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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. 5
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,4 COMMENTS 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. Comments 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) LINKS Robert Israel, Table of n, a(n) for n = 1..200 Peter Kagey, An illustration of A331423(4) = 7. MAPLE 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 MATHEMATICA 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]] CROSSREFS Cf. A091908, A332378. Bisections are A331425, A331428. Sequence in context: A262807 A169603 A022920 * A240825 A243773 A097604 Adjacent sequences:  A331420 A331421 A331422 * A331424 A331425 A331426 KEYWORD nonn,look AUTHOR César Eliud Lozada, Jan 16 2020 STATUS approved

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Last modified May 25 11:27 EDT 2020. Contains 334592 sequences. (Running on oeis4.)