Iterating on tanh(1) produces bursts with constant preamble term sum: In[1120]:= ColumnForm[ContinuedFraction[#, 11] & /@ NestList[FullSimplify[2*Ceiling[1/#] - 1 - 1/#] &, Tanh[1], 69]] Out[1120]={ {{0, 1, 3, 5, 7, 9, 11, 13, 15, 17, 19}}, {{1, 1, 2, 5, 7, 9, 11, 13, 15, 17, 19}}, {{0, 2, 2, 5, 7, 9, 11, 13, 15, 17, 19}}, {{2, 1, 1, 5, 7, 9, 11, 13, 15, 17, 19}}, {{0, 1, 1, 1, 1, 5, 7, 9, 11, 13, 15}}, {{1, 2, 1, 5, 7, 9, 11, 13, 15, 17, 19}}, {{0, 3, 1, 5, 7, 9, 11, 13, 15, 17, 19}}, {{3, 6, 7, 9, 11, 13, 15, 17, 19, 21, 23}}, {{0, 1, 2, 6, 7, 9, 11, 13, 15, 17, 19}}, {{1, 1, 1, 6, 7, 9, 11, 13, 15, 17, 19}}, {{0, 2, 1, 6, 7, 9, 11, 13, 15, 17, 19}}, {{2, 7, 7, 9, 11, 13, 15, 17, 19, 21, 23}}, {{0, 1, 1, 7, 7, 9, 11, 13, 15, 17, 19}}, {{1, 8, 7, 9, 11, 13, 15, 17, 19, 21, 23}}, {{0, 9, 7, 9, 11, 13, 15, 17, 19, 21, 23}}, {{9, 1, 6, 9, 11, 13, 15, 17, 19, 21, 23}}, {{0, 1, 8, 1, 6, 9, 11, 13, 15, 17, 19}}, {{1, 1, 7, 1, 6, 9, 11, 13, 15, 17, 19}}, {{0, 2, 7, 1, 6, 9, 11, 13, 15, 17, 19}}, {{2, 1, 6, 1, 6, 9, 11, 13, 15, 17, 19}}, {{0, 1, 1, 1, 6, 1, 6, 9, 11, 13, 15}}, {{1, 2, 6, 1, 6, 9, 11, 13, 15, 17, 19}}, {{0, 3, 6, 1, 6, 9, 11, 13, 15, 17, 19}}, {{3, 1, 5, 1, 6, 9, 11, 13, 15, 17, 19}}, {{0, 1, 2, 1, 5, 1, 6, 9, 11, 13, 15}}, {{1, 1, 1, 1, 5, 1, 6, 9, 11, 13, 15}}, {{0, 2, 1, 1, 5, 1, 6, 9, 11, 13, 15}}, {{2, 2, 5, 1, 6, 9, 11, 13, 15, 17, 19}}, {{0, 1, 1, 2, 5, 1, 6, 9, 11, 13, 15}}, {{1, 3, 5, 1, 6, 9, 11, 13, 15, 17, 19}}, {{0, 4, 5, 1, 6, 9, 11, 13, 15, 17, 19}}, {{4, 1, 4, 1, 6, 9, 11, 13, 15, 17, 19}}, {{0, 1, 3, 1, 4, 1, 6, 9, 11, 13, 15}}, {{1, 1, 2, 1, 4, 1, 6, 9, 11, 13, 15}}, {{0, 2, 2, 1, 4, 1, 6, 9, 11, 13, 15}}, {{2, 1, 1, 1, 4, 1, 6, 9, 11, 13, 15}}, {{0, 1, 1, 1, 1, 1, 4, 1, 6, 9, 11}}, {{1, 2, 1, 1, 4, 1, 6, 9, 11, 13, 15}}, {{0, 3, 1, 1, 4, 1, 6, 9, 11, 13, 15}}, {{3, 2, 4, 1, 6, 9, 11, 13, 15, 17, 19}}, {{0, 1, 2, 2, 4, 1, 6, 9, 11, 13, 15}}, {{1, 1, 1, 2, 4, 1, 6, 9, 11, 13, 15}}, {{0, 2, 1, 2, 4, 1, 6, 9, 11, 13, 15}}, {{2, 3, 4, 1, 6, 9, 11, 13, 15, 17, 19}}, {{0, 1, 1, 3, 4, 1, 6, 9, 11, 13, 15}}, {{1, 4, 4, 1, 6, 9, 11, 13, 15, 17, 19}}, {{0, 5, 4, 1, 6, 9, 11, 13, 15, 17, 19}}, {{5, 1, 3, 1, 6, 9, 11, 13, 15, 17, 19}}, {{0, 1, 4, 1, 3, 1, 6, 9, 11, 13, 15}}, {{1, 1, 3, 1, 3, 1, 6, 9, 11, 13, 15}}, {{0, 2, 3, 1, 3, 1, 6, 9, 11, 13, 15}}, {{2, 1, 2, 1, 3, 1, 6, 9, 11, 13, 15}}, {{0, 1, 1, 1, 2, 1, 3, 1, 6, 9, 11}}, {{1, 2, 2, 1, 3, 1, 6, 9, 11, 13, 15}}, {{0, 3, 2, 1, 3, 1, 6, 9, 11, 13, 15}}, {{3, 1, 1, 1, 3, 1, 6, 9, 11, 13, 15}}, {{0, 1, 2, 1, 1, 1, 3, 1, 6, 9, 11}}, {{1, 1, 1, 1, 1, 1, 3, 1, 6, 9, 11}}, {{0, 2, 1, 1, 1, 1, 3, 1, 6, 9, 11}}, {{2, 2, 1, 1, 3, 1, 6, 9, 11, 13, 15}}, {{0, 1, 1, 2, 1, 1, 3, 1, 6, 9, 11}}, {{1, 3, 1, 1, 3, 1, 6, 9, 11, 13, 15}}, {{0, 4, 1, 1, 3, 1, 6, 9, 11, 13, 15}}, {{4, 2, 3, 1, 6, 9, 11, 13, 15, 17, 19}}, {{0, 1, 3, 2, 3, 1, 6, 9, 11, 13, 15}}, {{1, 1, 2, 2, 3, 1, 6, 9, 11, 13, 15}}, {{0, 2, 2, 2, 3, 1, 6, 9, 11, 13, 15}}, {{2, 1, 1, 2, 3, 1, 6, 9, 11, 13, 15}}, {{0, 1, 1, 1, 1, 2, 3, 1, 6, 9, 11}}, {{1, 2, 1, 2, 3, 1, 6, 9, 11, 13, 15}}} --rwg On Sat, Oct 5, 2013 at 6:20 PM, Bill Gosper <billgosper@gmail.com> wrote:
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