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%I #22 Jun 14 2021 21:04:10
%S 1,4,15,0,1,36,6,70,21,7,0,1,112,64,189,108,36,18,0,0,1,270,220,50,
%T 407,352,110,55,0,0,0,0,1,624,528,884,689,325,91,0,26,0,0,0,0,1,1162,
%U 1092,266,14,1530,1545,480,270,45,0,0,0,0,0,0,0,1,2080,2032,416,80
%N Irregular triangle read by rows: T(n,k) = number of k-sided polygons formed when every pair of vertices of a regular n-gon are joined by an infinite line, for k = 3, 4, ..., max_k.
%C See A344857 for examples and images of the polygons.
%F Sum of row(n) = A344857(n) = A344311(n) + A007678(n).
%e A pentagon with all vertices connected forms 10 triangles inside the pentagon, 5 triangles outside the pentagon, giving 15 triangles in all, and 1 smaller pentagon inside the pentagon, so row 3 is [15,0,1].
%e The table begins:
%e 1;
%e 4;
%e 15,0,1;
%e 36,6;
%e 70,21,7,0,1;
%e 112,64;
%e 189,108,36,18,0,0,1;
%e 270,220,50;
%e 407,352,110,55,0,0,0,0,1;
%e 624,528;
%e 884,689,325,91,0,26,0,0,0,0,1;
%e 1162,1092,266,14;
%e 1530,1545,480,270,45,0,0,0,0,0,0,0,1;
%e 2080,2032,416,80;
%e 2567,2754,1003,374,17,68,0,0,0,0,0,0,0,0,1;
%e 3402,3366,180,18,18;
%e 3952,4807,1672,475,95,76,0,19,0,0,0,0,0,0,0,0,1;
%e 5380,5360,1580,240,0,20;
%e 5943,7392,2583,1260,21,0,0,0,0,0,0,0,0,0,0,0,0,0,1;
%e 7590,9020,2310,132,132,66;
%e 9430,9775,4508,1518,253,46,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1;
%e 11304,12288,2280,144;
%e 13025,14650,6250,2375,200,75,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1;
%e 16042,16952,5954,728,260,52;
%e 17064,22464,7884,2700,567,189,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1;
%e 21616,24192,7056,2016,168,28;
%e 23751,29319,11281,3828,348,319,0,87,29,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1;
%e 29880,29010,4140,540;
%e 30814,39370,15314,5177,341,496,0,62,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1;
%e 37440,42624,14240,3008,544,64;
%e 41481,49335,19305,7854,891,363,66,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1;
%Y Cf. A344857 (total number of polygons), A344899 (number of edges), A146212 (number of vertices), A344866, A344311, A007678, A331450 (number of k-gons inside the regular n-gon).
%K nonn,tabf
%O 3,2
%A _Scott R. Shannon_, Jun 03 2021
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