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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. 3
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 (list; graph; refs; listen; history; text; internal format)
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: A230491 A067249 A288321 * A211171 A054605 A119576

Adjacent sequences:  A155188 A155189 A155190 * A155192 A155193 A155194

KEYWORD

nonn

AUTHOR

Rick L. Shepherd, Jan 21 2009

STATUS

approved

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Last modified October 22 17:54 EDT 2019. Contains 328319 sequences. (Running on oeis4.)