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

1,2

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.

Links

Felix Huber, Table of n, a(n) for n = 1..3800

Eric Weisstein's World of Mathematics, Quadratic Equation

Example

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.

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);

Crossrefs

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

Sequence in context: A279626 A238607 A143270 * A037338 A136486 A003203

Adjacent sequences: A379594 A379595 A379596 * A379598 A379599 A379600

Keyword

nonn

Author

Felix Huber, Feb 18 2025

Status

approved