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

[Rob Pratt]

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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