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

1, 4, 504, 387144514512, 134785660354544802902690364367892668197456173472

Offset: 0

Author

Kival Ngaokrajang, Jan 24 2015

Keywords

• nonn
• frac

Comments

Numerators are given in A253690.

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, A253690.

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(b, ", "))}

Formula

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

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

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

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