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Number of polygon edges formed when every pair of vertices of a regular (2n-1)-gon are joined by an infinite line.
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%I #34 Jan 14 2024 12:30:55

%S 0,3,30,189,684,1815,3978,7665,13464,22059,34230,50853,72900,101439,

%T 137634,182745,238128,305235,385614,480909,592860,723303,874170,

%U 1047489,1245384,1470075,1723878,2009205,2328564,2684559,3079890,3517353,3999840,4530339,5111934,5747805,6441228,7195575

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

%C 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 A bisection of A344899. - _N. J. A. Sloane_, Sep 12 2021

%C See A344857 for other examples and images of the polygons.

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).

%F a(n) = 4*n^4 - 22*n^3 + 44*n^2 - 35*n + 9 (see Sidorenko link in A344857 for proof).

%F From _Stefano Spezia_, Jun 10 2021: (Start)

%F G.f.: 3*x^2*(1 + 5*x + 23*x^2 + 3*x^3)/(1 - x)^5.

%F 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 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 (Python)

%o def A344907(n): return n*(n*(n*(4*n - 22) + 44) - 35) + 9 # _Chai Wah Wu_, Sep 12 2021

%Y Cf. A344899 (number of edges for all n-gons), A344866 (number of polygon), A146212, A344857, A344311, A007678, A331450, A344938.

%Y See also A347322.

%K nonn,easy

%O 1,2

%A _Scott R. Shannon_, Jun 02 2021