5 Oct
2013
5 Oct
'13
7:20 p.m.
In[1106]:= NestList[2*Ceiling[1/#] - 1 - 1/# &, 3/8, 22] Out[1106]= {3/8, 7/3, 4/7, 5/4, 1/5, 4, 3/4, 5/3, 2/5, 5/2, 3/5, 4/3, 1/4, 3, 2/3, 3/2, 1/3, 2, 1/2, 1, 0, Indeterminate, Indeterminate} So 3/8 is the "20th rational". The 68th iterate on √2: In[1127]:= ContinuedFraction[ Nest[Simplify[2*Ceiling[1/#] - 1 - 1/#] &, Sqrt[2], 68]] Out[1127]= {1, 1, 1, 1, 1, 1, 1, 1, 1, {2}} I.e., it tries to disguise √2 as the golden ratio by sticking nine 1s on the front. Many other iterates of this process produce CFs with only 1s and 2s. --rwg