Signature Sequence

Let $\theta$ be an Irrational Number, define $S(\theta)=\{c+d\theta: c,d\in\Bbb{N}\}$, and let $c_n(\theta)+d_n \theta(\theta)$ be the sequence obtained by arranging the elements of $S(\theta)$ in increasing order. A sequence $x$ is said to be a signature sequence if there Exists a Positive Irrational Number $\theta$ such that $x=\{c_n(\theta)\}$, and $x$ is called the signature of $\theta$.

The signature of an Irrational Number is a Fractal Sequence. Also, if $x$ is a signature sequence, then the Lower-Trimmed Subsequence is $V(x)=x$.

References

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