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

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.

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.

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

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

From Elmo R. Oliveira, Apr 17 2026: (Start)

E.g.f.: (3/4)*x^2*(10 + 14*x + 3*x^2)*exp(x).

a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5). (End)

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).

See also A091908, A092098, A331782.

Sequence in context: A383979 A202255 A243212 * A232124 A232204 A232117

Adjacent sequences: A365926 A365927 A365928 * A365930 A365931 A365932

Keyword

nonn,easy

Author

Vijay Srinivas Balaji, Sep 23 2023

Status

approved