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A342721
a(n) is the number of concave integer quadrilaterals (up to congruence) with integer side lengths a,b,c,d with n=Max(a,b,c,d), integer diagonals e,f and integer area.
5
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 0, 2, 0, 0, 0, 3, 1, 1, 0, 0, 1, 3, 0, 0, 0, 2, 1, 0, 6, 0, 4, 4, 2, 1, 0, 0, 1, 0, 0, 6, 0, 2, 8, 6, 2, 0, 1, 2, 0, 2, 0, 9, 0, 0, 2, 0, 13, 1, 0, 4, 0, 3, 0, 3, 5, 10, 11
OFFSET
1,17
COMMENTS
Without loss of generality we assume that a is the largest side length and that the diagonal e divides the concave quadrilateral into two triangles with sides a,b,e and c,d,e. Then e < a is a necessary condition for concavity. The triangle inequality further implies e > a-b and abs(e-c) < d < e+c.
EXAMPLE
a(66)=1 because the only concave integer quadrilateral with longest edge length 66 and integer area has sides a=66, b=55, c=12, d=65, diagonals e=55, f=65 and area 1650.
MATHEMATICA
an={};
area[a_, b_, c_, d_, e_, f_]:=(1/4)Sqrt[(4e^2 f^2-(a^2+c^2-b^2-d^2)^2)]
he[a_, b_, e_]:=(1/(2 e))Sqrt[(-((a-b-e) (a+b-e) (a-b+e) (a+b+e)))];
paX[e_]:={e, 0} (*vertex A coordinate*)
pbX[a_, b_, e_]:={(-a^2+b^2+e^2)/(2 e), he[a, b, e]}(*vertex B coordinate*)
pc={0, 0}; (*vertex C coordinate*)
pdX[c_, d_, e_]:={(c^2-d^2+e^2)/(2 e), -he[c, d, e]}(*vertex D coordinate*)
concaveQ[{bx_, by_}, {dx_, dy_}, e_]:=If[by dx-bx dy<0||by dx-bx dy>(by-dy) e, True, False]
gQ[x_, y_]:=Module[{z=x-y, res=False}, Do[If[z[[i]]>0, res=True; Break[], If[z[[i]]<0, Break[]]], {i, 1, 4}]; res]
canonicalQ[{a_, b_, c_, d_}]:=Module[{m={a, b, c, d}}, If[(gQ[{b, a, d, c}, m]||gQ[{d, c, b, a}, m]||gQ[{c, d, a, b}, m]), False, True]]
Do[cnt=0;
Do[pa=paX[e]; pb=pbX[a, b, e]; pd=pdX[c, d, e];
If[(f=Sqrt[(pb-pd).(pb-pd)]; IntegerQ[f])&&(ar=area[a, b, c, d, e, f]; IntegerQ[ar])&&concaveQ[pb, pd, e]&&canonicalQ[{a, b, c, d}], cnt++
(*; Print[{{a, b, c, d, e, f, ar}, Graphics[Line[{pa, pb, pc, pd, pa}]]}]*)],
{b, 1, a}, {e, a-b+1, a-1}, {c, 1, a}, {d, Abs[e-c]+1, Min[a, e+c-1]}];
AppendTo[an, cnt], {a, 1, 75}
]
an
CROSSREFS
Cf. A340858 for trapezoids, A342720 for concave integer quadrilaterals with arbitrary area.
Sequence in context: A329265 A130209 A109127 * A263456 A002284 A016424
KEYWORD
nonn
AUTHOR
Herbert Kociemba, Mar 19 2021
STATUS
approved