Scatterplot Question

[franktaw at netscape.net]

What you are looking at is basically a graphical representation of
the continued fraction for pi (or more precisely, 2*pi).  Robert's
graph is, I think, showing the famous good approximation pi ~
355/113; Leroy's graph seems to represent several convergents
between 22/7 and 355/113.  At other scales, you will see
other patterns.

Note that comparisons of sin n, as used for A141330, are
equivalent to comparisons with a saw-tooth wave; so we really
are just looking at n mod 2pi.

Franklin T. Adams-Watters

-----Original Message-----
From: Robert Israel <israel at math.ubc.ca>

It does seem to be a real mathematical effect, a sort of "Moire 
pattern".
But this is only a hint of a pattern that emerges at the next scale.

Here's the plot I obtained when I extended the sequence A141330 to 
10000
terms.

<http://www.math.ubc.ca/~israel/problems/A141330.gif>

Cheers,

Robert Israel

On Fri, 1 Aug 2008, Leroy Quet wrote:
> ... I used the EIS' plot function on these sequences. And I was 
rather surprised.
>
> Are the irregular hexagons that appear in the scatterplots of these 
sequences:
> (a) just an interference artifact of the plotting software or of my 
browser?
> Or (b) a result purely of the mathematics of the sequences?
>
> (If the answer is (a), then the plots may very well look different to 
other people viewing them than they do to me.)
>
> Hopefully the answer is (b), but I doubt it.
>
> Thanks,
> Leroy Quet





A comparison of A122527 and A121231 shows that they

http://research.att.com/~njas/sequences/?q=id:A122527|id:A121231

Is this some coincidence enforced by the odd-prime dimension
of the row count? (See also A052264) Can these numbers
be smaller than those in A055084 for the 6xn matrices although
the latter are counted with restrictions?

Richard