A253690 Numerators of approximation to 2^(1/3) by Halley's method after n iterations.

1, 5, 635, 487771523185, 169819290704671870437365746682881808313592465345

Offset: 0

Author

Kival Ngaokrajang, Jan 24 2015

Keywords

• nonn
• frac

Comments

Denominators are given in A253904.

Examples

			Approximations to 2^(1/3):
n = 1: 5/4 = 1.25; error = -0.00992104...
n = 2: 635/504 = 1.2599206...; error = -0.00000041...
n = 3: 487771523185/387144514512 = 1.2599210...; error = -3.001136... * 10^-20.
		

Links

• Eric Weisstein's World of Mathematics, Halley's method
• Wikipedia, Cube root

Crossrefs

Cf. A002580, A248041, A248042, A253904.

Programs

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(a, ", "))}

Formula

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

where x(n) = a(n-1)*(a(n-1)^3n + 4*A253904(n-1)^3)

and y(n) = 2*(a(n-1)^3 + A253904(n-1)^3);

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