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

1, 5, 635, 487771523185, 169819290704671870437365746682881808313592465345

Offset

0,2

Comments

Denominators are given in A253904.

Links

Table of n, a(n) for n=0..4.

Eric Weisstein's World of Mathematics, Halley's method

Wikipedia, Cube root

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.

Example

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.

Prog

(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, ", "))}

Crossrefs

Cf. A002580, A248041, A248042, A253904.

Sequence in context: A348081 A256070 A203339 * A068421 A142535 A203475

Adjacent sequences: A253687 A253688 A253689 * A253691 A253692 A253693

Keyword

nonn,frac

Author

Kival Ngaokrajang, Jan 24 2015

Status

approved