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%I #30 Jan 01 2024 22:46:58
%S 1,4,11,25,50,91,154,234,375,550,769,1079,1456,1783,2500,3196,3987,
%T 5016,6175,7348,9086,10879,12836,15250,17875,20682,24129,27811,31419,
%U 36425,41416,46664,52921,59500,66489,74481,82954,91807,102050,112750,123700,136654
%N Number of regions in a polygon whose boundary consists of n+2 equally spaced points around the arc of a semicircle. See Comments for precise definition.
%C A semicircular polygon with n+2 points is created by placing n+2 equally spaced vertices along a semicircle's arc, which includes the two end vertices. Now connect every pair of vertices by a straight line segment. The sequence gives the number of regions in the resulting figure.
%C Note that there is a curious relationship between the terms of this sequence and the number of regions in the 'general position' polygon given in A006522. They are a match except for every third term starting at a(8) = 234. Examining the images for n = 8,11,14,17 shows that these polygons have interior points at which three or more lines intersect, while the other n values have no such intersection points. Such multi-line intersection points will reduce the number of regions as compared to the general position polygon which has no multi-line intersection points. This is reflected by the terms in this sequence being lower than the corresponding value in A006522 for n = 8,11,14,... . Why every third value of n in this sequence starting at n = 8 leads to polygons having multiple line intersection points while other values of n do not is currently not known.
%H Lars Blomberg, <a href="/A333643/b333643.txt">Table of n, a(n) for n = 1..80</a>
%H Scott R. Shannon, <a href="/A333643/a333643_4.png">Illustration for n = 2</a>.
%H Scott R. Shannon, <a href="/A333643/a333643_5.png">Illustration for n = 3</a>.
%H Scott R. Shannon, <a href="/A333643/a333643_6.png">Illustration for n = 4</a>.
%H Scott R. Shannon, <a href="/A333643/a333643_7.png">Illustration for n = 5</a>.
%H Scott R. Shannon, <a href="/A333643/a333643_8.png">Illustration for n = 7</a>.
%H Scott R. Shannon, <a href="/A333643/a333643_9.png">Illustration for n = 10</a>.
%H Scott R. Shannon, <a href="/A333643/a333643_10.png">Illustration for n = 12</a>.
%H Scott R. Shannon, <a href="/A333643/a333643_11.png">Illustration for n = 15</a>.
%H Scott R. Shannon, <a href="/A333643/a333643_14.png">Illustration for n = 17</a>.
%H Scott R. Shannon, <a href="/A333643/a333643_15.png">Illustration for n = 19</a>.
%H Scott R. Shannon, <a href="/A333643/a333643_17.png">Illustration for n = 20</a>.
%H Scott R. Shannon, <a href="/A333643/a333643_12.png">Illustration for n = 10 with random distance-based coloring</a>.
%H Scott R. Shannon, <a href="/A333643/a333643_13.png">Illustration for n = 15 with random distance-based coloring</a>.
%H Scott R. Shannon, <a href="/A333643/a333643_16.png">Illustration for n = 19 with random distance-based coloring</a>.
%H Scott R. Shannon, <a href="/A333643/a333643_18.png">Illustration for n = 20 with random distance-based coloring</a>.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Semicircle">Semicircle</a>.
%Y Cf. A333642, A333519, A007678, A290865, A092867, A331452, A331929, A331931.
%K nonn
%O 1,2
%A _Scott R. Shannon_ and _N. J. A. Sloane_, Mar 31 2020
%E More terms from _Lars Blomberg_, Apr 20 2020
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