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A356119 Irregular table read by rows: T(n,k) is the number of k-sided polygons formed, for k>=3, in an equilateral triangle when straight line segments connect the three corner vertices to the points dividing the sides into n equal parts.

%I A356119 #17 Jul 27 2022 10:33:01
%S A356119 1,6,12,3,3,1,24,6,36,9,9,7,48,24,6,72,21,15,19,84,48,12,6,108,51,33,
%T A356119 25,132,78,18,18,168,69,51,43,180,120,48,18,216,135,57,61,252,156,66,
%U A356119 36,294,159,105,67,312,234,84,48,372,225,117,103,408,264,138,60,456,291,159,121,486,372,138,84
%N A356119 Irregular table read by rows: T(n,k) is the number of k-sided polygons formed, for k>=3, in an equilateral triangle when straight line segments connect the three corner vertices to the points dividing the sides into n equal parts.
%C A356119 The maximum sided k-gon up to n = 250 is the 6-gon; it is likely this is the maximum sided k-gon for all n.
%C A356119 See A092098 for more images of the triangle.
%H A356119 Scott R. Shannon, <a href="/A356119/a356119.txt">Table for n = 1..250</a>.
%H A356119 Scott R. Shannon, <a href="/A356119/a356119.jpg">Image for n = 20</a>.
%F A356119 Sum of row(n) = A092098(n).
%e A356119 The table begins:
%e A356119 1;
%e A356119 6;
%e A356119 12,  3,   3,   1;
%e A356119 24,  6;
%e A356119 36,  9,   9,   7;
%e A356119 48,  24,  6;
%e A356119 72,  21,  15,  19;
%e A356119 84,  48,  12,  6;
%e A356119 108, 51,  33,  25;
%e A356119 132, 78,  18,  18;
%e A356119 168, 69,  51,  43;
%e A356119 180, 120, 48,  18;
%e A356119 216, 135, 57,  61;
%e A356119 252, 156, 66,  36;
%e A356119 294, 159, 105, 67;
%e A356119 312, 234, 84,  48;
%e A356119 372, 225, 117, 103;
%e A356119 408, 264, 138, 60;
%e A356119 456, 291, 159, 121;
%e A356119 486, 372, 138, 84;
%e A356119 .
%e A356119 .
%e A356119 See the attached text file for more examples.
%Y A356119 Cf. A092098 (regions), vertices (A331782), edges (A332376), A335413.
%K A356119 nonn,tabf
%O A356119 1,2
%A A356119 _Scott R. Shannon_, Jul 27 2022