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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.

%I A379597 #20 Mar 12 2025 04:16:16
%S A379597 1,4,12,24,50,80,134,192,276,366,510,632,834,1018,1262,1502,1858,2136,
%T A379597 2584,2956,3448,3910,4576,5076,5834,6488,7320,8066,9136,9892,11118,
%U A379597 12114,13358,14482,15978,17108,18862,20272,22024,23532,25700,27216,29600,31486,33746
%N 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.
%C A379597 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.
%C A379597 a(n) is even for n >= 2.
%H A379597 Felix Huber, <a href="/A379597/b379597.txt">Table of n, a(n) for n = 1..3800</a>
%H A379597 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/QuadraticEquation.html">Quadratic Equation</a>
%e A379597 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.
%p A379597 A379597:=proc(n)
%p A379597    option remember;
%p A379597    local a,u,v,w;
%p A379597    if n=1 then
%p A379597       1
%p A379597    else
%p A379597       a:=0;	
%p A379597       for u to n-1 do
%p A379597          for v from 0 to n-u do
%p A379597             w:=n-u-v;
%p A379597             if igcd(u,v,w)=1 then
%p A379597                if v=0 then
%p A379597                   a:=a+1
%p A379597                elif w=0 or w>=v^2/(4*u) then
%p A379597                   a:=a+2
%p A379597                else
%p A379597                   a:=a+4
%p A379597                fi
%p A379597             fi
%p A379597          od
%p A379597       od;
%p A379597       a+procname(n-1)
%p A379597    fi;
%p A379597 end proc;
%p A379597 seq(A379597(n),n=1..45);
%Y A379597 Cf. A067274, A091626, A091627, A364384, A364385, A365876, A365877, A381710, A381711.
%K A379597 nonn
%O A379597 1,2
%A A379597 _Felix Huber_, Feb 18 2025