[seqfan] Re: Two missing sequences based on the "semicircles on a line" problem A290447

[Allan Wechsler]

If you look at the diagram for n = 9 on Hasler's web-app, and just trace
your finger down the axis of symmetry, you'll count 9 regions straddling
the axis. That implies that a(9) is odd (because there are certainly an
even number of off-axis regions). That in turn means that A055795 can't be
right, because A055795(9) = 162 is even.

Indeed, my eyeball count for n = 9 (please, somebody repeat my work!) is
161. I suspect that A055795 is correct for a set of semicircles that are
perturbed to prevent intersections of more than two; the 9-point case, as
has been noted above, has a single triple point. If this conjecture is
true, then A055795(n) - a(n) is an interesting measure of "multiplicity"; 0
up to n = 8, 1 for n = 9, and (eyeballing again) 5 for n = 10.

If all this guesswork is correct, then a(10) = A055795(10) - 5 = 250.

I have a vague outline for a program to automate the count, and would be
happy to talk to any would-be coders off-list.

On Fri, Aug 11, 2017 at 5:12 PM, Rob Pratt <Rob.Pratt at sas.com> wrote:

> The first match is:
> http://oeis.org/A055795
>
> It has formula:
> a(n) = A000127(n)-1
>
> And http://oeis.org/A000127 has this description:
> A000127   Maximal number of regions obtained by joining n points around a
> circle by straight lines. Also number of regions in 4-space formed by n-1
> hyperplanes.
> (Formerly M1119 N0427)
>
> Does "unrolling" the circle to a straight line change the straight lines
> to semicircles and lose one region that becomes unbounded, so that accounts
> for the -1?
>
> And does uniform spacing yield the maximum number of regions?
>
> -----Original Message-----
> From: SeqFan [mailto:seqfan-bounces at list.seqfan.eu] On Behalf Of Allan
> Wechsler
> Sent: Friday, August 11, 2017 4:21 PM
> To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
> Subject: [seqfan] Re: Two missing sequences based on the "semicircles on a
> line" problem A290447
>
> EXTERNAL
>
> For Neil's region-counting question, I used Hasler's web-app at A290447
> (Lovely -- we need more of this sort of thing!) and just counted regions by
> eye, so my results may not be reliable. For n=7 I get 56 regions; for n=8 I
> get 98 regions.
>
> 0,1,3,7,15,30,56,98 still has 2 matches. I don't know if I trust myself to
> count the regions for n=9.
>
> On Fri, Aug 11, 2017 at 11:38 AM, Neil Sloane <njasloane at gmail.com> wrote:
>
> > Dear Seq Fans,
> > As before, start with n (labeled)
> > points equally spaced along a line,
> > and draw (upwards) semicircles between pairs of these points.
> >
> > If you join every pair of points by a semicircle, the number of
> > regions is A290447, as discussed earlier.
> > This is an analog of A006561, which has n equally spaced points on a
> > circle.
> >
> > But what if we count the (closed) regions? This will be the analog of
> > A7678. I get, for n >= 1, 0,1,3,7,15,30. (Too many matches, need help
> > to extend it)
> >
> > And what if we ask for the analog of the Motzkin numbers, A001006? In
> > this context it would be:
> > The total number of ways of drawing k (upwards) semicircles so that
> > they don't intersect each other (except at the baseline), summed over
> > k from 0 to n-choose-2.
> > For n=3 there are 8 possibilities, since any combination of the 3
> > semicircles works.
> >
> > If anyone cares to add these two sequences to the OEIS, please do so
> > (and post the A-numbers here)
> >
> > --
> > Seqfan Mailing list - http://list.seqfan.eu/
> >
>
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