Fractal Sequence

Given an infinitive sequence  with associative array , then  is said to be a fractal sequence

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.

REFERENCES:

Kimberling, C. "Fractal Sequences and Interspersions." Ars Combin. 45, 157-168, 1997.