Fractal Sequence

Given an infinitive sequence {x_n} with associative array a(i,j), then {x_n} is said to be a fractal sequence

1. If i+1=x_n, then there exists m<n such that i=x_m,

2. If h<i, then, for every j, there is exactly one k such that a(i,j)<a(h,k)<a(i,j+1).

(As i and j range through N, the array A=a(i,j), called the associative array of x, ranges through all of N.) An example of a fractal sequence is 1, 1, 1, 1, 2, 1, 2, 1, 3, 2, 1, 3, 2, 1, 3, ....

If {x_n} is a fractal sequence, then the associated array is an interspersion. If x is a fractal sequence, then the upper-trimmed subsequence is given by lambda(x)=x, and the lower-trimmed subsequence V(x) is another fractal sequence. The signature of an irrational number is a fractal sequence.

See also

Associative Array, Infinitive Sequence

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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.

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