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

[Allan Wechsler]

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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>