A155193 Number of cubic equations ax^3 + bx^2 + cx + d = 0 with integer coefficients |a|,|b|,|c|,|d| <= n, a <> 0, having one real root and two conjugate complex roots.

0, 38, 384, 1614, 4592, 10506, 20828, 37358, 62132, 97574, 146308, 211418, 296032, 403918, 538784, 704918, 906720, 1149162, 1437036, 1776038, 2171556, 2629790, 3156924, 3759698, 4444648, 5219390, 6091008, 7067614, 8157088, 9368178

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

Table of n, a(n) for n=0..29.

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, A155192.

Sequence in context: A249711 A220918 A187078 * A159943 A221634 A251056

Adjacent sequences: A155190 A155191 A155192 * A155194 A155195 A155196

Keyword

nonn

Author

Rick L. Shepherd, Jan 21 2009

Status

approved