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Irregular table read by rows: T(n,k) is the number of k-sided polygons, for k>=3, in a square when straight line segments connect the n-1 points between each corner that divide each edge into n equal parts to the n-1 points on the edge on the opposite side of the square.
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%I #17 Jan 04 2024 14:29:31

%S 0,1,0,4,12,12,56,32,16,156,124,24,8,0,4,384,228,72,28,716,648,144,68,

%T 8,4,1312,1144,240,112,8,2244,1912,528,256,3528,3072,696,360,16,5012,

%U 5536,1296,524,48,28,7696,6596,1960,572,16,10340,11448,2968,1028,160,24,14520,14428,3872,1156,104,8

%N Irregular table read by rows: T(n,k) is the number of k-sided polygons, for k>=3, in a square when straight line segments connect the n-1 points between each corner that divide each edge into n equal parts to the n-1 points on the edge on the opposite side of the square.

%C Up to n = 50 the maximum sided k-gon created is the 8-gon. It is plausible this is the maximum sided k-gon for all n, although this is unknown.

%C See A355798 for more images of the square.

%C The keyword "look" is for the n = 10 image. - _N. J. A. Sloane_, Jul 21 2022

%H Scott R. Shannon, <a href="/A355801/a355801.jpg">Image for n = 10</a>.

%e The table begins:

%e 0, 1;

%e 0, 4;

%e 12, 12;

%e 56, 32, 16;

%e 156, 124, 24, 8, 0, 4;

%e 384, 228, 72, 28;

%e 716, 648, 144, 68, 8, 4;

%e 1312, 1144, 240, 112, 8;

%e 2244, 1912, 528, 256;

%e 3528, 3072, 696, 360, 16;

%e 5012, 5536, 1296, 524, 48, 28;

%e 7696, 6596, 1960, 572, 16;

%e 10340, 11448, 2968, 1028, 160, 24;

%e 14520, 14428, 3872, 1156, 104, 8;

%e 19588, 19156, 5296, 2052, 160, 8;

%e 25392, 26112, 7160, 2152, 208, 24;

%e 31820, 37244, 9936, 3240, 488, 64;

%e .

%e .

%Y Cf. A355798 (regions), A355799 (vertices), A355800 (edges), A355801 (k-gons), A255011 (all vertices), A290131, A331452, A335678.

%K nonn,tabf,look

%O 1,4

%A _Scott R. Shannon_, Jul 17 2022