A342720 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) and integer diagonals e,f.

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 1, 3, 1, 2, 2, 3, 1, 4, 2, 4, 2, 5, 3, 7, 1, 2, 4, 3, 13, 7, 20, 12, 5, 3, 7, 10, 3, 8, 2, 14, 12, 10, 15, 17, 8, 11, 10, 20, 13, 15, 10, 45, 9, 18, 25, 46, 38, 18, 2, 25, 20, 30, 18, 32, 17, 32, 43

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(15)=1 because the only concave integer quadrilateral with longest edge length 15 has a=15, b=13, c=13, d=15 and diagonals e=4 and f=24. a(20)=3 because there are three solutions (a,b,c,d,e,f): (20,13,15,18,9,26), (20,13,13,20,11,24) and {20,15,15,20,7,24}.

Mathematica

an={};
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])&&concaveQ[pb, pd, e]&&canonicalQ[{a, b, c, d}], cnt++
(*; Print[{{a, b, c, d, e, f}, 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, A342721 for concave integer quadrilaterals with integer area.

Sequence in context: A239928 A114912 A231723 * A029274 A239498 A079219

Adjacent sequences: A342717 A342718 A342719 * A342721 A342722 A342723

Keyword

nonn

Author

Herbert Kociemba, Mar 19 2021

Status

approved