Quaternion "loop" sequence

[Creighton Dement]

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> Date: Mon, 13 Jun 2005 03:00:12 +0200
> Subject: Re: Quaternion "loop" sequence
> From: Gerald McGarvey <Gerald.McGarvey at comcast.net>
> To: "Creighton Dement" <crowdog at crowdog.de>,seqfan at ext.jussieu.fr

> 
> For this sequence, a(n) - a(n-6) has some patterns, especially at the
> end: -8 -4 4 8 4 -4 -8 -4 4 8 4 -4 -8 -4 4 8 4 -4
> also sum(a(n)+...+a(n-5)) has this pattern at the end:
> -2 -10 -14 -10 -2 2 -2 -10 -14 -10 -2 2 -2 -10 -14 -10 -2 2 -2
> 
 
Hi Gerald and seqfans, 

Thank you for that interesting observation which I probably wouldn't
have found on my own.  I placed the first 1000 terms of the sequences
(a(n)) and (a(n)-a(n-6)) under
http://www.crowdog.de/Triton/QuaLoop6Grid.html

In addition to the patterns formed, I would  also be 
interested in stating the following informal conjecture mathematically
precise: only a finite number of "essentially different" "plot patterns"
can be formed when using elements of the form 
a'i + b'j + c'k + de as a seed under same 
the rules (a(n)) (see previous post). Of course, before the conjecture
can be made precise, one has to define "essentially different" as well
as "plot pattern". 

Sorry if it takes me a while to respond- I'm having difficulties after a
computer crash.

Sincerely, 
Creighton