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A246919 The length of the longest nontrivial integral cevian of an equilateral triangle of side n that divides an edge into two integral parts, or 0 if no such cevian exists.

%I A246919 #26 Dec 10 2016 19:35:35
%S A246919 0,0,7,0,19,14,37,13,61,38,91,28,127,74,169,49,217,122,271,76,331,182,
%T A246919 397,109,469,254,547,148,631,338,721,193,817,434,919,244,1027,542,
%U A246919 1141,301,1261,662,1387,364,1519,794,1657,433,1801,938,1951,508,2107
%N A246919 The length of the longest nontrivial integral cevian of an equilateral triangle of side n that divides an edge into two integral parts, or 0 if no such cevian exists.
%C A246919 A cevian is a line segment which joins a vertex of a triangle with a point on the opposite side (or its extension).
%C A246919 A nontrivial cevian is one that does not coincide with a side of the triangle.
%C A246919 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.
%H A246919 Colin Barker, <a href="/A246919/b246919.txt">Table of n, a(n) for n = 1..10000</a>
%H A246919 Wikipedia, <a href="http://en.wikipedia.org/wiki/Cevian">Cevian</a>
%F A246919 Conjectures from _Colin Barker_, Jun 06 2016: (Start)
%F A246919 a(n) = 3*a(n-4)-3*a(n-8)+a(n-12) for n>14.
%F A246919 G.f.: x^3*(7 +19*x^2 +14*x^3 +16*x^4 +13*x^5 +4*x^6 -4*x^7 +x^8 -11*x^9 +x^10 +2*x^11 +4*x^13) / ((1 -x)^3*(1 +x)^3*(1 +x^2)^3).
%F A246919 (End)
%t A246919 Rest@ CoefficientList[Series[x^3 (7 + 19 x^2 + 14 x^3 + 16 x^4 + 13 x^5 + 4 x^6 - 4 x^7 + x^8 - 11 x^9 + x^10 + 2 x^11 + 4 x^13)/((1 - x)^3 (1 + x)^3 (1 + x^2)^3), {x, 0, 53}], x] (* _Michael De Vlieger_, Jun 06 2016 *)
%o A246919 (PARI)
%o A246919 \\ Returns the length of the longest integral cevian of an equilateral triangle of side n.
%o A246919 longest(n) = {
%o A246919   s=[];
%o A246919   m=12*n^2;
%o A246919   fordiv(m, f,
%o A246919     g=m\f;
%o A246919     if(f<=g && (f+g)%2==0,
%o A246919       x=(f+g)\2;
%o A246919       if(x%4==0,
%o A246919         s=concat(s, x\4)
%o A246919       )
%o A246919     )
%o A246919   );
%o A246919   if(#s==1, return(0));
%o A246919   for(i=1, #s, if(s[i]!=n, return(s[i])))
%o A246919 }
%o A246919 vector(100, n, longest(n))
%Y A246919 Cf. A229839, A246918, A246920.
%K A246919 nonn
%O A246919 1,3
%A A246919 _Colin Barker_, Sep 07 2014