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

Offset

1,2

Links

Table of n, a(n) for n=1..12.

Joseph-Alfred Serret, Section 512, Cours d'algèbre supérieure, Gauthier-Villars.

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 * A359367 A321760 A083203

Adjacent sequences: A316181 A316182 A316183 * A316185 A316186 A316187

Keyword

nonn,more

Author

Greg Dresden, Jun 25 2018

Status

approved