A365929 - OEIS

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.

0, 0, 15, 108, 396, 1050, 2295, 4410, 7728, 12636, 19575, 29040, 41580, 57798, 78351, 103950, 135360, 173400, 218943, 272916, 336300, 410130, 495495, 593538, 705456, 832500, 975975, 1137240, 1317708, 1518846, 1742175, 1989270, 2261760, 2561328, 2889711, 3248700, 3640140, 4065930, 4528023

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OFFSET

0,3

COMMENTS

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.

Comments from N. J. A. Sloane, Oct 29 2023: (Start)

"In general position" means that all interior intersection points are simple. No three-way or higher intersections are permitted.

If the 3*n+3 boundary points are included in the count, there are 3*A366478 points.

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.

(End)

REFERENCES

Vijay Srinivas Balaji, Formulating A Conjecture For Intersections Created From Crossing Lines Within Different Polygons, International School of Helsingborg, 2023.

LINKS

Paolo Xausa, Table of n, a(n) for n = 0..10000

Vijay Srinivas Balaji, Diagram of Intersections for a Triangle.

Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Scott R. Shannon, Image for n = 2.

Scott R. Shannon, Image for n = 4.

Scott R. Shannon, Image for n = 7.

Scott R. Shannon, Image for n = 10.

FORMULA

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]

G.f.: 3*x^2*(5 + 11*x + 2*x^2)/(1 - x)^5. - Stefano Spezia, Sep 24 2023

EXAMPLE

a(5) = (3/4) * 5^2 * (3*5^2 - 4*5 + 1) = 1050.

MAPLE

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;

MATHEMATICA

LinearRecurrence[{5, -10, 10, -5, 1}, {0, 0, 15, 108, 396}, 50] (* or *)

A365929[n_]:=3n^2(n-1)(3n-1)/4; Array[A365929, 50, 0] (* Paolo Xausa, Nov 07 2023 *)

CROSSREFS

Cf. A367015 (number of regions), A366932 (number of edges), A366478 (vertices including boundary points), A033428 (number of chords).