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A246919
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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.
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4
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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
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OFFSET
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1,3
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COMMENTS
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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.
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LINKS
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FORMULA
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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)
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MATHEMATICA
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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 *)
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PROG
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(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))
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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