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A253904 Denominators of approximation to 2^(1/3) by Halley's method after n iterations.

%I A253904 #33 Feb 16 2025 08:33:24
%S A253904 1,4,504,387144514512,134785660354544802902690364367892668197456173472
%N A253904 Denominators of approximation to 2^(1/3) by Halley's method after n iterations.
%C A253904 Numerators are given in A253690.
%H A253904 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/HalleysMethod.html">Halley's method</a>
%H A253904 Wikipedia, <a href="http://en.wikipedia.org/wiki/Cube_root">Cube root</a>
%F A253904 a(n) = y(n)/gcd(x(n),y(n))
%F A253904 where x(n) = A253690(n-1)*(A253690(n-1)^3n + 4*a(n-1)^3)
%F A253904 and y(n) = 2*(A253690(n-1)^3 + a(n-1)^3);
%F A253904 x(0) = y(0) = 1.
%e A253904 Approximations to 2^(1/3):
%e A253904 n = 1: 5/4 = 1.25; error = -0.00992104...
%e A253904 n = 2: 635/504 = 1.2599206...; error = -0.00000041...
%e A253904 n = 3: 487771523185/387144514512 = 1.2599210...; error = -3.001136... * 10^-20.
%o A253904 (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 A253904 Cf. A002580, A248041, A248042, A253690.
%K A253904 nonn,frac
%O A253904 0,2
%A A253904 _Kival Ngaokrajang_, Jan 24 2015