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A345459 Number of polygons formed when connecting all 4n points on the perimeter of an n X n square by infinite lines.

%I A345459 #18 Jun 22 2021 12:25:55
%S A345459 0,4,80,568,2024,6052,12144,26976,45024,76724,116840,191128,245976,
%T A345459 388452,501888,661476,870168,1199724,1402096,1911384,2188320,2739280,
%U A345459 3371264,4224288,4617224,5801372,6780568
%N A345459 Number of polygons formed when connecting all 4n points on the perimeter of an n X n square by infinite lines.
%C A345459 The width/height of the entire figure grows as ~ 2*n^3 for large n. See the Formula section below.
%H A345459 Scott R. Shannon, <a href="/A345459/a345459.gif">Image for n = 2</a>. In this and other images the square's points are highlighted as white dots while the outer open regions, which are not counted, are darkened. The key for the edge-number coloring is shown at the top-left of the image.
%H A345459 Scott R. Shannon, <a href="/A345459/a345459_1.gif">Image for n = 3</a>.
%H A345459 Scott R. Shannon, <a href="/A345459/a345459_2.gif">Image for n = 4</a>.
%H A345459 Scott R. Shannon, <a href="/A345459/a345459_3.gif">Image for n = 5</a>.
%H A345459 Scott R. Shannon, <a href="/A345459/a345459_4.gif">Image for n = 6</a>.
%F A345459 a(n) = A345650(n) - A345649(n) + 1.
%F A345459 Assuming the square is of size n x n centered on the origin the x (or y) offset for the eight outermost vertices is n^3 - 2*n^2 + 3*n/2, which have a corresponding y (or x) offset of n^2 - 3*n/2 + 1. The total distance from the origin of these vertices is sqrt(n^6 - 4*n^5 + 8*n^4 - 9*n^3 + 13*n^2/2 - 3*n + 1).
%e A345459 a(2) = 80. Connecting the 8 perimeter points results in the creation of forty-eight 3-gons and eight 4-gons inside the square while creating twenty-four 3-gons outside the square, giving eighty polygons in total. See the linked images.
%Y A345459 Cf. A255011 (number inside the square), A345648 (number outside the square), A345649 (number of vertices), A345650 (number of edges), A344993, A344857, A092098, A007678.
%K A345459 nonn,more
%O A345459 0,2
%A A345459 _Scott R. Shannon_ and _N. J. A. Sloane_, Jun 20 2021