[seqfan] Extending signature sequences for transcendental numbers

[Kerry Mitchell]

I've extended the idea of the signature sequence for transcendental
numbers.  To get the signature sequence of x, form y values:

y = a0 + a1x

for positive integers a0 and a1.  Sort the list by the value of y and the
sequence of a0's is the signature sequence.  If x is irrational, then the
sequence is a fractal sequence.

Transcendental numbers are not solutions to any polynomial equation with
integer coefficients.  I used that idea to create a series of sequences:

S1(x) is the sequence of a0's when sorting y = a0 + a1x.  (S1 is the
standard signature sequence.)
S2(x) is the sequence of a0's when sorting y = a0 + a1x + a2x^2.
S3(x) is the sequence of a0's when sorting y = a0 + a1x + a2x^2 + a3x^3.
etc.
Sn(x) is the sequence of a0's when sorting y = a0 + a1x + a2x^2 + ... +
anx^n.

Then, I defined T(x) to be the limit of Sn(x) as n goes to infinity.

For the few values I've investigated, it looks like T(x) is well defined and
is similar to, but different from, the regular signature sequence S1(x).

Is this a well known problem?  I didn't find any of my sequences in the
OEIS.  Can anyone provide some pointers?

Thanks,
Kerry Mitchell
-- 
lkmitch at gmail.com
www.kerrymitchellart.com
http://spacefilling.blogspot.com/