OFFSET
1,3
COMMENTS
In a drawing the distinction between simple and multiple intersection points may be difficult due to near-coincidences. E.g. there are no coincident intersections for n=7.
Note that 3 divides a(2k)-1 and a(2k+1). - T. D. Noe, Jun 29 2005
The interior intersection points only can be the result of the concurrency of 2 or 3 segments by construction. It is easy to see that the total number of 2-intersections N2 is 3*(n-1)^2 (which includes every 3-intersection as two 2-intersections) by symmetry. But we are interested in excluding the concurrency of more than 2. By Ceva's theorem necessary and sufficient condition for 3 concurrent segments that connect the edges with the opposite side, the number of 3-intersections N3 is the same as the number of (i,j,k) belonging to [1,n-1]x[1,n-1]x[1,n-1] such that (i/(n-i))*(j/(n-j))*(k/(n-k))=1. Thus the terms a(n)=N2-2*N3. - Herman Jamke (hermanjamke(AT)fastmail.fm), Oct 26 2006
If n is even then a(n) < 3*(n-1)^2; if n is odd then a(n) = 3*(n-1)^2 except for n in A332378. - N. J. A. Sloane, Feb 14 2020
LINKS
EXAMPLE
a(3)=12 because the 3*2 line segments intersect each other in 12 distinct points (see pictures given at link)
a(4)=13 because the 27 intersections form 6 two line intersection points and 7 three line intersection points.
PROG
(PARI) for(n=1, 70, conc=0; for(i=1, n-1, for(j=1, n-1, for(k=1, n-1, if(i*j*k/((n-i)*(n-j)*(n-k))==1, conc++)))); print1(3*(n-1)^2-2*conc, ", ")) \\ Herman Jamke (hermanjamke(AT)fastmail.fm), Oct 26 2006
CROSSREFS
Cf. A091910 = radial locations of intersection points, A092098 = number of regions that the line segments cut the triangle into, A006561.
KEYWORD
nonn
AUTHOR
Hugo Pfoertner, Feb 19 2004
EXTENSIONS
More terms from T. D. Noe, Jun 29 2005
More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), Oct 26 2006
STATUS
approved