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

Neil Sloane njasloane at gmail.com
Fri Aug 11 17:38:13 CEST 2017

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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