A155191 Number of cubic equations ax^3 + bx^2 + cx + d = 0 with integer coefficients |a|,|b|,|c|,|d| <= n, a <> 0, having three distinct real roots.

0, 6, 84, 374, 1108, 2606, 5264, 9522, 15972, 25242, 38132, 55322, 77816, 106510, 142588, 187078, 241228, 306318, 383912, 475266, 582100, 706010, 848788, 1012050, 1197920, 1408190, 1645268, 1910854, 2207436, 2537118, 2902896, 3306402

Offset

0,2

Comments

Clearly each term is even as ax^3 + bx^2 + cx + d = 0 and -ax^3 - bx^2 - cx - d = 0 have the same roots.

The variable D in the PARI program below is the discriminant of the reduced form y^3 + py + q = 0.

References

Jan Gullberg, Mathematics, From the Birth of Numbers, W. W. Norton & Co., NY, pages 318-9.

Links

Gheorghe Coserea, Table of n, a(n) for n = 0..128

Prog

(PARI) {for(n=0, 32, c=0; forvec(xx=[[ -n, n], [ -n, n], [ -n, n], [ -n, n]],
if(xx[1]==0, next, z=Pol(xx); x=y-xx[2]/(3*xx[1]);
zz=eval(z); if(polcoeff(zz, 3)<>1, zz=zz/polcoeff(zz, 3));
p=polcoeff(zz, 1); q=polcoeff(zz, 0); D=(q/2)^2+(p/3)^3;
if(D<0, c++))); print1(c, ", "))}
(PARI)
Delta(a, b, c, d) = b^2*c^2 - 4*a*c^3 - 4*b^3*d - 27*a^2*d^2 + 18*a*b*c*d;
seq(n) = {
  my(a = vector(n));
  forvec(v=[[1, n], [-n, n], [-n, n], [-n, n]],
         if (Delta(v[1], v[2], v[3], v[4]) > 0, a[vecmax(abs(v))]++));
  for (i = 2, #a, a[i] += a[i-1]);
  return(concat(0, 2*a));
};
seq(31) \\ Gheorghe Coserea, Apr 09 2016

Crossrefs

Cf. A155192, A155193.

Sequence in context: A351178 A350947 A288321 * A211171 A338405 A054605

Adjacent sequences: A155188 A155189 A155190 * A155192 A155193 A155194

Keyword

nonn

Author

Rick L. Shepherd, Jan 21 2009

Status

approved