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

Neil Sloane njasloane at gmail.com
Fri Aug 11 23:31:30 CEST 2017

Rob,  I don't think they can be the same, because of the basic fact
that for the semicircles problem, as the number
of points increases, there are points where x circles meet, where x
can be arbitrarily large. The effect of this is to reduce the number
of little regions.

I say this with confidence, because I have been trying to go the other
way: that is, take the arrangement of the semicircles, and try to curl
up the baseline to make a circle (or even a horseshoe)  in which the
incidences are unchanged, but the lines get "straighter".  And it
seems to be a dead end, nothing nice seems to happen.

So I expect that for my first question, the "no. of regions", it will
be less than A055795 = A000127 - 1 after a while. Starting probably at
n=9, where - as Allan will confirm - one would need very tiny eyeballs
to do the count by eye!  (n=9 is the first time there is a triple

Best regards

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 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
> 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)
>> --
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