Editing Lucas number (section)

== Continued fractions for powers of the golden ratio ==
Close [[Diophantine approximation|rational approximations]] for powers of the golden ratio can be obtained from their [[continued fraction]]s.

For positive integers ''n'', the continued fractions are:
:<math> \varphi^{2n-1} = [L_{2n-1}; L_{2n-1}, L_{2n-1}, L_{2n-1}, \ldots] </math>
:<math> \varphi^{2n} = [L_{2n}-1; 1, L_{2n}-2, 1, L_{2n}-2, 1, L_{2n}-2, 1, \ldots] </math>.

For example:
:<math> \varphi^5 = [11; 11, 11, 11, \ldots] </math>
is the limit of
:<math> \frac{11}{1}, \frac{122}{11}, \frac{1353}{122}, \frac{15005}{1353}, \ldots </math>
with the error in each term being about 1% of the error in the previous term; and
:<math> \varphi^6 = [18 - 1; 1, 18 - 2, 1, 18 - 2, 1, 18 - 2, 1, \ldots] = [17; 1, 16, 1, 16, 1, 16, 1, \ldots] </math>
is the limit of
:<math> \frac{17}{1}, \frac{18}{1}, \frac{305}{17}, \frac{323}{18}, \frac{5473}{305}, \frac{5796}{323}, \frac{98209}{5473}, \frac{104005}{5796}, \ldots </math>
with the error in each term being about 0.3% that of the ''second'' previous term.