Signature sequences
N. J. A. Sloane
njas at research.att.com
Wed May 28 23:23:56 CEST 2003
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
>
More information about the SeqFan
mailing list