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%I #30 Jul 21 2021 09:08:44
%S 7,30,144,474,1324,2934,5797,10614,17424,27480,41845,61602,85711,
%T 120120,159213,207798,269668,349272,434878,545496,661764,804582,
%U 973471,1174980,1374646,1631304,1908768,2218254,2560198,2976486,3378985,3887796,4405671,4995240,5617689,6322878
%N Number of regions formed by infinite lines when connecting all vertices and all points that divide the sides of an equilateral triangle into n equal parts.
%C The count of regions includes both the closed bounded polygons and the open unbounded areas surrounding these polygons with two edges that go to infinity. The number of unbounded areas appears to equal 6*(n^2 - n + 1).
%C See A344279 for further examples and images of the regions.
%H Scott R. Shannon, <a href="/A343755/a343755.gif">Image for n = 1</a>. In this and other images the triangle's vertices are highlighted as white dots while the outer open regions are cross-hatched. The key for the edge-number coloring is shown at the top-left of the image. Note the edge count for open areas also includes the two infinite edges
%H Scott R. Shannon, <a href="/A343755/a343755_1.gif">Image for n = 2</a>.
%H Scott R. Shannon, <a href="/A343755/a343755_2.gif">Image for n = 3</a>.
%H Scott R. Shannon, <a href="/A343755/a343755_6.gif">Image for n = 4</a>.
%H Scott R. Shannon, <a href="/A343755/a343755_3.gif">Image for n = 5</a>.
%H Scott R. Shannon, <a href="/A343755/a343755_5.gif">Image for n = 6</a>.
%F Conjectured formula: a(n) = A344279(n) + 6*(n^2 - n + 1).
%F Conjectured formula: a(n) = A344279(n) + A121205(n-1), for n>=7.
%e a(1) = 7 as the three connected vertices of a triangle form one polygon along with six outer unbounded areas, seven regions in total.
%e a(2) = 30 as when the three vertices and three edges points are connected they form twelve polygons, all inside the triangle, along with eighteen outer unbounded areas, thirty regions in total.
%e a(2) = 144 as when the three vertices and six edges points are connected they form one hundred two polygons, seventy-five inside the triangle and twenty-seven outside, along with forty-two outer unbounded areas, one hundred forty-four regions in total.
%Y Cf. A344279 (number of polygons), A344657 (number of vertices), A344896 (number of edges), A346446 (number of k-gons), A092867 (number polygons inside the triangle), A121205, A345025.
%K nonn
%O 1,1
%A _Scott R. Shannon_, Jun 28 2021
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