[seqfan] Re: fractal sequences

franktaw at netscape.net franktaw at netscape.net
Sun Nov 1 08:40:19 CET 2009

I  have a proof that I am writing up that such a sequence must be an
extended signature sequence.

As you know, the definition of a signature sequence is the values of i
in the sorted numbers of the form i + j theta for a fixed, irrational
theta, with i and j positive integers.  For an extended signature
sequence, we also allow theta to be rational.  There are two extended
signature sequences for each positive rational theta; one with the i's
in increasing order, the other with them in decreasing order.  These
can be regarded as signature sequences for theta - epsilon and theta +
epsilon, respectively, where epsilon is an infinitesimal.

-----Original Message-----
From: Kimberling, Clark <ck6 at evansville.edu>

I'm hoping someone can shed some light on fractal sequences - here's the
question:

If V(x) is the lower trim of a fractal sequence x, and if V(x)=x,
what can be said about x?

It is known that V(x)=x when x is a signature sequence of a positive
irrational number.

Recall that the UPPER trim of x is x itself, but the LOWER trim of x,
although
fractal sequence, need not be identical to x.

Example:  A0084531 is the signature sequence of the golden ratio:
1, 2, 1, 3, 2, 4, 1, 3, 5, 2, 4, 1, 6, 3, 5, 2, 7,...
The upper trim is what's left after deleting the first occurrence of
each n.
The lower trim is what's left after subtracting 1 from all terms and
then deleting all 0s.

Clark Kimberling