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A371373 Place n equally spaced points around the circumference of a circle and then, for each pair of points, draw two distinct circles, whose radii are the same as the first circle, such that both points lie on their circumferences. The sequence gives the total number of vertices formed.

%I A371373 #26 Jun 02 2025 22:33:40
%S A371373 1,2,4,4,25,19,140,144,460,500,1210,901,2587,2758,4696,5136,8687,7831,
%T A371373 14136,14600,21610,22572,32246,31033,46125,47450,63748,65772,86565,
%U A371373 82051,114824,117760,148930,152796,190820,189973,241055,247038,300028,306840,369943,367711,451586,459448
%N A371373 Place n equally spaced points around the circumference of a circle and then, for each pair of points, draw two distinct circles, whose radii are the same as the first circle, such that both points lie on their circumferences. The sequence gives the total number of vertices formed.
%C A371373 The number of circles that cross to form the intersections follows a similar pattern to that seen in A371254; see that sequence for further information. The details of the crossing counts are given in A371377.
%H A371373 Scott R. Shannon, <a href="/A371373/a371373_5.jpg">Image for n = 3</a>. In this and other images the initial circle on which the n points are placed is drawn in a thicker white line for clarity.
%H A371373 Scott R. Shannon, <a href="/A371373/a371373_6.jpg">Image for n = 4</a>.
%H A371373 Scott R. Shannon, <a href="/A371373/a371373_7.jpg">Image for n = 5</a>.
%H A371373 Scott R. Shannon, <a href="/A371373/a371373_8.jpg">Image for n = 6</a>.
%H A371373 Scott R. Shannon, <a href="/A371373/a371373_9.jpg">Image for n = 7</a>.
%H A371373 Scott R. Shannon, <a href="/A371373/a371373_10.jpg">Image for n = 8</a>.
%H A371373 Scott R. Shannon, <a href="/A371373/a371373_11.jpg">Image for n = 9</a>.
%H A371373 Scott R. Shannon, <a href="/A371373/a371373_12.jpg">Image for n = 10</a>.
%H A371373 Scott R. Shannon, <a href="/A371373/a371373_13.jpg">Image for n = 11</a>.
%H A371373 Scott R. Shannon, <a href="/A371373/a371373_14.jpg">Image for n = 12</a>. Note the 6 circle intersections shown in blue.
%H A371373 Scott R. Shannon, <a href="/A371373/a371373_16.jpg">Image for n = 15</a>. Note the 5 circle intersections shown in green.
%H A371373 Scott R. Shannon, <a href="/A371373/a371373_15.jpg">Image for n = 20</a>.
%H A371373 Scott R. Shannon, <a href="/A371373/a371373_17.jpg">Image for n = 24</a>.
%H A371373 Scott R. Shannon, <a href="/A371373/a371373_18.jpg">Image for n = 30</a>. Note the 9 circle intersections shown in violet.
%F A371373 a(n) = A371375(n) - A371374(n) + 1 by Euler's formula.
%Y A371373 Cf. A371374 (regions), A371375 (edges), A371376 (k-gons), A371377 (vertex crossings), A371254, A007569, A358746, A331702.
%K A371373 nonn
%O A371373 1,2
%A A371373 _Scott R. Shannon_, Mar 20 2024