A332378 -id:A332378 - 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.

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

A331782 Total number of vertices in graph formed 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

3, 7, 21, 25, 63, 67, 129, 133, 219, 223, 333, 337, 471, 475, 621, 637, 819, 823, 1029, 1021, 1263, 1267, 1521, 1477, 1803, 1807, 2109, 2113, 2439, 2431, 2793, 2797, 3171, 3175, 3549, 3577, 3999, 4003, 4449, 4417, 4923, 4903, 5421, 5425, 5859, 5947, 6489, 6397, 7059, 7063, 7653, 7657, 8271, 8275, 8889

Offset: 1

N. J. A. Sloane, Feb 13 2020

nonn

a(n) <= 3*(n^2-n+1), with equality iff n is odd and not a member of A332378. A331423 gives the difference between a(n) and the upper bound.

N. J. A. Sloane, Table of n, a(n) for n = 1..1000

Cf. A091908, A092098 (number of cells), A332376 (number of edges), A332378, A331423.

a(n) = A091908(n) + 3*n.

A331423 Divide each side of a triangle into n>=1 equal parts and trace the corresponding cevians, i.e., join every point, except for the first and last ones, with the opposite vertex. a(n) is the number of points at which three cevians meet.

Original entry on oeis.org

0, 1, 0, 7, 0, 13, 0, 19, 0, 25, 0, 31, 0, 37, 6, 43, 0, 49, 0, 61, 0, 61, 0, 91, 0, 73, 0, 79, 0, 91, 0, 91, 0, 97, 12, 103, 0, 109, 0, 133, 0, 133, 0, 127, 42, 133, 0, 187, 0, 145, 0, 151, 0, 157, 12, 175, 0, 169, 0, 235, 0, 181, 48, 187, 6, 205, 0, 199, 0

Offset: 1

César Eliud Lozada, Jan 16 2020

Denote the cevians by a0, a1,...,an, b0, b1,...,bn, c0, c1,...,cn. For any given n, the indices (i,j,k) of (ai, bj, ck) meeting at a point are the integer solutions of:
n^3 - (i + j + k)*n^2 + (j*k + k*i + i*j)*n - 2*i*j*k = 0,  with 0 < i, j, k < n
or, equivalently and shorter,
(n-i)*(n-j)*(n-k) - i*j*k = 0, with 0 < i, j, k < n.
From N. J. A. Sloane, Feb 14 2020: (Start)
Stated another way, a(n) = number of triples (i,j,k) in [1,n-1] X [1,n-1] X [1,n-1] such that (i/(n-i))*(j/(n-j))*(k/(n-k)) = 1.
This is the quantity N3 mentioned in A091908.
Indices of zeros are precisely all odd numbers except those listed in A332378.
(End)

Antti Karttunen, Table of n, a(n) for n = 1..1045 (first 200 terms from Robert Israel)
Peter Kagey, An illustration of A331423(4) = 7.
Hugo Pfoertner, Visualization of diagonal intersections in an equilateral triangle.
Jim Propp and Adam Propp-Gubin, Counting Triangles in Triangles, arXiv:2409.17117 [math.CO], 2024. See p. 9.

Cf. A091908, A332378. Bisections are A331425, A331428.

Ceva:= proc(n) local a, i, j, k; a:=0;
for i from 1 to n-1 do
for j from 1 to n-1 do
for k from 1 to n-1 do
if i*j*k/((n-i)*(n-j)*(n-k)) = 1 then a:=a+1; fi;
od: od: od: a; end;
t1:=[seq(Ceva(n), n=1..80)];  # N. J. A. Sloane, Feb 14 2020

CevIntersections[n_] := Length[Solve[(n - i)*(n - j)*(n - k) - i*j*k == 0 && 0 < i < n &&  0 < j < n && 0 < k < n, {i, j, k}, Integers]];
Map[CevIntersections[#] &, Range[50]]

A331423(n) = sum(i=1, n-1, sum(j=1, n-1, sum(k=1, n-1, (1==(i*j*k)/((n-i)*(n-j)*(n-k)))))); \\ (After the Maple program) - Antti Karttunen, Dec 12 2021

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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.

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A331782 Total number of vertices in graph formed 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.

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A331423 Divide each side of a triangle into n>=1 equal parts and trace the corresponding cevians, i.e., join every point, except for the first and last ones, with the opposite vertex. a(n) is the number of points at which three cevians meet.

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