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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.

%I A331423 #48 Sep 29 2024 03:05:25
%S A331423 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,
%T A331423 0,91,0,91,0,97,12,103,0,109,0,133,0,133,0,127,42,133,0,187,0,145,0,
%U A331423 151,0,157,12,175,0,169,0,235,0,181,48,187,6,205,0,199,0
%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.
%C A331423 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:
%C A331423 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
%C A331423 or, equivalently and shorter,
%C A331423 (n-i)*(n-j)*(n-k) - i*j*k = 0, with 0 < i, j, k < n.
%C A331423 From _N. J. A. Sloane_, Feb 14 2020: (Start)
%C A331423 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.
%C A331423 This is the quantity N3 mentioned in A091908.
%C A331423 Indices of zeros are precisely all odd numbers except those listed in A332378.
%C A331423 (End)
%H A331423 Antti Karttunen, <a href="/A331423/b331423.txt">Table of n, a(n) for n = 1..1045</a> (first 200 terms from Robert Israel)
%H A331423 Peter Kagey, <a href="/A331423/a331423.png">An illustration of A331423(4) = 7</a>.
%H A331423 Hugo Pfoertner, <a href="http://www.randomwalk.de/sequences/a091908.pdf">Visualization of diagonal intersections in an equilateral triangle.</a>
%H A331423 Jim Propp and Adam Propp-Gubin, <a href="https://arxiv.org/abs/2409.17117">Counting Triangles in Triangles</a>, arXiv:2409.17117 [math.CO], 2024. See p. 9.
%p A331423 Ceva:= proc(n) local a, i, j, k; a:=0;
%p A331423 for i from 1 to n-1 do
%p A331423 for j from 1 to n-1 do
%p A331423 for k from 1 to n-1 do
%p A331423 if i*j*k/((n-i)*(n-j)*(n-k)) = 1 then a:=a+1; fi;
%p A331423 od: od: od: a; end;
%p A331423 t1:=[seq(Ceva(n), n=1..80)];  # _N. J. A. Sloane_, Feb 14 2020
%t A331423 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]];
%t A331423 Map[CevIntersections[#] &, Range[50]]
%o A331423 (PARI) 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
%Y A331423 Cf. A091908, A332378. Bisections are A331425, A331428.
%K A331423 nonn,look
%O A331423 1,4
%A A331423 _César Eliud Lozada_, Jan 16 2020