# Greetings from The On-Line Encyclopedia of Integer Sequences! http://oeis.org/ Search: id:a316184 Showing 1-1 of 1 %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, Section 512, 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 # Content is available under The OEIS End-User License Agreement: http://oeis.org/LICENSE