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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.
3
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
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
Sequence in context: A239834 A337148 A202804 * A229720 A350108 A024933
KEYWORD
nonn
AUTHOR
Rick L. Shepherd, Jan 21 2009
STATUS
approved