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

Charles Greathouse charles.greathouse at case.edu
Tue May 13 17:21:29 CEST 2014

```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
>
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>
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>
```