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Denominators of approximation to 2^(1/3) by Halley's method after n iterations.
1

%I #32 Aug 01 2019 00:19:06

%S 1,4,504,387144514512,134785660354544802902690364367892668197456173472

%N Denominators of approximation to 2^(1/3) by Halley's method after n iterations.

%C Numerators are given in A253690.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/HalleysMethod.html">Halley's method</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Cube_root">Cube root</a>

%F a(n) = y(n)/gcd(x(n),y(n))

%F where x(n) = A253690(n-1)*(A253690(n-1)^3n + 4*a(n-1)^3)

%F and y(n) = 2*(A253690(n-1)^3 + a(n-1)^3);

%F x(0) = y(0) = 1.

%e Approximations to 2^(1/3):

%e n = 1: 5/4 = 1.25; error = -0.00992104...

%e n = 2: 635/504 = 1.2599206...; error = -0.00000041...

%e n = 3: 487771523185/387144514512 = 1.2599210...; error = -3.001136... * 10^-20.

%o (PARI) {a=1; b=1; print1(b, ", "); for(n=1, 5, x=a*(a^3+4*b^3); y=2*b*(a^3+b^3); a=x/gcd(x, y); b=y/gcd(x, y); print1(b, ", "))}

%Y Cf. A002580, A248041, A248042, A253690.

%K nonn,frac

%O 0,2

%A _Kival Ngaokrajang_, Jan 24 2015