[seqfan] Re: Comtets triangles in convex polygons
Ray Chandler
rayjchandler at sbcglobal.net
Tue Dec 4 22:50:26 CET 2018
Richard,
It looks to me that sequence A217748 gives the desired terms for Exercise 8(5) although I can't make sense of the formula yet.
a(3) = 1 since by definition, every side of the interior triangles is either a side or a diagonal - there need not be any diagonals to have a qualifying triangle.
There also appears to be some confusion about the n_3 notation. Referring to page 6 in the document, it is clear that it is intended to be a falling factorial rather than a rising factorial or Pochhammer symbol. This does not explain what the formula represents.
Ray
> -----Original Message-----
> From: SeqFan [mailto:seqfan-bounces at list.seqfan.eu] On Behalf Of Neil
> Sloane
> Sent: Thursday, November 29, 2018 8:31 AM
> To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
> Subject: [seqfan] Re: Comtets triangles in convex polygons
>
> In my copy of Comtet's 1974 book I have a pencilled note in the margin saying
> "not integral"
>
> All the other sequences have A-numbers written next to them, but because
> these were not integers I did not add them to the OEIS.
>
> I also have the original 2-volume French edition, Analyse combinatoire,
> which has the same Problem 8, but it does not have the sub-problem 8(5).
>
>
>
> Best regards
> Neil
>
> Neil J. A. Sloane, President, OEIS Foundation.
> 11 South Adelaide Avenue, Highland Park, NJ 08904, USA.
> Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ.
> Phone: 732 828 6098; home page: http://NeilSloane.com
> Email: njasloane at gmail.com
>
>
>
> On Thu, Nov 29, 2018 at 5:14 AM Richard J. Mathar <mathar at mpia-
> hd.mpg.de>
> wrote:
>
> > Comtet's book of 1974 has a dubious expression on page 75, exercise
> > 8(5), as spelled out in
> >
> http://oeis.org/wiki/Sequences_from_Comtet%27s_Advanced_Combinatori
> cs .
> > This deals with triangles inside convex polygons constructed by diagonals.
> >
> > What is the correct result? Do we just have to wrap the entire
> > polynomial in a round() to get correct numbers, which would have been
> > written as ||n_3(n^3+18*n^2+43*n+60)/6!|| with two double bars in the
> > book, to get 1, 7, 31, 97, 247,... for n>=1? This is not correct,
> > because for n<=3 there are no diagonals in the polygons, so there are
> > none of these triangles.
> > That sort of error could be guessed from a similar observation in A321988.
> > Or are there other typos, as in A321986?
> > I'd expect to be 4 of these triangles in the 4-gon, defined by the two
> > ways of defining a diagonal, and each of these has such a triangle on
> > either side. This naively produces n*(n-3)/2 = A000096(n-3) for n>4,
> > since there are n points for the start of a diagonal, n-3 candidates
> > for the other end of the diagonal, a factor of 2 for the two sides of
> > the diagonal if n=4, and a division through 2 for the double counting
> > of the two directions of the diagonal. This is obviously not what
> > Comtet had in mind.
> >
> > (Link to the book: at the end of that OEIS wiki page.
> > The Pochhammer symbol (n)_3 is introduced on page 6.
> > Related sequences A217753, A217754, A277652)
> >
> > --
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> >
>
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