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.

0, 0, 7, 0, 19, 14, 37, 13, 61, 38, 91, 28, 127, 74, 169, 49, 217, 122, 271, 76, 331, 182, 397, 109, 469, 254, 547, 148, 631, 338, 721, 193, 817, 434, 919, 244, 1027, 542, 1141, 301, 1261, 662, 1387, 364, 1519, 794, 1657, 433, 1801, 938, 1951, 508, 2107

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.

Links

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

Crossrefs

Cf. A229839, A246918, A246920.

Programs

Mathematica

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 *)

PARI

\\ Returns the length of the longest integral cevian of an equilateral triangle of side n.
longest(n) = {
  s=[];
  m=12*n^2;
  fordiv(m, f,
    g=m\f;
    if(f<=g && (f+g)%2==0,
      x=(f+g)\2;
      if(x%4==0,
        s=concat(s, x\4)
      )
    )
  );
  if(#s==1, return(0));
  for(i=1, #s, if(s[i]!=n, return(s[i])))
}
vector(100, n, longest(n))

Formula

Conjectures from Colin Barker, Jun 06 2016: (Start)

a(n) = 3*a(n-4)-3*a(n-8)+a(n-12) for n>14.

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).

(End)