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

1,3

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

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)

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 *)

Prog

(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))

Crossrefs

Cf. A229839, A246918, A246920.

Sequence in context: A225949 A067152 A052440 * A167299 A340952 A282071

Adjacent sequences: A246916 A246917 A246918 * A246920 A246921 A246922

Keyword

nonn

Author

Colin Barker, Sep 07 2014

Status

approved