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.

See also Associative Array ,

Infinitive
Sequence
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References Kimberling, C. "Fractal Sequences and Interspersions." Ars Combin. 45 , 157-168, 1997. Referenced on Wolfram|Alpha Fractal Sequence
Cite this as:
Weisstein, Eric W. "Fractal Sequence."
From MathWorld --A Wolfram Web Resource. https://mathworld.wolfram.com/FractalSequence.html

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