A379597 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 nonnegative discriminant.

1, 4, 12, 24, 50, 80, 134, 192, 276, 366, 510, 632, 834, 1018, 1262, 1502, 1858, 2136, 2584, 2956, 3448, 3910, 4576, 5076, 5834, 6488, 7320, 8066, 9136, 9892, 11118, 12114, 13358, 14482, 15978, 17108, 18862, 20272, 22024, 23532, 25700, 27216, 29600, 31486, 33746

Offset: 1

Author

Felix Huber, Feb 18 2025

Keywords

• nonn

Comments

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

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

Examples

			a(3) = 12 because there are 12 equations with abs(u) + abs(v) + abs(w) <= 3 and distinct solution set having a nonnegative discriminant: (u, v, w) = (1, 0, 0), (1, -1, 0), (1, 1, 0), (1, 0, -1), (1, -1, -1), (1, 1, -1), (1, -2, 0), (1, 2, 0), (1, 0, -2), (2, -1, 0), (2, 1, 0), and (2, 0, -1). Multiplied equations like 2*(1, 0, 0) = (2, 0, 0) or (-1)*(1, -1, 0) = (-1, 1, 0) do not have a distinct solution set.
		

Links

• Felix Huber, Table of n, a(n) for n = 1..3800
• Eric Weisstein's World of Mathematics, Quadratic Equation

Crossrefs

Cf. A067274, A091626, A091627, A364384, A364385, A365876, A365877, A381710, A381711.

Programs

Maple

A379597:=proc(n)
   option remember;
   local a,u,v,w;
   if n=1 then
      1
   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=0 or w>=v^2/(4*u) then
                  a:=a+2
               else
                  a:=a+4
               fi
            fi
         od
      od;
      a+procname(n-1)
   fi;
end proc;
seq(A379597(n),n=1..45);