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%I #16 Dec 15 2022 06:47:00
%S 3,0,7,0,16,29,0,30,35,1,0,90,96,0,105,126,35,1,0,272,304,48,32,0,1,0,
%T 315,324,81,0,0,0,1,0,460,940,60,40,0,0,0,1,0,671,858,264,88,11,0,0,0,
%U 1,0,960,1656,108,48,0,1144,1807,559,130,13,0,0,0,0,0,1,0,1960,3136,448,168,0,14,0,0,0,0,0,1
%N Irregular table read by rows: T(n,k) is the number of k-gons formed, k>=2, among all circles that can be constructed on vertices of an n-sided regular polygon, using only a compass.
%C See A331702 and A359046 for further details and images.
%C Conjecture: the only value for n which leads to the creation of 2-gons is n = 2. Despite values for n mod 6 = 0 forming intersecting arcs at the center of the n-gon, these are cut by other circles and thus create 3-gons or 4-gons. This is in contrast to values of n mod 4 = 0 in A359009 which do lead to the creation of 2-gons at the center of the figure from similar arcs.
%H Scott R. Shannon, <a href="/A359061/a359061_2.jpg">Image for n = 17</a>.
%H Scott R. Shannon, <a href="/A359061/a359061.jpg">Image for n = 23</a>.
%H Scott R. Shannon, <a href="/A359061/a359061_1.jpg">Image for n = 24</a>.
%F Sum of row n = A359046(n).
%e The table begins:
%e 3;
%e 0, 7;
%e 0, 16, 29;
%e 0, 30, 35, 1;
%e 0, 90, 96;
%e 0, 105, 126, 35, 1;
%e 0, 272, 304, 48, 32, 0, 1;
%e 0, 315, 324, 81, 0, 0, 0, 1;
%e 0, 460, 940, 60, 40, 0, 0, 0, 1;
%e 0, 671, 858, 264, 88, 11, 0, 0, 0, 1;
%e 0, 960, 1656, 108, 48;
%e 0, 1144, 1807, 559, 130, 13, 0, 0, 0, 0, 0, 1;
%e 0, 1960, 3136, 448, 168, 0, 14, 0, 0, 0, 0, 0, 1;
%e 0, 2100, 3270, 945, 180, 15, 0, 0, 0, 0, 0, 0, 0, 1;
%e 0, 3088, 5584, 896, 368, 16, 16, 0, 0, 0, 0, 0, 0, 0, 1;
%e 0, 3400, 5814, 1513, 493, 85, 34, 0, 0, 0, 0, 0, 0, 0, 0, 1;
%e 0, 4536, 8712, 1224, 288, 54, 36;
%e 0, 5586, 8797, 2774, 665, 76, 152, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
%e 0, 7940, 12480, 2440, 960, 100, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
%e 0, 7833, 14175, 3486, 1050, 147, 63, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
%e 0, 10428, 19448, 3850, 1408, 22, 44, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
%Y Cf. A331702 (vertices), A359046 (regions), A359047 (edges), A359009, A358782, A007678.
%K nonn,tabf
%O 2,1
%A _Scott R. Shannon_, Dec 14 2022
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