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A344907 Number of polygon edges formed when every pair of vertices of a regular (2n-1)-gon are joined by an infinite line.

%I A344907 #34 Jan 14 2024 12:30:55
%S A344907 0,3,30,189,684,1815,3978,7665,13464,22059,34230,50853,72900,101439,
%T A344907 137634,182745,238128,305235,385614,480909,592860,723303,874170,
%U A344907 1047489,1245384,1470075,1723878,2009205,2328564,2684559,3079890,3517353,3999840,4530339,5111934,5747805,6441228,7195575
%N A344907 Number of polygon edges formed when every pair of vertices of a regular (2n-1)-gon are joined by an infinite line.
%C A344907 This sequences gives the number of polygon edges formed when connecting every pair of vertices of a regular polygon, with an odd number of vertices, by an infinite line.
%C A344907 A bisection of A344899. - _N. J. A. Sloane_, Sep 12 2021
%C A344907 See A344857 for other examples and images of the polygons.
%H A344907 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).
%F A344907 a(n) = 4*n^4 - 22*n^3 + 44*n^2 - 35*n + 9 (see Sidorenko link in A344857 for proof).
%F A344907 From _Stefano Spezia_, Jun 10 2021: (Start)
%F A344907 G.f.: 3*x^2*(1 + 5*x + 23*x^2 + 3*x^3)/(1 - x)^5.
%F A344907 a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) for n > 5. (End)
%e A344907 a(3) = 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 unbounded regions are also formed.
%o A344907 (Python)
%o A344907 def A344907(n): return n*(n*(n*(4*n - 22) + 44) - 35) + 9 # _Chai Wah Wu_, Sep 12 2021
%Y A344907 Cf. A344899 (number of edges for all n-gons), A344866 (number of polygon), A146212, A344857, A344311, A007678, A331450, A344938.
%Y A344907 See also A347322.
%K A344907 nonn,easy
%O A344907 1,2
%A A344907 _Scott R. Shannon_, Jun 02 2021