>>14
I don't know if they have a name. The pattern of irrational number approximations in the Stern-Brocot tree is similar to the continued fraction, a more common notation than the Stern-Brocot tree sequences.
sqrt(2) = [1;2,2,2,2,2,...] (algebraic)
phi = [1;1,1,1,1,1,1,...] (algebraic)
e = [2;1,2,1,1,4,1,1,6,1,1,8,...] (transcendental)
cube root of 2 = [1;3,1,5,1,1,4,1,1,8,1,14,1,10,2,1,4,12,2,3,2...] (algebraic, non periodic CF)
π = [3;7,15,1,292,1,1,1,2,1,3,1,...] (transcendental, no known pattern)
My guess is that there's an infinite but countable number of regular patterns you can use to generate an irrational number. Since the set of irrational numbers is uncountable almost all continued fractions of irrational numbers have no pattern.