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 *)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Vijay Srinivas Balaji, Sep 23 2023
STATUS
approved