cp's OEIS Frontend

A344899 Number of polygon edges formed when every pair of vertices of a regular n-gon are joined by an infinite line.

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