A155192 Number of cubic equations ax^3 + bx^2 + cx + d = 0 with integer coefficients |a|,|b|,|c|,|d| <= n, a <> 0, having three real roots, of which at least two are equal.

0, 10, 32, 70, 132, 198, 272, 370, 504, 646, 780, 934, 1152, 1330, 1520, 1734, 2036, 2270, 2560, 2818, 3184, 3494, 3788, 4110, 4584, 4970, 5328, 5782, 6284, 6686, 7128, 7554, 8192

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..512

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, ", "))}

Crossrefs

Cf. A155191, A155193.

Sequence in context: A239834 A337148 A202804 * A229720 A350108 A024933

Adjacent sequences: A155189 A155190 A155191 * A155193 A155194 A155195

Keyword

nonn

Author

Rick L. Shepherd, Jan 21 2009

Status

approved