A359009 - OEIS

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 #23 Dec 13 2022 10:14:15

%S 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 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 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 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 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 See A358782 for more images of the k-gons.

%C The author thanks Zach Shannon some of whose code was used in the generation of this sequence.

%H Scott R. Shannon, <a href="/A359009/a359009.jpg">Image for n = 13</a>.

%H Scott R. Shannon, <a href="/A359009/a359009_1.jpg">Image for n = 21</a>.

%F Sum of row n = A358782(n).

%e The table begins:

%e 1;

%e 0, 7;

%e 8, 4;

%e 0, 40, 20, 6;

%e 6, 72, 6, 0, 0, 0, 0, 0, 0, 0, 1;

%e 0, 133, 98, 42, 7, 1;

%e 16, 184, 56, 0, 8;

%e 0, 342, 306, 99, 54, 0, 0, 1;

%e 10, 510, 220, 60, 10, 10, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;

%e 0, 693, 858, 231, 88, 11, 11, 0, 0, 1;

%e 24, 924, 612, 120, 60;

%e 0, 1469, 1560, 455, 299, 13, 0, 0, 13, 0, 0, 1;

%e 14, 1806, 1428, 350, 98, 28, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, \\

%e 0, 0, 0, 1;

%e 0, 2550, 2910, 870, 405, 75, 60, 0, 0, 0, 0, 0, 0, 1;

%e 32, 3280, 2000, 768, 352, 0, 16;

%e 0, 4301, 4862, 1734, 680, 102, 34, 0, 17, 0, 17, 0, 0, 0, 0, 1;

%e 18, 4878, 4482, 1332, 324, 54, 36, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, \\

%e 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;

%e 0, 6517, 7847, 2565, 1045, 190, 133, 19, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;

%e 80, 7340, 7040, 1920, 700, 0, 80;

%e 0, 9723, 11487, 4515, 1491, 210, 168, 21, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;

%e .

%Y Cf. A358782 (regions), A358746 (vertices), A358783 (edges), A331451, A344938.

%K nonn,tabf

%O 2,3

%A _Scott R. Shannon_, Dec 12 2022