

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
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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

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=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

Cf. A155191, A155193.
Sequence in context: A063926 A239834 A202804 * A229720 A024933 A198646
Adjacent sequences: A155189 A155190 A155191 * A155193 A155194 A155195


KEYWORD

nonn


AUTHOR

Rick L. Shepherd, Jan 21 2009


STATUS

approved



