A114984 Coefficients of cubic equations in the form w^2=4*x^3-g2*x-g3 Weierstrass elliptic form whose solutions approximate zeta zeros.

200, 199, 442, 441, 684, 683, 926, 925, 1168, 1167, 1410, 1409, 1652, 1651, 1894, 1893, 2136, 2135, 2378, 2377, 2620, 2619, 2862, 2861, 3104, 3103, 3346, 3345, 3588, 3587, 3830, 3829, 4072, 4071, 4314, 4313, 4556, 4555, 4798, 4797, 5040, 5039, 5282

Offset

0,1

Comments

Other good approximation functions are: 1/2+I*b[n]->1/2-I/LogIntegral[1/(2.85*n)] 1/2+I*b[n]->1/2-I/LogIntegral[1/(4.05*Sqrt[n])] These types of functions gives the root finding function in Mathematica a place to start in iterations.

Links

Table of n, a(n) for n=0..42.

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

Formula

f[n = 200 + 242*(n - 1); g2/4=(1-f[n]);g3/4=f[n]; (a(n),a)(n+1) = {g3/4,g2/4}.

a(n) = (139+61*(-1)^n+121*n). a(n) = a(n-1)+a(n-2)-a(n-3). G.f.: (43*x^2-x+200) / ((x-1)^2*(x+1)). [Colin Barker, Dec 26 2012]

Example

f[n_] = 200 + 242*(n - 1);
b = Table[Table[x /. NSolve[x^3 - (1 - f[n])*x + f[n] == 0, x][[m]], {m, 1, 3}], {n, 1, 25}]
gives roots like:
{-1., 0.5 - 14.1333*I, 0.5+ 14.1333*I},
{-1., 0.5 - 21.0178*I, 0.5 + 21.0178*I}

Mathematica

f[n_] = 200 + 242*(n - 1); a = Flatten[Table[Abs[Coefficient[x^3 - (1 - f[n])*x + f[n], x, m]], {n, 1, 50}, {m, 0, 1}]]
LinearRecurrence[{1, 1, -1}, {200, 199, 442}, 50] (* Harvey P. Dale, Jun 29 2015 *)

Crossrefs

Sequence in context: A035881 A239525 A180104 * A192545 A118118 A124472

Adjacent sequences: A114981 A114982 A114983 * A114985 A114986 A114987

Keyword

nonn,uned,easy

Author

Roger L. Bagula, Feb 22 2006

Status

approved