An infinite simple continued fraction that has a repeating pattern is the root of a quadratic equation.
For instance, X^2-2 = 0 has the root X = √2 which can be expressed as X = [1;2,2,2,...]
The golden mean is the positive root of X^2-X-1 and can be expressed as X = [1;1,1,1,...]

HAKMEM has an interesting section on simple continued fractions, including algorithms for performing arithmetic operations on them which may be viewed at