

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
(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 3189.


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=yxx[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



KEYWORD

nonn


AUTHOR



STATUS

approved



