A342722 a(n) is the number of convex 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, 2, 2, 1, 5, 7, 8, 5, 7, 13, 14, 11, 15, 31, 18, 14, 18, 30, 25, 24, 22, 64, 42, 35, 51, 58, 34, 48, 37, 87, 71, 46, 69, 74, 51, 53, 74, 110, 53, 72, 61, 96, 106, 73, 60, 181, 102, 103, 125, 134, 79, 118, 133, 215, 141, 82, 82, 221

Offset: 1

Author

Herbert Kociemba, Apr 25 2021

Keywords

• nonn

Comments

Without loss of generality we assume that a is the largest side length and that the diagonal e divides the convex quadrilateral into two triangles with sides a,b,e and c,d,e. The triangle inequality implies e > a-b and abs(e-c) < d < e+c.

Examples

			a(6)=1 because the only convex integer quadrilateral with longest edge length 6 is a trapezoid with sides a=6, b=5, c=4, d=5 and diagonals e=f=7.
		

Crossrefs

Cf. A340858 for trapezoids, A342720 and A342721 for concave integer quadrilaterals, A342723 for convex integer quadrilaterals with integer area.

Programs

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*)
convexQ[{bx_,by_},{dx_,dy_},e_]:=If[(by-dy) e>by dx-bx dy>0,True,False]
(*define order on tuples*)
gQ[x_,y_]:=Module[{z=x-y,res=False},Do[If[z[[i]]>0,res=True;Break[],If[z[[i]]<0,Break[]]],{i,1,6}];res]
(*check if tuple is canonical*)
canonicalQ[{a_,b_,c_,d_,e_,f_}]:=Module[{x={a,b,c,d,e,f}},If[(gQ[{b,a,d,c,e,f},x]||gQ[{d,c,b,a,e,f},x]||gQ[{c,d,a,b,e,f},x]||gQ[{b,c,d,a,f,e},x]||gQ[{a,d,c,b,f,e},x]||gQ[{c,b,a,d,f,e},x]||gQ[{d,a,b,c,f,e},x]),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])&&convexQ[pb,pd,e]&&canonicalQ[{a,b,c,d,e,f}],cnt++
(*;Print[{{a,b,c,d,e,f},Graphics[Line[{pa,pb,pc,pd,pa}]]}]*)],
{b,1,a},{e,a-b+1,a+b-1},{c,1,a},{d,Abs[e-c]+1,Min[a,e+c-1]}];
AppendTo[an,cnt],{a,1 ,60}
]
an