, ....
If is a fractal sequence, then the associated array is an interspersion. If is a fractal sequence, then the upper-trimmed subsequence is given by , and the lower-trimmed subsequence is another fractal sequence. The signature of an irrational number is a fractal sequence.