cp's OEIS Frontend

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.

%I A316184 #11 Jul 31 2018 22:49:31
%S A316184 1,7,9,35,37,91,183,189,341,559,845,855
%N 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.
%H A316184 Joseph-Alfred Serret, <a href="https://gallica.bnf.fr/ark:/12148/bpt6k29114h/f245.image">Section 512</a>, Cours d'algèbre supérieure, Gauthier-Villars.
%e A316184 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.
%t A316184 SetOfQRs = {};
%t A316184 M = 1000;
%t A316184 Do[
%t A316184   If[Divisible[3 (a^2 - a + 1), c^2] &&
%t A316184     Divisible[(2 a - 1) (a^2 - a + 1), c^3] &&
%t A316184     3 (a^2 - a + 1)/c^2 <=  M,
%t A316184    SetOfQRs =
%t A316184     Union[SetOfQRs, { { (3 (a^2 - a + 1))/
%t A316184        c^2, ((2 a - 1) (a^2 - a + 1))/c^3}}   ]],
%t A316184   {c, 1, M/3 + 1, 2}, {a, 1, Sqrt[M c^2/3 + 3/4] + 1/2}];
%t A316184 Print[SetOfQRs // MatrixForm];
%Y A316184 Cf. A039921, A316157.
%K A316184 nonn,more
%O A316184 1,2
%A A316184 _Greg Dresden_, Jun 25 2018