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A344866 Number of polygons formed when every pair of vertices of a regular (2n-1)-gon are joined by an infinite line.

%I A344866 #54 Jun 01 2025 13:59:21
%S A344866 0,1,16,99,352,925,2016,3871,6784,11097,17200,25531,36576,50869,68992,
%T A344866 91575,119296,152881,193104,240787,296800,362061,437536,524239,623232,
%U A344866 735625,862576,1005291,1165024,1343077,1540800,1759591,2000896,2266209,2557072,2875075,3221856,3599101,4008544,4451967
%N A344866 Number of polygons formed when every pair of vertices of a regular (2n-1)-gon are joined by an infinite line.
%C A344866 This is the odd-indexed subsequence of A344857. See A344857 for images of the polygons.
%H A344866 Harvey P. Dale, <a href="/A344866/b344866.txt">Table of n, a(n) for n = 1..1000</a>
%H A344866 John Elias, <a href="/A344866/a344866.png">Illustration of initial terms: Cubic Vertex Fractal</a>
%H A344866 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 A344866 Alexander Sidorenko, <a href="/A344857/a344857.txt">Explicit Formulas for Odd-Indexed Terms in A344899, A146212, and A344857</a>.
%H A344866 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).
%F A344866 a(n) = 2*n^4 - 11*n^3 + 23*n^2 - 21*n + 7.
%F A344866 G.f.: x^2*(1 + 11*x + 29*x^2 + 7*x^3)/(1 - x)^5. - _Stefano Spezia_, Jun 04 2021
%e A344866 a(3) = 16 as the five connected vertices form eleven polygons inside the regular pentagon while also forming five triangles outside the pentagon, giving sixteen polygons in total.
%t A344866 LinearRecurrence[{5,-10,10,-5,1},{0,1,16,99,352},40] (* _Harvey P. Dale_, Jun 01 2025 *)
%o A344866 (Python)
%o A344866 def A344866(n): return n*(n*(n*(2*n - 11) + 23) - 21) + 7 # _Chai Wah Wu_, Sep 12 2021
%Y A344866 Cf. A344857 (number for even and odd n), A344311, A344938, A007678, A341735 (number inside the n-gon), A344899 (number of edges).
%Y A344866 See also A347320.
%K A344866 nonn,easy
%O A344866 1,3
%A A344866 _Scott R. Shannon_, Jun 01 2021
%E A344866 Edited by _N. J. A. Sloane_, Sep 12 2021