A091910 -id:A091910 - cp's OEIS frontend

A092098 Number of regions that the line segments in A091908(n) cut the equilateral triangle into.

1, 6, 19, 30, 61, 78, 127, 150, 217, 246, 331, 366, 469, 510, 625, 678, 817, 870, 1027, 1080, 1261, 1326, 1519, 1566, 1801, 1878, 2107, 2190, 2437, 2520, 2791, 2886, 3169, 3270, 3559, 3678, 3997, 4110, 4447, 4548, 4921, 5034, 5419, 5550, 5899, 6078, 6487

Offset: 1

Hugo Pfoertner, Feb 19 2004

nonn

Number of chambers in an n-sected triangle. That is, n sectors are extended from each vertex to the opposite edge of the triangle. - Eric Gottlieb, Jun 26 2005
How many chambers does the edge n-sected simplex with m vertices have? We have given just the first few terms of the case m = 3. This question is natural in the context of central hyperplane arrangements as it generalizes the braid arrangement. Mike Ackerman, Sul-Young Choi, Peter Coughlin, Japheth Wood and I originally encountered the question in the context of voting theory, where we were exploring ways to tabulate votes when voters' preferences are partially ordered. Unfortunately, it turns out that the chambers of the 3-sected simplex with n vertices are not in correspondence with the set of posets on n letters as the chain with three elements and a fourth incomparable element illustrates. - Eric Gottlieb, Jun 26 2005
"Equilateral" is not needed: the sequence counts regions correctly for any triangle with n-sected sides. Ceva's Theorem is used to deduct vanishing regions from the naive count. The first deduction is at n=15 for n odd and n=20 for n even. - Len Smiley and Brian Wick (mathclub(AT)math.uaa.alaska.edu), Jul 04 2005

			E.g. the number of chambers in the bisected triangle is six, the number of permutations on 3 letters. The number of chambers in the trisected triangle is equal to 19, the number of posets on 3 elements. - _Eric Gottlieb_, Jun 26 2005
a(2)=6: The 3 line segments cut the equilateral triangle into 6 triangles.
a(3)=19: The 3*2 line segments form 12 triangles, 3 quadrilaterals, 3 pentagons and 1 central non-regular hexagon. See pictures at Pfoertner link.

Hugo Pfoertner, Table of n, a(n) for n = 1..1000
Hugo Pfoertner, Visualization of diagonal intersections in an equilateral triangle.
Hugo Pfoertner, Visualization of diagonal intersections in an equilateral triangle. [Local copy]
Scott R. Shannon, Image for n = 6.
Scott R. Shannon, Image for n = 7.
Scott R. Shannon, Image for n = 12.
Scott R. Shannon, Image for n = 13.
Scott R. Shannon, Image for n = 100.
Scott R. Shannon, Image for n = 101.
Sequences formed by drawing all diagonals in regular polygon

Cf. A091908 (number of intersections), A091910 (radial locations of intersection points), A006533.

regions:=proc(n::nonnegint)
   local j,k,l,a;
   a:=0;
   if (n mod 2<>0) then
      a:=3*n^2-3*n+1
   else
      a:=3*n^2-6*n+6
   fi;
   for l from 1 to floor(n/2)-1 do
      for k from 1 to floor(n/2)-1 do
         for j from 1 to floor(n/2)-1 do
            if((n-k)*l*j=k*(n-l)*(n-j)) then
               a:=a-6
            fi
         od
      od
   od;
   return a
end proc;
seq(regions(i),i=1..100);  # Len Smiley and Brian Wick, Jun 30 2005

regions[n_]:=
If[Mod[n,2] == 0, 3n^2-6n+6, 3n^2-3n+1]-
   6*Count[
  Flatten@
   Table[
    Abs[(n-k)l*j - k(n-l)(n-j)],
    {j,1,Floor[n/2]-1},
    {k,1,Floor[n/2]-1},
    {l,1,Floor[n/2]-1}],
  0] (* Ethan Beihl, Oct 13 2016 *)

for(n=1,100,regions=0;if(n%2!=0,regions=3*n^2-3*n+1,regions=3*n^2-6*n+6);for(l=1,floor(n/2)-1,for(k=1,floor(n/2)-1,for(j=1,floor(n/2)-1,if((n-k)*l*j==k*(n-l)*(n-j),regions-=6))));print1(regions,",")) \\ Herman Jamke (hermanjamke(AT)fastmail.fm), Oct 27 2006

Note that 3 divides a(2k) and a(2k+1)-1. - T. D. Noe, Jun 29 2005

More terms from T. D. Noe, Jun 29 2005
Further terms from Brian Wick (mathclub(AT)math.uaa.alaska.edu), Jun 30 2005
More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), Oct 27 2006

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.

Original entry on oeis.org

0, 1, 12, 13, 48, 49, 108, 109, 192, 193, 300, 301, 432, 433, 576, 589, 768, 769, 972, 961, 1200, 1201, 1452, 1405, 1728, 1729, 2028, 2029, 2352, 2341, 2700, 2701, 3072, 3073, 3444, 3469, 3888, 3889, 4332, 4297, 4800, 4777, 5292, 5293, 5724, 5809, 6348

Offset: 1

Hugo Pfoertner, Feb 19 2004

nonn

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

			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.

Hugo Pfoertner, Table of n, a(n) for n = 1..1000
Hugo Pfoertner, Visualization of diagonal intersections in an equilateral triangle.
Index entries for sequences formed by drawing all diagonals in regular polygon

Cf. A091910 = radial locations of intersection points, A092098 = number of regions that the line segments cut the triangle into, A006561.

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

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

More terms from T. D. Noe, Jun 29 2005

More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), Oct 26 2006

cp's OEIS Frontend

A092098 Number of regions that the line segments in A091908(n) cut the equilateral triangle into.

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