[seqfan] Re: Aaron David Fairbanks : Pappus Chain Sequence (by way of moderator)

Aaron David Fairbanks unknownthingamabob at gmail.com
Wed May 14 03:14:10 CEST 2014


Okay, neat.

I submitted it at A242412 <https://oeis.org/A242412> in case anyone wants
to take a look if/when it's approved. Once again, this is my first time
doing any of this, so it might not be perfect.

The wording for the title was a little tricky; I ended up going with: "In a
Pappus chain, where r_U = r_V/2, the numerator of the simplified fraction:
r_V / the radius of an inscribed circle drawn tangent to C_U, the nth
circle in the chain, and the (n-1)th circle in the chain," which makes
little to no sense without a visual aid, so I added a "(see link)" and
added a link to this Wolfram Mathworld image:
http://mathworld.wolfram.com/images/eps-gif/PappusTangentChain_800.gif
Hopefully
that kind of reliance on an outside image is allowed?

It looks like my first message was caught in a filter and a moderator had
to approve it. Maybe it was the image I tried to include. Here is what it
looked like: http://i.imgur.com/jUlC9lq.png Sorry for the trouble anyway!

-Aaron Fairbanks


On Tue, May 13, 2014 at 11:21 AM, Charles Greathouse <
charles.greathouse at case.edu> wrote:

> Yes, this sequence should be submitted, and yes it does end up being a
> quadratic (and so I assume the one you give is correct). I did the
> calculations a few weeks ago.
>
> It would be great to get pictures of the original sangaku inspiring the
> problem.
>
> Charles Greathouse
> Analyst/Programmer
> Case Western Reserve University
>
>
> On Tue, May 13, 2014 at 9:21 AM, Olivier Gerard <olivier.gerard at gmail.com
> >wrote:
>
> > From: Aaron David Fairbanks <unknownthingamabob at gmail.com>
> > To: Seqfan <seqfan at list.seqfan.eu>
> > Date: Sat, 10 May 2014 18:54:19 -0400
> > Subject: Pappus Chain Sequence
> >
> > Hi Seqfans,
> >
> > I couldn't help but notice that the sequence exhibited by this
> Numberphile
> > video (http://www.youtube.com/watch?v=sG_6nlMZ8f4) has not been
> submitted
> > to the OEIS:
> >
> > 15, 23, 39, 63, 95...
> >
> > It is defined as the denominators of the sequence of ratios between blue
> > circles' radii and the outer circle's radius in the following diagram (a
> > Pappus chain where the radius of the inner circle is 1/2 the radius of
> the
> > outer circle):
> >
> > i.e. the largest blue circle has 1/15 the radius of the enclosing circle,
> > the 2nd largest has 1/23, and so on. If you can't view this image, the
> > Numberphile video does a good job explaining it.
> > ​
> >
> > I am a completely amateur math enthusiast, and I'm not at all familiar
> with
> > this field of math, but I was interested by this video. Is the sequence
> > worth noting? OEIS tells me it "appears to be + 4x^2 - 4x + 15," but I
> > don't know if this is true, or how I would go about proving it. Should I
> > work on submitting it anyway? If so, how would I give a more technical
> > definition of these "blue circles"?
> >
> > I'm sure there are many other related sequences in the same vein as this
> > one, that might be interesting to explore. I haven't looked into any of
> > them on my own yet, but feel free to give it a go if you see some
> > potential.
> >
> > This is my first exchange with the Seqfan mailing list, so let me know if
> > I'm doing anything completely wrong.
> >
> > Thanks everyone,
> > Aaron Fairbanks
> >
> > _______________________________________________
> >
> > Seqfan Mailing list - http://list.seqfan.eu/
> >
>
> _______________________________________________
>
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>



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