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A114984 Coefficients of cubic equations in the form w^2=4*x^3-g2*x-g3 Weierstrass elliptic form whose solutions approximate zeta zeros. 0
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified October 21 11:30 EDT 2019. Contains 328295 sequences. (Running on oeis4.)