Signature Sequence
Let 
be an irrational number, define ![S(theta)=<span style=]()
{c+dtheta:c,d in N}" src="https://mathworld.wolfram.com/images/equations/SignatureSequence/Inline2.gif" style="height:15px; width:146px" />, and let 
be the sequence obtained by arranging the elements of 
in increasing order. A sequence 
is said to be a signature sequence if there exists a positive irrational number 
such that ![x=<span style=]()
{c_n(theta)}" src="https://mathworld.wolfram.com/images/equations/SignatureSequence/Inline7.gif" style="height:15px; width:63px" />, and 
is called the signature of 
.
One can also define two extended signature sequences for positive rational 
by taking the 
in increasing order or decreasing order. These can be considered signature sequences for 
and 
, respectively, where 
is an infinitesimal.
The signature of an irrational number or either signature of a rational number is a fractal sequence. Also, if 
is a signature or extended signature sequence, then the lower-trimmed subsequence is 
. It has been conjectured that every sequence with both of these properties is a signature or extended signature sequence.
If every initial subsequence of a sequence 
is an initial subsequence of some signature sequence, then 
is either a signature sequence, an extended signature sequence, or one of the two limiting cases: all 1's, or the natural numbers (which could be regarded as signature sequences for zero and infinity).
REFERENCES:
Kimberling, C. "Fractal Sequences and Interspersions." Ars Combin. 45, 157-168, 1997.
