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A331423
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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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5
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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
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OFFSET
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1,4
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COMMENTS
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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.
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)
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LINKS
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MAPLE
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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;
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MATHEMATICA
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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]]
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PROG
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(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
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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