A small factors sequence

[Creighton Dement]

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> Date: Sun,  4 Dec 2005 21:08:50 +0100
> Subject: Re: A small factors sequence
> From: Gerald McGarvey <Gerald.McGarvey at comcast.net>
> To: "Creighton Dement" <crowdog at crowdog.de>,seqfan at ext.jussieu.fr

> An observation about the sequence a(n):
> If a difference 'triangle' is formed by repeatedly taking the
> differences b(n+6) = a(n+6) - a(n)
> c(n+12) = b(n+12) - b(n+6) etc.
> then the 7th iteration of these differences are
> 279936, -279936, -559872, -279936, 279936, 559872, 279936, -279936 the
> 8th differences are zero, and the 6th iterations also have small
> factors, but not all the terms in this 'triangle' have small factors.
> I don't know if this
> can help at all to explain the small factors, and I'm not following
> how the sequence comes from the formula (I don't have Maple).
> A curiosity: no term before a(43) is divisible by 7, but a(43) through
> a(49) are all divisible by 7.
> 
> Sincerely,
> Gerald

Dear Gerald and Seqfans, 

That's neat! Thanks very much. Regarding the definition of the sequence:
sorry, I should have just said "the sequence is the expansion of 
(1-16*x+56*x^3-140*x^4+28*x^6-16*x^7+28*x^2+56*x^5+x^8)/(x^2-x+1)^8 "
instead of making it look like it was some code written especially for
Maple.

You've pointed to one curiousity about the terms of the sequence before
a(43) not being divisible by 7 and then 7 terms in a row are divisible
by 7. I think there are several of these "curiousities" regarding other
numbers as well- though I haven't had the chance to look at it much in
detail.  It is perhaps also interesting to note that FAMP actually
returned a different sequence:

1, -8, -144, -720, 7920, 190080, 1235520, -17297280, -518918400,
-4151347200, 70572902400, 2540624486400

With exponential generating function:
(1-16*x+56*x^3-140*x^4+28*x^6-16*x^7+28*x^2+56*x^5+x^8)/(x^2-x+1)^8

i.e. same as above (the n-th term is multiplied by n!) 

Note to Gerald: actually, it's interesting that *you* should respond: I
was using the same element (1/2)('i + 'j + 'k + e) to generate the
sequence that we used before when graphing your "diamonds". The
algorithm is very easy to implement- if you wish, I'll send it to you
when I have more time on my hands. 


Sincerely, 
Creighton



> At 07:57 PM 12/3/2005, Creighton Dement wrote:
> 
> > Dear Seqfans,
> > 
> > I came across a sequence given in Maple by
> >
seriestolist(series((1-16*x+56*x^3-140*x^4+28*x^6-16*x^7+28*x^2+56*x^5+x^8)/(x^2-x+1)^8,
> > x=0,50));
> > 
> > 1, -8, -72, -120, 330, 1584, 1716, -3432, -12870, -11440, 19448,
> > 63648, 50388, -77520, -232560, -170544, 245157, 692208, 480700,
> > -657800, -1776060, -1184040, 1560780, 4071600, 2629575, -3365856,
> > -8544096, -5379616, 6724520, 16695360, 10295472, -12620256,
> > -30761874, -18643560, 22481940, 53956656, 32224114, -38320568,
> > -90759240, -53524680, 62891499, 147258144, 85900584, -99884400,
> > -231550200, -133784560, 154143080, 354201120, 202927725, -231917400
> > 
> > By chance, I noticed that each term of the above sequence (in the
> > range given) has small factors.  Pardon me for asking a vague
> > question, but does the above sequence illustrate anything of
> > interest in particular? 
> > p.s. I have visitors at the moment and, unfortunately, won't be able
> > to look at the above again in detail until next week (though I am
> > quite fortunate to have visitors!)
> > 
> > Sincerely,
> > Creighton
> > 
> > -It's a shame when the girl of your dreams would still rather be
> > with someone else when you're actually in a dream.
> > 
> 
>