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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. 2
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 (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

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

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Last modified June 18 13:15 EDT 2021. Contains 345112 sequences. (Running on oeis4.)