A264263 The number of distinct nontrivial integral cevians of an isosceles triangle, with base of length 1 and legs of length n, that divide the base into two integral parts.

0, 1, 1, 2, 2, 1, 3, 3, 1, 3, 3, 2, 5, 3, 1, 3, 7, 3, 3, 3, 1, 5, 5, 2, 5, 3, 3, 7, 3, 1, 5, 11, 3, 3, 3, 1, 5, 11, 3, 4, 4, 3, 7, 3, 3, 7, 7, 3, 5, 5, 1, 7, 7, 1, 3, 3, 3, 11, 11, 5, 5, 7, 3, 3, 3, 3, 15, 7, 1, 3, 7, 7, 11, 5, 1, 5, 11, 3, 3, 7, 3, 7, 7, 2

Offset: 1

Author

Colin Barker, Nov 10 2015

Keywords

• nonn
• easy

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.

If a(n) = 1 then the length of the unique cevian is n^2.

It seems that a(n) = 1 if and only if n is the average of twin prime pairs divided by 2 (A040040).

Examples

			a(4) = 2 because for legs of length 4 there are two cevians, of length 6 and 16, that divide the base into two integral parts.
		

Links

• Colin Barker, Table of n, a(n) for n = 1..1000
• Wikipedia, Cevian
• Wikipedia, Isosceles triangle

Crossrefs

Cf. A040040, A264264.

Programs

PARI

ceviso(n) = {
  my(d, L=List());
  for(k=1, n^2,
    if(issquare(n^2+k^2-k, &d) && d!=n,
      listput(L, d)
    )
  );
  Vec(L)
}
vector(100, n, #ceviso(n))