A114798 Cubic polynomial coefficients such that an elliptical term is zero.

3, 2, 12, 16, 27, 54, 48, 128, 75, 250, 108, 432, 147, 686, 192, 1024, 243, 1458, 300, 2000, 363, 2662, 432, 3456, 507, 4394, 588, 5488, 675, 6750, 768, 8192, 867, 9826, 972, 11664, 1083, 13718, 1200, 16000, 1323, 18522, 1452, 21296, 1587, 24334, 1728

Offset

0,1

Comments

I had noticed that the elliptical term: j=g2[n]^3/(g2[n]^2-27*g3[n]^2) was singular for a kind of polynomial with three real roots: (x+n)^2*(x-2*n) This table gives all zeros: Table[((4*a[[2*n + 1]])^3 - 27*(4*a[[2*n + 2]])^2)/(4*a[[2*n + 1]])^3, {n, 0, 49}]

Apparently pairs (a(2*n), a(2*n+1)) such that x^3 - a(2*n)*x + a(2*n+1) = (x-(2*n+2)) * (x+(n+1))^2. [Joerg Arndt, Mar 15 2013]

Links

Table of n, a(n) for n=0..46.

Index entries for linear recurrences with constant coefficients, signature (0,4,0,-6,0,4,0,-1).

Formula

w^2=4*z^3-g2[n]*z-g3[n] a(n) = {g2[n],g3[n]}/4

a(n) = (3*n^2+12*n+12)/4 for n even. a(n) = (n^3+3*n^2+3*n+1)/4 for n odd. G.f.: (2*x^5-3*x^4+8*x^3+2*x+3) / ((x-1)^4*(x+1)^4). - Colin Barker, Mar 15 2013

Example

x^3-3*x-2
x^3-12*x-16
x^3-27*x-54

Mathematica

a = Flatten[Table[Abs[Coefficient[Expand[(x + n)^2*(x - 2*n)], x, 1 - m]], {n, 1, 50}, {m, 0, 1}]]

Crossrefs

Sequence in context: A220883 A253246 A152550 * A167639 A113205 A136657

Adjacent sequences: A114795 A114796 A114797 * A114799 A114800 A114801

Keyword

nonn,uned,easy

Author

Roger L. Bagula, Feb 18 2006

Status

approved