

A316184


Positive integers R such that there is a cubic x^3  Qx + R that has three real roots whose continued fraction expansion have common tails.


1



1, 7, 9, 35, 37, 91, 183, 189, 341, 559, 845, 855
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OFFSET

1,2


LINKS

Table of n, a(n) for n=1..12.
JosephAlfred Serret, Section 512, Cours d'algèbre supérieure, GauthierVillars.


EXAMPLE

For the first entry of R=1, we have the polynomial x^3  3x + 1. Its roots, expressed as continued fractions, all have a common tail of 3, 2, 3, 1, 1, 6, 11, ... The next examples are R=7 with the polynomial x^3  7x + 7, then R=9 with the polynomial x^3  9x + 9, and Q=35 with the polynomial x^3  21x + 35. Note that for the R=7 example, we get the common tail of 2, 3, 1, 6, 10, 5, ... which is contained in A039921.


MATHEMATICA

SetOfQRs = {};
M = 1000;
Do[
If[Divisible[3 (a^2  a + 1), c^2] &&
Divisible[(2 a  1) (a^2  a + 1), c^3] &&
3 (a^2  a + 1)/c^2 <= M,
SetOfQRs =
Union[SetOfQRs, { { (3 (a^2  a + 1))/
c^2, ((2 a  1) (a^2  a + 1))/c^3}} ]],
{c, 1, M/3 + 1, 2}, {a, 1, Sqrt[M c^2/3 + 3/4] + 1/2}];
Print[SetOfQRs // MatrixForm];


CROSSREFS

Cf. A039921, A316157.
Sequence in context: A261961 A177030 A189974 * A321760 A083203 A082536
Adjacent sequences: A316181 A316182 A316183 * A316185 A316186 A316187


KEYWORD

nonn,more


AUTHOR

Greg Dresden, Jun 25 2018


STATUS

approved



