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A344938 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.

%I A344938 #22 Jun 14 2021 21:04:10
%S A344938 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 A344938 407,352,110,55,0,0,0,0,1,624,528,884,689,325,91,0,26,0,0,0,0,1,1162,
%U A344938 1092,266,14,1530,1545,480,270,45,0,0,0,0,0,0,0,1,2080,2032,416,80
%N A344938 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 A344938 See A344857 for examples and images of the polygons.
%F A344938 Sum of row(n) = A344857(n) = A344311(n) + A007678(n).
%e A344938 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 A344938 The table begins:
%e A344938 1;
%e A344938 4;
%e A344938 15,0,1;
%e A344938 36,6;
%e A344938 70,21,7,0,1;
%e A344938 112,64;
%e A344938 189,108,36,18,0,0,1;
%e A344938 270,220,50;
%e A344938 407,352,110,55,0,0,0,0,1;
%e A344938 624,528;
%e A344938 884,689,325,91,0,26,0,0,0,0,1;
%e A344938 1162,1092,266,14;
%e A344938 1530,1545,480,270,45,0,0,0,0,0,0,0,1;
%e A344938 2080,2032,416,80;
%e A344938 2567,2754,1003,374,17,68,0,0,0,0,0,0,0,0,1;
%e A344938 3402,3366,180,18,18;
%e A344938 3952,4807,1672,475,95,76,0,19,0,0,0,0,0,0,0,0,1;
%e A344938 5380,5360,1580,240,0,20;
%e A344938 5943,7392,2583,1260,21,0,0,0,0,0,0,0,0,0,0,0,0,0,1;
%e A344938 7590,9020,2310,132,132,66;
%e A344938 9430,9775,4508,1518,253,46,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1;
%e A344938 11304,12288,2280,144;
%e A344938 13025,14650,6250,2375,200,75,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1;
%e A344938 16042,16952,5954,728,260,52;
%e A344938 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 A344938 21616,24192,7056,2016,168,28;
%e A344938 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 A344938 29880,29010,4140,540;
%e A344938 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 A344938 37440,42624,14240,3008,544,64;
%e A344938 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 A344938 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 A344938 nonn,tabf
%O A344938 3,2
%A A344938 _Scott R. Shannon_, Jun 03 2021