I add a Delphi-procedure, which calculates the possible solutions in the form (x, y, z) if you use only positive integers.

procedure TForm1.Button2Click(Sender: TObject);

var x,y,z:integer;

begin

for x:= 1 to 50 do

for y:= 1 to 45 do

for z := 1 to 40 do

begin

if (x*x+2*y*y+3*z*z=1417) then

begin

form1.memo1.lines.add(inttostr(x));

form1.memo2.lines.add(inttostr(y));

form1.memo3.lines.add(inttostr(z));

end;

end;

end;

(1, 18, 16)

(10, 15, 17)

(19, 12, 16)

(34, 3, 9)