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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.

%I A342720 #13 Apr 16 2021 00:12:59
%S A342720 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,
%T A342720 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,
%U A342720 9,18,25,46,38,18,2,25,20,30,18,32,17,32,43
%N 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.
%C A342720 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.
%e A342720 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}.
%t A342720 an={};
%t A342720 he[a_,b_,e_]:=1/(2 e) Sqrt[(-((a-b-e) (a+b-e) (a-b+e) (a+b+e)))]
%t A342720 paX[e_]:={e,0} (*vertex A coordinate*)
%t A342720 pbX[a_,b_,e_]:={(-a^2+b^2+e^2)/(2 e),he[a,b,e]}(*vertex B coordinate*)
%t A342720 pc={0,0};(*vertex C coordinate*)
%t A342720 pdX[c_,d_,e_]:={(c^2-d^2+e^2)/(2 e),-he[c,d,e]}(*vertex D coordinate*)
%t A342720 concaveQ[{bx_,by_},{dx_,dy_},e_]:=If[by dx-bx dy<0||by dx-bx dy>(by-dy) e,True,False]
%t A342720 gQ[x_,y_]:=Module[{z=x-y,res=False},Do[If[z[[i]]>0,res=True;Break[],
%t A342720   If[z[[i]]<0,Break[]]],{i,1,4}];res]
%t A342720 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]]
%t A342720 Do[cnt=0;
%t A342720 Do[pa=paX[e];pb=pbX[a,b,e];pd=pdX[c,d,e];
%t A342720 If[(f=Sqrt[(pb-pd).(pb-pd)];IntegerQ[f])&&concaveQ[pb,pd,e]&&canonicalQ[{a,b,c,d}],cnt++
%t A342720 (*;Print[{{a,b,c,d,e,f},Graphics[Line[{pa,pb,pc,pd,pa}]]}]*)],
%t A342720 {b,1,a},{e,a-b+1,a-1},{c,1,a},{d,Abs[e-c]+1,Min[a,e+c-1]}];
%t A342720 AppendTo[an,cnt],
%t A342720 {a,1,75}
%t A342720 ]
%t A342720 an
%Y A342720 Cf. A340858 for trapezoids, A342721 for concave integer quadrilaterals with integer area.
%K A342720 nonn
%O A342720 1,17
%A A342720 _Herbert Kociemba_, Mar 19 2021