Signature sequences

[Rob Arthan]

Dear All,


In case it's of interest to people interested in these signature sequences, 
implicit in the discussion of my sequence A064034 is an efficient algorithm 
for testing whether i + j*x > i' + j'*x for arbitrary integral i and j and 
for x any quadratic surd. The paper referenced there gives more details in 
the case when x = sqrt(2) and gives a hint about the general case in a 
footnote. I can supply more information to anyone who wants it.

This may look like a shameless plug, but that's because it is!

Regards,

Rob.

On Wednesday 28 May 2003 10:23 pm, N. J. A. Sloane wrote:
> Kerry Mitchell just posted a message to the math fun list
> which relates to sequences, and I will append it below.
>
> The OEIS prsently contains these examples:
>
> signature sequences (1): A007336 A007337 A023115 A023116 A023117 A023118
> A023119 A023120 A023121 A023122 A023123 A023124 signature sequences (2):
> A023125 A023126 A023127 A023128 A023129 A023130 A023131 A023132 A023133
> A023134 A035796
>
> But the sig. seq. for phi seems to be missing - maybe someone
> could send it in.
>
> NJAS
>
> >Date: Wed, 28 May 2003 18:57:04 +0000
> >
> >Lately, I've been playing with the signature of irrational numbers.  My
> >understanding, from what I've read on the web, is this:  For a positive
> >irrational number x, form the numbers y = i + j*x, where i and j are both
> >positive integers.  Since x is irrational, no 2 y values will be the same
> > for different i and j.  Arrange the ys by size, then the sequence of i
> > values is a fractal sequence, and the signature of x.
> >
> >For x = phi ~ 1.618034, the first few entries are:
> >i	j	y
> >1	1	2.618033989
> >2	1	3.618033989
> >1	2	4.236067977
> >3	1	4.618033989
> >2	2	5.236067977
> >4	1	5.618033989
> >1	3	5.854101966
> >3	2	6.236067977
> >5	1	6.618033989
> >
> >And the signature begins: 1, 2, 1, 3, 2, 4, 1, 3, 5.  If you strike the
> > first occurence of every integer in the sequence, you get the original
> > sequence back, which makes this a fractal sequence.
> >
> >My questions are:
> >
> >- What about the j sequence?  From what I've seen experimentally, it seems
> > to be a fractal sequence, too.  Why is the signature the i sequence as
> > opposed to the j sequence?  What's known about the relation of the j
> > sequence to the i sequence?
> >
> >- For a limited set of integers (both i and j run from 1 to 50), I plotted
> > i vs. j, and the result was very interesting (I thought).  You can find
> > the picture here:
> >
> >http://www.fractalus.com/kerry/sigofphi.html
> >
> >The plot is one continuous zig-zag line which seems to never cross itself.
> >But, the angle of the line changes slightly, causing some areas to bunch
> > up and appear darker, and others to spread out and appear lighter.  The
> > overall effect is of a series of rectangles drawn in different shades of
> > gray.  Can anyone point me to other work that has been done on this?
> >
> >Thanks,
> >Kerry Mitchell
> >--
> >lkmitch at att.net
> >www.fractalus.com/kerry