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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.
2

%I #13 Dec 10 2016 19:39:43

%S 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,

%T 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,

%U 15,7,1,3,7,7,11,5,1,5,11,3,3,7,3,7,7,2

%N 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.

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

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

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

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

%H Colin Barker, <a href="/A264263/b264263.txt">Table of n, a(n) for n = 1..1000</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Cevian">Cevian</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Isosceles_triangle">Isosceles triangle</a>

%e 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.

%o (PARI)

%o ceviso(n) = {

%o my(d, L=List());

%o for(k=1, n^2,

%o if(issquare(n^2+k^2-k, &d) && d!=n,

%o listput(L, d)

%o )

%o );

%o Vec(L)

%o }

%o vector(100, n, #ceviso(n))

%Y Cf. A040040, A264264.

%K nonn,easy

%O 1,4

%A _Colin Barker_, Nov 10 2015