%I #39 Sep 12 2021 12:33:49
%S 0,1,3,8,30,78,189,320,684,1010,1815,2052,3978,4718,7665,8576,13464,
%T 12546,22059,23720,34230,36542,50853,47928,72900,76466,101439,105560,
%U 137634,115230,182745,188672,238128,245378,305235,294948,385614,395390,480909,491840,592860,544950,723303,737528
%N Number of polygon edges formed when every pair of vertices of a regular n-gon are joined by an infinite line.
%C See A344857 for other examples and images of the polygons.
%H J. F. Rigby, <a href="https://doi.org/10.1007/BF00147438">Multiple intersections of diagonals of regular polygons, and related topics</a>, Geom. Dedicata 9 (1980), 207-238.
%H Alexander Sidorenko, <a href="/A344857/a344857.txt">Explicit Formulas for Odd-Indexed Terms in A344899, A146212, and A344857.</a>
%F Conjectured formula odd n: a(n) = (n^4 - 7*n^3 + 17*n^2 - 11*n)/4 = (n-1)*n*(n^2-6*n+11)/4.
%F This formula is correct: see the Sidorenko link. - _N. J. A. Sloane_, Sep 12 2021
%F See also A344907.
%F a(n) = A344857(n) + A146212(n) - 1 (Euler's theorem.).
%e 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.
%e 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.
%Y Cf. A344907 (number of edges for odd n), A344857 (number of polygons), A146212 (number of vertices), A344866, A344311, A007678, A331450, A344938.
%Y Bisections: A344907, A347322.
%K nonn
%O 1,3
%A _Scott R. Shannon_, Jun 02 2021