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

[Charles Greathouse]

This is a great first submission!

I wonder if there's a way to emphasize, in the name or comments, the (IMO
amazing or at least nonobvious) fact that the areas are all unit fractions
-- not members of some quadratic extension nor even general fractions.

Charles Greathouse
Analyst/Programmer
Case Western Reserve University


On Tue, May 13, 2014 at 9:14 PM, Aaron David Fairbanks <
unknownthingamabob at gmail.com> wrote:

> 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
> > >
> > > _______________________________________________
> > >
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> > >
> >
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> >
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> >
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