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
Keywords
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
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
Comments