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Signature Sequence


Let theta be an irrational number, define S(theta)={c+dtheta:c,d in N}, and let c_n(theta)+thetad_n(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.

One can also define two extended signature sequences for positive rational theta by taking the c_n in increasing order or decreasing order. These can be considered signature sequences for theta-epsilon and theta+epsilon, respectively, where epsilon is an infinitesimal.

The signature of an irrational number or either signature of a rational number is a fractal sequence. Also, if x is a signature or extended signature sequence, then the lower-trimmed subsequence is V(x)=x. 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 S is an initial subsequence of some signature sequence, then S 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).


Portions of this entry contributed by Franklin T. Adams-Watters

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References

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

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Signature Sequence

Cite this as:

Adams-Watters, Franklin T. and Weisstein, Eric W. "Signature Sequence." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/SignatureSequence.html

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