cp's OEIS Frontend

A091908 Number of interior intersection points made by the straight line segments connecting the edges of an equilateral triangle with the n-1 points resulting from a subdivision of the sides into n equal pieces, counting coinciding intersection points only once.

%I A091908 #28 Nov 07 2023 19:53:52
%S A091908 0,1,12,13,48,49,108,109,192,193,300,301,432,433,576,589,768,769,972,
%T A091908 961,1200,1201,1452,1405,1728,1729,2028,2029,2352,2341,2700,2701,3072,
%U A091908 3073,3444,3469,3888,3889,4332,4297,4800,4777,5292,5293,5724,5809,6348
%N A091908 Number of interior intersection points made by the straight line segments connecting the edges of an equilateral triangle with the n-1 points resulting from a subdivision of the sides into n equal pieces, counting coinciding intersection points only once.
%C A091908 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.
%C A091908 Note that 3 divides a(2k)-1 and a(2k+1). - _T. D. Noe_, Jun 29 2005
%C A091908 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
%C A091908 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
%H A091908 Hugo Pfoertner, <a href="/A091908/b091908.txt">Table of n, a(n) for n = 1..1000</a>
%H A091908 Hugo Pfoertner, <a href="http://www.randomwalk.de/sequences/a091908.pdf">Visualization of diagonal intersections in an equilateral triangle.</a>
%H A091908 <a href="/index/Pol#Poonen">Index entries for sequences formed by drawing all diagonals in regular polygon</a>
%e A091908 a(3)=12 because the 3*2 line segments intersect each other in 12 distinct points (see pictures given at link)
%e A091908 a(4)=13 because the 27 intersections form 6 two line intersection points and 7 three line intersection points.
%o A091908 (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
%Y A091908 Cf. A091910 = radial locations of intersection points, A092098 = number of regions that the line segments cut the triangle into, A006561.
%Y A091908 For the basic properties of the underlying graph, see A092098 (cells), A331782 (vertices), A331782 (vertices), A332376 & A332377 (edges). - _N. J. A. Sloane_, Feb 14 2020
%K A091908 nonn
%O A091908 1,3
%A A091908 _Hugo Pfoertner_, Feb 19 2004
%E A091908 More terms from _T. D. Noe_, Jun 29 2005
%E A091908 More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), Oct 26 2006