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

[David Applegate]

For the analog of the Motzkin numbers (A001006), the answer is A001006 - 
the reason Neil got 8 for n=3 is that he was allowing the endpoints to 
meet, but A001006 doesn't.  The analogous sequence allowing endpoints to 
meet is A054726.

For any figure that doesn't depend on the even spacing of the points, a 
figure of half-circles on the line is equivalent to a corresponding 
figure of chords of a circle.  They can be viewed as figures in the 
Poincare half-plane and Beltrami-Klein disk models of hyperbolic 
geometry.  And figures in one can be projected onto figures in the other 
via (for example) projection onto the Riemann sphere (see 
https://en.wikipedia.org/wiki/Hyperbolic_geometry, for example).  This 
doesn't apply to A290447 and A006561 because the projection doesn't 
preserve "equal spacing".
-Dave

On 8/11/2017 11:38 AM, Neil Sloane 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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