[seqfan] Re: Two missing sequences based on the "semicircles on a line" problem A290447
David Applegate
david at bcda.us
Sat Aug 12 04:34:09 CEST 2017
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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