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A371374 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 regions formed.

%I A371374 #18 Jun 02 2025 22:33:57
%S A371374 1,1,9,9,51,48,211,217,612,651,1475,1248,3017,3193,5415,5793,9623,
%T A371374 9000,15429,15901,23352,24311,34501,33840,49001,50337,67365,69385,
%U A371374 91003,87720,120219,123169,155430,159291,198521,198792,250121,256121,310635,317441,382203,382032,465691,473573
%N A371374 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 regions formed.
%C A371374 See A371373 and A371254 for further information. The details of the number of regions with k sides is given in A371376.
%H A371374 Scott R. Shannon, <a href="/A371374/a371374.jpg">Image for n = 3</a>. In this and other images the initial circle and n points are shown in white for clarity.
%H A371374 Scott R. Shannon, <a href="/A371374/a371374_1.jpg">Image for n = 4</a>.
%H A371374 Scott R. Shannon, <a href="/A371374/a371374_2.jpg">Image for n = 5</a>.
%H A371374 Scott R. Shannon, <a href="/A371374/a371374_3.jpg">Image for n = 6</a>.
%H A371374 Scott R. Shannon, <a href="/A371374/a371374_4.jpg">Image for n = 7</a>.
%H A371374 Scott R. Shannon, <a href="/A371374/a371374_5.jpg">Image for n = 8</a>.
%H A371374 Scott R. Shannon, <a href="/A371374/a371374_6.jpg">Image for n = 9</a>.
%H A371374 Scott R. Shannon, <a href="/A371374/a371374_7.jpg">Image for n = 10</a>.
%H A371374 Scott R. Shannon, <a href="/A371374/a371374_8.jpg">Image for n = 11</a>.
%H A371374 Scott R. Shannon, <a href="/A371374/a371374_9.jpg">Image for n = 12</a>.
%H A371374 Scott R. Shannon, <a href="/A371374/a371374_10.jpg">Image for n = 15</a>.
%H A371374 Scott R. Shannon, <a href="/A371374/a371374_11.jpg">Image for n = 20</a>.
%H A371374 Scott R. Shannon, <a href="/A371374/a371374_12.jpg">Image for n = 24</a>.
%H A371374 Scott R. Shannon, <a href="/A371374/a371374_13.jpg">Image for n = 30</a>.
%F A371374 a(n) = A371375(n) - A371373(n) + 1 by Euler's formula.
%Y A371374 Cf. A371373 (vertices), A371375 (edges), A371376 (k-gons), A371377 (vertex crossings), A371254, A371253, A006533, A358782, A359046.
%K A371374 nonn
%O A371374 1,3
%A A371374 _Scott R. Shannon_, Mar 20 2024