Wolfram's iteration is an algorithm for computing the square root of a rational number 
using properties of the binary
representation of 
.
The algorithm begins with 
, and then iterates
 |  |  |
(1)
|
 |  |  |
(2)
|
Interpreted as a binary number, 
then converges to 
.
For example, for 
(i.e., Pythagoras's constant), 
is given by 2, 4, 16, 28, 28, 112, 92, 368, 28, ... (OEIS
A095803), and 
by 0, 4, 8, 20, 44, 88, 180, 360, 724, ... (OEIS A095804).
The binary representation of successive terms of 
(with the "binary" point shifted after the first
term) are then
 |
(3)
|
as illustrated above, which can be seen to produce increasing numbers of digits in the binary representation of 
,
 |
(4)
|
(OEIS A004539). Interpreting the binary fractions produced at each step gives the sequence of approximations 1, 1, 5/4, 11/8, 11/8, 45/32, 45/32, 181/128, 181/128, ... (OEIS A095805 and A095806).
