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

[Neil Sloane]

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
intersection)

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 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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