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A359009 Irregular table read by rows: T(n,k) is the number of k-gons formed, k>=2, when every pair of n points, placed at the vertices of a regular n-gon, are connected by a circle and where the points lie at the ends of the circle's diameter.

%I A359009 #23 Dec 13 2022 10:14:15
%S A359009 1,0,7,8,4,0,40,20,6,6,72,6,0,0,0,0,0,0,0,1,0,133,98,42,7,1,16,184,56,
%T A359009 0,8,0,342,306,99,54,0,0,1,10,510,220,60,10,10,0,0,0,0,0,0,0,0,0,0,0,
%U A359009 0,1,0,693,858,231,88,11,11,0,0,1,24,924,612,120,60,0,1469,1560,455,299,13,0,0,13,0,0,1
%N A359009 Irregular table read by rows: T(n,k) is the number of k-gons formed, k>=2,  when every pair of n points, placed at the vertices of a regular n-gon, are connected by a circle and where the points lie at the ends of the circle's diameter.
%C A359009 Conjectures: for odd values of n all vertices are simple, other than those defining the diameters of the circles. For n > 2 and (n-2) mod 4 = 0, T(n,2) = n. For n mod 4 = 0, T(n,2) = k*n, k>=2. For odd n, T(n,2) = 0.
%C A359009 See A358782 for more images of the k-gons.
%C A359009 The author thanks Zach Shannon some of whose code was used in the generation of this sequence.
%H A359009 Scott R. Shannon, <a href="/A359009/a359009.jpg">Image for n = 13</a>.
%H A359009 Scott R. Shannon, <a href="/A359009/a359009_1.jpg">Image for n = 21</a>.
%F A359009 Sum of row n = A358782(n).
%e A359009 The table begins:
%e A359009 1;
%e A359009 0, 7;
%e A359009 8, 4;
%e A359009 0, 40, 20, 6;
%e A359009 6, 72, 6, 0, 0, 0, 0, 0, 0, 0, 1;
%e A359009 0, 133, 98, 42, 7, 1;
%e A359009 16, 184, 56, 0, 8;
%e A359009 0, 342, 306, 99, 54, 0, 0, 1;
%e A359009 10, 510, 220, 60, 10, 10, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
%e A359009 0, 693, 858, 231, 88, 11, 11, 0, 0, 1;
%e A359009 24, 924, 612, 120, 60;
%e A359009 0, 1469, 1560, 455, 299, 13, 0, 0, 13, 0, 0, 1;
%e A359009 14, 1806, 1428, 350, 98, 28, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, \\
%e A359009                 0, 0, 0, 1;
%e A359009 0, 2550, 2910, 870, 405, 75, 60, 0, 0, 0, 0, 0, 0, 1;
%e A359009 32, 3280, 2000, 768, 352, 0, 16;
%e A359009 0, 4301, 4862, 1734, 680, 102, 34, 0, 17, 0, 17, 0, 0, 0, 0, 1;
%e A359009 18, 4878, 4482, 1332, 324, 54, 36, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, \\
%e A359009                 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
%e A359009 0, 6517, 7847, 2565, 1045, 190, 133, 19, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
%e A359009 80, 7340, 7040, 1920, 700, 0, 80;
%e A359009 0, 9723, 11487, 4515, 1491, 210, 168, 21, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
%e A359009 .
%e A359009 .
%Y A359009 Cf. A358782 (regions), A358746 (vertices), A358783 (edges), A331451, A344938.
%K A359009 nonn,tabf
%O A359009 2,3
%A A359009 _Scott R. Shannon_, Dec 12 2022