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Number of regions after n iterations of constructing circles from all current vertices using only a compass, starting with one vertex. See the Comments.
7

%I #45 Jun 28 2024 03:42:00

%S 0,1,3,21,7169

%N Number of regions after n iterations of constructing circles from all current vertices using only a compass, starting with one vertex. See the Comments.

%C See A359569 for further details and images.

%H Scott R. Shannon, <a href="/A359570/a359570_3.jpg">Image for 3-region figure</a> (black background). In this and other images the vertices used to construct the circles are shown as white dots.

%H Scott R. Shannon, <a href="/A359570/a359570_7.jpg">Image for 3-region figure</a> (white background).

%H Scott R. Shannon, <a href="/A359570/a359570_1.jpg">Image for 21-region figure</a> (black background).

%H Scott R. Shannon, <a href="/A359570/a359570_4.jpg">Image for 21-region figure</a> (white background).

%H Scott R. Shannon, <a href="/A359570/a359570_2.jpg">Image for 7169-region figure</a> (black background).

%H Scott R. Shannon, <a href="/A359570/a359570_6.jpg">Image for 7169-region figure</a> (white background).

%H N. J. A. Sloane, New Gilbreath Conjectures, Sum and Erase, Dissecting Polygons, and Other New Sequences, Doron Zeilberger's Exper. Math. Seminar, Rutgers, Sep 14 2023: <a href="https://vimeo.com/866583736?share=copy">Video</a>, <a href="http://neilsloane.com/doc/EMSep2023.pdf">Slides</a>, <a href="http://neilsloane.com/doc/EMSep2023.Updates.txt">Updates</a>. (Mentions this sequence.)

%F For n >= 3, a(n) = A359571(n) - A359569(n) + 1 by Euler's formula.

%Y Cf. A359569 (vertices), A359571 (edges), A359619 (k-gons), A359253, A359046, A358782.

%K nonn,more,hard

%O 1,3

%A _Scott R. Shannon_, Jan 06 2023