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A342723 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), integer diagonals e,f and integer area.

%I A342723 #10 Apr 27 2021 02:45:23
%S A342723 0,0,0,1,1,0,0,1,0,1,0,2,1,1,3,1,2,0,0,4,3,0,0,4,7,2,0,5,2,5,0,1,0,3,
%T A342723 4,3,4,0,6,7,3,4,0,3,4,0,0,5,0,9,10,9,3,0,5,8,0,4,0,17,4,0,9,1,19,2,0,
%U A342723 6,2,7,0,7,7,7,23,2,8,12,0,10,0,5,0,15,27
%N A342723 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), integer diagonals e,f and integer area.
%C A342723 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 A342723 a(4)=1 is the smallest possible solution and is a rectangle with a=c=4, b=d=3, e=f=5 and area 12. a(24)=4 includes the smallest possible solution with all sides a,b,c,d different and a=24, b=20, c=15, d=7, e=20, f=25 and area 234. Furthermore there are three rectangles with a=24,b=7, a=24,b=10 and a=24,b=18.
%t A342723 an={};
%t A342723 area[a_,b_,c_,d_,e_,f_]:=1/4 Sqrt[4e^2 f^2-(a^2+c^2-b^2-d^2)^2];
%t A342723 he[a_,b_,e_]:=1/(2 e) Sqrt[-(a-b-e) (a+b-e) (a-b+e) (a+b+e)];
%t A342723 paX[e_]:={e,0} (*vertex A coordinate*)
%t A342723 pbX[a_,b_,e_]:={(-a^2+b^2+e^2)/(2 e),he[a,b,e]}(*vertex B coordinate*)
%t A342723 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*)
%t A342723 convexQ[{bx_,by_},{dx_,dy_},e_]:=If[(by-dy) e>by dx-bx dy>0,True,False]
%t A342723 (*define order on tuples*)
%t A342723 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 A342723 (*check if tuple is canonical*)
%t A342723 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 A342723 Do[cnt=0;
%t A342723 Do[pa=paX[e];pb=pbX[a,b,e];pd=pdX[c,d,e];
%t A342723 If[(f=Sqrt[(pb-pd).(pb-pd)];IntegerQ[f])&&(ar=area[a,b,c,d,e,f]; IntegerQ[ar])&&convexQ[pb,pd,e]&&canonicalQ[{a,b,c,d,e,f}],cnt++
%t A342723 (*;Print[{{a,b,c,d,e,f,ar},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]}];
%t A342723 AppendTo[an,cnt],{a,1,85}]
%t A342723 an
%Y A342723 Cf. A340858 for trapezoids, A342720 and A342721 for concave integer quadrilaterals, A342722 for convex integer quadrilaterals with arbitrary area.
%K A342723 nonn
%O A342723 1,12
%A A342723 _Herbert Kociemba_, Apr 25 2021