A391527 - OEIS

%I #47 Feb 02 2026 10:27:15

%S 4,18,44,88,138,212,296,406,524,674,814,1024,1194,1416,1662,1946,2180,

%T 2526,2794,3176,3552,3938,4278,4808,5232,5706,6204,6776,7206,7936,

%U 8416,9072,9708,10342,11032,11950,12532,13246,14026,15010,15664,16718,17418,18402,19470

%N a(n) is the number of quadratic polynomials with coefficients in {-n, ..., n}, positive leading coefficient, and having two rational roots.

%C a(n) is also the number of quadratic polynomials with coefficients in {-n, ..., n}, positive leading coefficient, and whose splitting field is Q.

%e For n = 1, the a(1) = 4 polynomials with coefficients in {-1, 0, 1} are: x^2-x, x^2-1, x^2, x^2+x.

%e For n = 2, the a(2) = 18 polynomials with coefficients in {-2, -1, 0, 1, 2} are: x^2-2x, x^2-2x+1, x^2-x-2, x^2-x, x^2-1, x^2, x^2+x-2, x^2+x, x^2+2x, x^2+2x+1, 2x^2-2x, 2x^2-2, 2x^2, 2x^2+2x, 2x^2-x-1, 2x^2-x, 2x^2+x-1, 2x^2+x.

%p a := proc(n)

%p    local d, i, j, k:

%p    d := 0:

%p    for i from 1 to n do

%p       for j from -n to n do

%p          for k from -n to n do

%p             if issqr(j^2-4*i*k) then d := d+1: fi: od: od: od:

%p    d: end:

%p seq(a(r), r = 1 .. 45);

%t a[n_]:=Count[Flatten @ Table[j^2 - 4 i k, {i,1,n}, {j,-n,n}, {k,-n,n}],m_/; IntegerQ[Sqrt[m]] && m >= 0 ];Array[a,45] (* _James C. McMahon_, Feb 02 2026 *)

%Y Cf. A391597 (quadratic, integer), A391108 (cubic, integer), A392944 (cubic, rational).

%K nonn

%O 1,1

%A _Lorenzo Sauras Altuzarra_, Jan 15 2026