[seqfan] Comtets triangles in convex polygons
Richard J. Mathar
mathar at mpia-hd.mpg.de
Thu Nov 29 11:13:57 CET 2018
Comtet's book of 1974 has a dubious expression on page 75,
exercise 8(5), as spelled out in
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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