A391527 a(n) is the number of quadratic polynomials with coefficients in {-n, ..., n}, positive leading coefficient, and having two rational roots.

4, 18, 44, 88, 138, 212, 296, 406, 524, 674, 814, 1024, 1194, 1416, 1662, 1946, 2180, 2526, 2794, 3176, 3552, 3938, 4278, 4808, 5232, 5706, 6204, 6776, 7206, 7936, 8416, 9072, 9708, 10342, 11032, 11950, 12532, 13246, 14026, 15010, 15664, 16718, 17418, 18402, 19470

Offset

1,1

Comments

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

Links

Table of n, a(n) for n=1..45.

Example

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.
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.

Maple

a := proc(n)
   local d, i, j, k:
   d := 0:
   for i from 1 to n do
      for j from -n to n do
         for k from -n to n do
            if issqr(j^2-4*i*k) then d := d+1: fi: od: od: od:
   d: end:
seq(a(r), r = 1 .. 45);

Mathematica

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 *)

Crossrefs

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

Sequence in context: A023618 A366985 A396180 * A115077 A278046 A258634

Adjacent sequences: A391524 A391525 A391526 * A391528 A391529 A391530

Keyword

nonn

Author

Lorenzo Sauras Altuzarra, Jan 15 2026

Status

approved