TOPICS

# 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.

Associative Array, Infinitive Sequence

## Explore with Wolfram|Alpha

More things to try:

## References

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

Fractal Sequence

## Cite this as:

Weisstein, Eric W. "Fractal Sequence." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/FractalSequence.html