Fractal Sequence

Given an Infinitive Sequence $\{x_n\}$ with associated 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 Infinitive Sequence

References

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