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.

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.

Links

Table of n, a(n) for n=1..75.

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

Adjacent sequences: A342718 A342719 A342720 * A342722 A342723 A342724

Keyword

nonn

Author

Herbert Kociemba, Mar 19 2021

Status

approved