%I #11 May 16 2021 05:52:34

%S 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,

%T 51,58,34,48,37,87,71,46,69,74,51,53,74,110,53,72,61,96,106,73,60,181,

%U 102,103,125,134,79,118,133,215,141,82,82,221

%N 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.

%C 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.

%e 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.

%t an={};

%t he[a_,b_,e_]:=1/(2 e) Sqrt[-(a-b-e) (a+b-e) (a-b+e) (a+b+e)];

%t paX[e_]:={e,0} (*vertex A coordinate*)

%t pbX[a_,b_,e_]:={(-a^2+b^2+e^2)/(2 e),he[a,b,e]}(*vertex B coordinate*)

%t pc={0,0};(*vertex C coordinate*)

%t pdX[c_,d_,e_]:={(c^2-d^2+e^2)/(2 e),-he[c,d,e]}(*vertex D coordinate*)

%t convexQ[{bx_,by_},{dx_,dy_},e_]:=If[(by-dy) e>by dx-bx dy>0,True,False]

%t (*define order on tuples*)

%t 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]

%t (*check if tuple is canonical*)

%t 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]]

%t Do[cnt=0;

%t Do[pa=paX[e];pb=pbX[a,b,e];pd=pdX[c,d,e];

%t If[(f=Sqrt[(pb-pd).(pb-pd)];IntegerQ[f])&&convexQ[pb,pd,e]&&canonicalQ[{a,b,c,d,e,f}],cnt++

%t (*;Print[{{a,b,c,d,e,f},Graphics[Line[{pa,pb,pc,pd,pa}]]}]*)],

%t {b,1,a},{e,a-b+1,a+b-1},{c,1,a},{d,Abs[e-c]+1,Min[a,e+c-1]}];

%t AppendTo[an,cnt],{a,1 ,60}

%t ]

%t an

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

%K nonn

%O 1,4

%A _Herbert Kociemba_, Apr 25 2021