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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.

%I A264263 #13 Dec 10 2016 19:39:43
%S A264263 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 A264263 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 A264263 15,7,1,3,7,7,11,5,1,5,11,3,3,7,3,7,7,2
%N 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.
%C A264263 A cevian is a line segment which joins a vertex of a triangle with a point on the opposite side (or its extension).
%C A264263 A nontrivial cevian is one that does not coincide with a side of the triangle.
%C A264263 If a(n) = 1 then the length of the unique cevian is n^2.
%C A264263 It seems that a(n) = 1 if and only if n is the average of twin prime pairs divided by 2 (A040040).
%H A264263 Colin Barker, <a href="/A264263/b264263.txt">Table of n, a(n) for n = 1..1000</a>
%H A264263 Wikipedia, <a href="http://en.wikipedia.org/wiki/Cevian">Cevian</a>
%H A264263 Wikipedia, <a href="http://en.wikipedia.org/wiki/Isosceles_triangle">Isosceles triangle</a>
%e A264263 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 A264263 (PARI)
%o A264263 ceviso(n) = {
%o A264263   my(d, L=List());
%o A264263   for(k=1, n^2,
%o A264263     if(issquare(n^2+k^2-k, &d) && d!=n,
%o A264263       listput(L, d)
%o A264263     )
%o A264263   );
%o A264263   Vec(L)
%o A264263 }
%o A264263 vector(100, n, #ceviso(n))
%Y A264263 Cf. A040040, A264264.
%K A264263 nonn,easy
%O A264263 1,4
%A A264263 _Colin Barker_, Nov 10 2015