Signature sequences

[N. J. A. Sloane]

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
>