A381710 a(n) is the number of distinct solution sets to the quadratic equations u*x^2 + v*x + w = 0 with integer coefficients u, v, w, abs(u) + abs(v) + abs(w) <= n having a negative discriminant.

0, 1, 5, 11, 25, 39, 69, 99, 143, 189, 265, 327, 437, 529, 653, 777, 965, 1107, 1343, 1531, 1783, 2021, 2367, 2619, 3013, 3343, 3771, 4153, 4707, 5087, 5721, 6229, 6865, 7437, 8197, 8767, 9677, 10391, 11279, 12043, 13155, 13919, 15147, 16101, 17249, 18301, 19763

Offset: 1

Author

Felix Huber, Mar 06 2025

Keywords

• nonn

Comments

Quadratic equations u*x^2 + v*x + w = 0 with real coefficients u, v, w and negative discriminant v^2 - 4*u*w have two complex solutions.

a(n) is odd for n >= 2.

Examples

			a(3) = 5 because there are 5 equations with abs(u) + abs(v) + abs(w) <= 3 and distinct solution set having a negative discriminant: (u, v, w) = (1, 0, 1), (1, -1, 1), (1, 1, 1), (1, 0, 2), (2, 0, 1). Multiplied equations like (-1)*(1, -1, 1) = (-1, 1, -1) do not have a distinct solution set.
		

Links

• Felix Huber, Table of n, a(n) for n = 1..5000
• Eric Weisstein's World of Mathematics, Quadratic Equation

Crossrefs

Cf. A067274, A091626, A091627, A364384, A364385, A365876, A365877, A379597, A381711.

Programs

Maple

A381710:=proc(n)
   option remember;
   local a,u,v,w;
      if n=1 then
      0
   else
      a:=0;
      for u to n-1 do
         for v from 0 to n-u do
            w:=n-u-v;
               if igcd(u,v,w)=1 then
                  if v=0 then
                     a:=a+1
                  elif w>v^2/(4*u) then
                    a:=a+2
               fi
            fi
         od
      od;
      a+procname(n-1)
   fi;
end proc;
seq(A381710(n),n=1..47);