A347321 A146212(2*n).
5, 37, 145, 471, 901, 2185, 3969, 5563, 11141, 17293, 21913, 36479, 50485, 51661, 90753, 118355, 139861, 191445, 238481
Offset: 2
This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
a(1) = a(2) = 0 as no polygon can be formed by one or two connected points. a(3) = 1 as the connected vertices form a triangle, while the six regions outside the triangle are open. a(4) = 4 as the four connected vertices form four triangles inside the square. Twelve open regions outside these polygons are also formed. a(5) = 16 as the five connected vertices form eleven polygons inside the regular pentagon while also forming five triangles outside the pentagon, giving sixteen polygons in total. Twenty open regions outside these polygons are also formed. a(6) = 42 as the six connected vertices form twenty-four polygons inside the regular hexagon while also forming eighteen polygons outside the hexagon, giving forty-two polygons in total. Thirty open regions outside these polygons are also formed. See the linked images above for further examples.
a(3) = 3 as the connected vertices form a triangle with three edges. Six infinite edges between the outer regions are also formed but these are not counted. a(5) = 30 as the five connected vertices form a pentagon with fives lines along the pentagon's edges, fifteen lines inside forming eleven polygons, and ten lines outside forming another five triangles. In all these sixteen polygons form thirty edges. Twenty infinite edges between the outer regions are also formed.
a(3) = 16 as the five connected vertices form eleven polygons inside the regular pentagon while also forming five triangles outside the pentagon, giving sixteen polygons in total.
LinearRecurrence[{5,-10,10,-5,1},{0,1,16,99,352},40] (* Harvey P. Dale, Jun 01 2025 *)
def A344866(n): return n*(n*(n*(2*n - 11) + 23) - 21) + 7 # Chai Wah Wu, Sep 12 2021
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]. The table begins: 1; 4; 15,0,1; 36,6; 70,21,7,0,1; 112,64; 189,108,36,18,0,0,1; 270,220,50; 407,352,110,55,0,0,0,0,1; 624,528; 884,689,325,91,0,26,0,0,0,0,1; 1162,1092,266,14; 1530,1545,480,270,45,0,0,0,0,0,0,0,1; 2080,2032,416,80; 2567,2754,1003,374,17,68,0,0,0,0,0,0,0,0,1; 3402,3366,180,18,18; 3952,4807,1672,475,95,76,0,19,0,0,0,0,0,0,0,0,1; 5380,5360,1580,240,0,20; 5943,7392,2583,1260,21,0,0,0,0,0,0,0,0,0,0,0,0,0,1; 7590,9020,2310,132,132,66; 9430,9775,4508,1518,253,46,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1; 11304,12288,2280,144; 13025,14650,6250,2375,200,75,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1; 16042,16952,5954,728,260,52; 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; 21616,24192,7056,2016,168,28; 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; 29880,29010,4140,540; 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; 37440,42624,14240,3008,544,64; 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;
a(2) = 2 as an infinite line connecting two points cuts space into two unbounded regions. a(3) = 7 as the three connected points of the 3-gon form one closed triangle along with six outer unbounded areas, seven regions in total. a(4) = 16 as the four connected points of the 4-gon form four closed triangle inside the square along with twelve outer unbounded areas, sixteen regions in total.
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