Sheep Talk
The sequence starts as:
A, B, AB, BAB, ABBAB, BABABBAB, ABBABBABABBAB,
BABABBABABBABBABABBAB,
ABBABBABABBABBABABBABABBABBABABBAB,
BABABBABABBABBABABBABABBABBAABABBABBABABBABABBABBABABBAB ,...
Number of A's: 1, 0, 1, 1, 2, 3, 5, 8, 13, 21 ,...
Number of B's: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, ...
Number of letters: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, ...
Let $a_n$ and $b_n$ be the number of A's and B's respectively in
the $nth$ word and $f_n$ the total number of letters in the $nth
word$. Note that in each word there is an A for every B in the
previous word so $$a_n = b_{n-1}\quad (1).$$ The number of B's is
given by the number of B's in the previous word plus the number of
A's in the previous word and so $$b_{n+1} = a_n + b_n\quad (2)$$.
Putting these two expressions together and substituting for $a_n$
in (2) we get $$b_{n+1} = b_{n-1}+ b_n$$ so the sequence $b_n$ is a
Fibonacci sequence and the pattern will continue. Similarly
substituting for $b_n$ in (2) we get $$a_{n+2}=a_n + a_{n+1}$$ so
the $a_n$ forn a Fibonaci sequence and the pattern will
continue.
Because the two sequences of numbers are the same apart from the
shift of one place the total number of letters is also a Fibonacci
sequence and the pattern will continue.
\begin{eqnarray} a_n &= b_{n-1} \\ f_n &= a_n + b_n =
b_{n-1} + b_n = b_{n+1} \\ f_{n-1}&= a_{n-1} + b_{n-1} =
a_{n-1} + a_n = a_{n+1} \\ f_{n+1}&= a_{n+1}+ b_{n+1} = f_{n-1}
+ f_n . \end{eqnarray}