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A355992 Irregular table read by rows: T(n,k) is the number of k-sided polygons formed, for k>=3, in a square when straight line segments connect the four corner vertices to the points dividing the sides into n equal parts.

%I A355992 #13 Jan 04 2024 14:10:23
%S A355992 4,24,8,56,28,12,96,80,8,4,144,140,36,12,216,216,24,4,272,332,76,24,8,
%T A355992 360,448,80,28,456,572,132,36,8,568,728,128,64,656,916,260,28,40,4,
%U A355992 792,1104,176,36,928,1308,316,128,32,4,1064,1568,304,128,16,1240,1772,396,88,32,4,1416,2032,432,156,32
%N A355992 Irregular table read by rows: T(n,k) is the number of k-sided polygons formed, for k>=3, in a square when straight line segments connect the four corner vertices to the points dividing the sides into n equal parts.
%C A355992 Up to n = 100 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 A355992 See A108914 for more images of the square.
%H A355992 Scott R. Shannon, <a href="/A355992/a355992.jpg">Image for n = 13</a>.
%e A355992 The table begins:
%e A355992 4;
%e A355992 24,   8;
%e A355992 56,   28,   12;
%e A355992 96,   80,   8,   4;
%e A355992 144,  140,  36,  12;
%e A355992 216,  216,  24,  4;
%e A355992 272,  332,  76,  24,  8;
%e A355992 360,  448,  80,  28;
%e A355992 456,  572,  132, 36,  8;
%e A355992 568,  728,  128, 64;
%e A355992 656,  916,  260, 28,  40, 4;
%e A355992 792,  1104, 176, 36;
%e A355992 928,  1308, 316, 128, 32, 4;
%e A355992 1064, 1568, 304, 128, 16;
%e A355992 1240, 1772, 396, 88,  32, 4;
%e A355992 1416, 2032, 432, 156, 32;
%e A355992 .
%e A355992 .
%Y A355992 Cf. A108914 (regions), A355948 (edges), A355949 (vertices), A355841, A331452, A335678.
%K A355992 nonn,tabf
%O A355992 1,1
%A A355992 _Scott R. Shannon_, Jul 22 2022