A246920 The number of distinct lengths of nontrivial integral cevians of an equilateral triangle of side n that divide an edge into two integral parts.

0, 0, 1, 0, 2, 1, 2, 2, 2, 2, 2, 1, 2, 2, 5, 4, 2, 2, 2, 2, 5, 2, 2, 5, 4, 2, 3, 2, 2, 5, 2, 6, 5, 2, 8, 2, 2, 2, 5, 8, 2, 5, 2, 2, 8, 2, 2, 9, 4, 4, 5, 2, 2, 3, 8, 8, 5, 2, 2, 5, 2, 2, 8, 8, 8, 5, 2, 2, 5, 8, 2, 8, 2, 2, 9, 2, 8, 5, 2, 14, 4, 2, 2, 5, 8, 2

Offset: 1

Author

Colin Barker, Sep 07 2014

Keywords

• nonn

Comments

A cevian is a line segment which joins a vertex of a triangle with a point on the opposite side (or its extension).

A nontrivial cevian is one that does not coincide with a side of the triangle.

For an equilateral triangle of side n, the lengths of its cevians are the values of y in the solutions to x^2-y^2-n*x+n^2=0.

Examples

			a(15) = 5 because cevians of an equilateral triangle of side 15 have length 13, 21, 35, 57 or 169.
		

Links

• Colin Barker, Table of n, a(n) for n = 1..10000
• Wikipedia, Cevian

Crossrefs

Cf. A229839, A246918, A246919.

Programs

PARI

\\ Returns the number of cevians of an equilateral triangle of side n.
count(n) = {
  s=[];
  n=12*n^2;
  fordiv(n, f,
    g=n\f;
    if(f<=g && (f+g)%2==0,
      x=(f+g)\2;
      if(x%4==0,
        s=concat(s, x\4)
      )
    )
  );
  Colrev(s)~
}
vector(100, n, #count(n)-1)