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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.

%I A381710 #6 Mar 12 2025 18:51:51
%S A381710 0,1,5,11,25,39,69,99,143,189,265,327,437,529,653,777,965,1107,1343,
%T A381710 1531,1783,2021,2367,2619,3013,3343,3771,4153,4707,5087,5721,6229,
%U A381710 6865,7437,8197,8767,9677,10391,11279,12043,13155,13919,15147,16101,17249,18301,19763
%N 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.
%C A381710 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.
%C A381710 a(n) is odd for n >= 2.
%H A381710 Felix Huber, <a href="/A381710/b381710.txt">Table of n, a(n) for n = 1..5000</a>
%H A381710 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/QuadraticEquation.html">Quadratic Equation</a>
%e A381710 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.
%p A381710 A381710:=proc(n)
%p A381710    option remember;
%p A381710    local a,u,v,w;
%p A381710       if n=1 then
%p A381710       0
%p A381710    else
%p A381710       a:=0;
%p A381710       for u to n-1 do
%p A381710          for v from 0 to n-u do
%p A381710             w:=n-u-v;
%p A381710                if igcd(u,v,w)=1 then
%p A381710                   if v=0 then
%p A381710                      a:=a+1
%p A381710                   elif w>v^2/(4*u) then
%p A381710                     a:=a+2
%p A381710                fi
%p A381710             fi
%p A381710          od
%p A381710       od;
%p A381710       a+procname(n-1)
%p A381710    fi;
%p A381710 end proc;
%p A381710 seq(A381710(n),n=1..47);
%Y A381710 Cf. A067274, A091626, A091627, A364384, A364385, A365876, A365877, A379597, A381711.
%K A381710 nonn
%O A381710 1,3
%A A381710 _Felix Huber_, Mar 06 2025