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A365929 Number of intersections formed within a triangle by placing n points "in general position" on each of the three sides and connecting each point to each of the points on the other two sides using straight lines.

%I A365929 #64 Nov 07 2023 18:36:42
%S A365929 0,0,15,108,396,1050,2295,4410,7728,12636,19575,29040,41580,57798,
%T A365929 78351,103950,135360,173400,218943,272916,336300,410130,495495,593538,
%U A365929 705456,832500,975975,1137240,1317708,1518846,1742175,1989270,2261760,2561328,2889711,3248700,3640140,4065930,4528023
%N A365929 Number of intersections formed within a triangle by placing n points "in general position" on each of the three sides and connecting each point to each of the points on the other two sides using straight lines.
%C A365929 There are n points on each of the three sides (not counting the vertices of the triangle). Each point must be connected to every point on the other two sides. A033428(n) = 3*n^2 gives the number of lines.
%C A365929 Comments from _N. J. A. Sloane_, Oct 29 2023: (Start)
%C A365929 "In general position" means that all interior intersection points are simple. No three-way or higher intersections are permitted.
%C A365929 If the 3*n+3 boundary points are included in the count, there are 3*A366478 points.
%C A365929 For the configurations where the boundary points are equally spaced and every pair of boundary points is joined by a chord, see A091908, A092098, A331782.
%C A365929 (End)
%D A365929 Vijay Srinivas Balaji, Formulating A Conjecture For Intersections Created From Crossing Lines Within Different Polygons, International School of Helsingborg, 2023.
%H A365929 Paolo Xausa, <a href="/A365929/b365929.txt">Table of n, a(n) for n = 0..10000</a>
%H A365929 Vijay Srinivas Balaji, <a href="https://1drv.ms/w/s!AoQIlZk01hoDg1swUpygaDpzRlQC">Diagram of Intersections for a Triangle.</a>
%H A365929 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).
%H A365929 Scott R. Shannon, <a href="/A365929/a365929.png">Image for n = 2</a>.
%H A365929 Scott R. Shannon, <a href="/A365929/a365929_1.png">Image for n = 4</a>.
%H A365929 Scott R. Shannon, <a href="/A365929/a365929_2.png">Image for n = 7</a>.
%H A365929 Scott R. Shannon, <a href="/A365929/a365929_3.png">Image for n = 10</a>.
%F A365929 a(n) = (3/4)*n^2*(n-1)*(3*n-1). [Proof: For intersection points defined by two points on two opposite sides, the number is 3*C(n,2)^2; for intersection points defined by two points on one side and one point on each of the other two sides, the number is 3*C(n,2)*n^2. - _N. J. A. Sloane_, Nov 07 2023]
%F A365929 G.f.: 3*x^2*(5 + 11*x + 2*x^2)/(1 - x)^5. - _Stefano Spezia_, Sep 24 2023
%e A365929 a(5) = (3/4) * 5^2 * (3*5^2 - 4*5 + 1) = 1050.
%p A365929 p__3 := n -> 9/4*n^4 - 3*n^3 + 3/4*n^2; for n from 0 to 55 do p__3(n); end do;
%t A365929 LinearRecurrence[{5,-10,10,-5,1},{0,0,15,108,396},50] (* or *)
%t A365929 A365929[n_]:=3n^2(n-1)(3n-1)/4;Array[A365929,50,0] (* _Paolo Xausa_, Nov 07 2023 *)
%Y A365929 Cf. A367015 (number of regions), A366932 (number of edges), A366478 (vertices including boundary points), A033428 (number of chords).
%Y A365929 See also A091908, A092098, A331782.
%K A365929 nonn,easy
%O A365929 0,3
%A A365929 _Vijay Srinivas Balaji_, Sep 23 2023