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I've come up with an idea of an integer sequence. It can be formulated (perhaps a bit loosely) as follows: For n points N(n) is the number of configurations where each point either lies on some circle or is a center of some circle. Each point lying on a circle can belong to only 1 circle and each center point can be the center of only 1 circle.

Then N(1) = 2 N(2) = 5 N(3) = 10 N(4) = 20 N(5) = 36 (I double checked N(5) but there is no guarantee that it is the right number)

Depending on how N(0) is defined (N(0)=0 or N(0)=1) this sequence very well can be A000712. However it appears that my description is new, so it easily can be another sequence.

all configurations for N(1), N(2), N(3) are shown here

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    $\begingroup$ oeis.org/… $\endgroup$ – Mr. Palomar Aug 10 at 11:24
  • $\begingroup$ Number of configurations of what, exactly? of points? of circles? $\endgroup$ – Gerry Myerson Aug 10 at 12:15
  • $\begingroup$ configurations of points. $\endgroup$ – A Z Aug 10 at 12:19
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    $\begingroup$ I'm not sure I understand your construction, it's a little too-loosely defined. Can you be more precise? $\endgroup$ – Amir Sagiv Aug 10 at 14:35
  • $\begingroup$ Frankly I think it is more or less intuitively clear if you look at the examples: i.stack.imgur.com/gdrBi.png Another problem that arises from this construction is to count the number of circles used in all the configurations for each n. This sequence goes like 2,8,22,54... and it is not in the OEIS. I do not in any way insist that it should be added though. $\endgroup$ – A Z Aug 10 at 20:42
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Your sequence is the same as the linked OEIS sequence. This is the

Number of partitions of $n$ into parts of two kinds.

In your case, the two kinds are circles for which the centre is occupied and circles for which the centre is not occupied. See the "example" section in the OEIS entry where you can match with your worked out example.

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