Problem: Determine all infinite strings of letters with the following properties:
\begin{enumerate}
  \item[(a)] Each letter is either $T$ or $S$,
  \item[(b)] If position $i$ and $j$ both have the letter $T$,
    then position $i+j$ has the letter $S$,
  \item[(c)] There are infinitely many integers $k$ such that position $2k-1$ has the $k$th $T$.
\end{enumerate}

Solution: We wish to find all infinite sequences $a_1, a_2, \dots$
of positive integers satisfying the following properties:
 \begin{enumerate}[(a)]
  \item $a_1 < a_2 < a_3 < \dotsb$,
  \item there are no positive integers $i$, $j$, $k$, not necessarily distinct, such that $a_i+a_j=a_k$,
  \item there are infinitely many $k$ such that $a_k = 2k-1$.
 \end{enumerate}

If $a_k=2k-1$ for some $k>1$, let $A_k=\{a_1,a_2,\dots,a_k\}$.
By (b) and symmetry, we have
\[ 2k-1
  \ge \frac{|A_k-A_k|-1}{2}+|A_k|
  \ge \frac{2|A_k|-2}{2}+|A_k| = 2k-1. \]
But in order for $|A_k-A_k|=2|A_k|-1$,
we must have $A_k$ an arithmetic progression,
whence $a_n=2n-1$ for all $n$ by taking $k$ arbitrarily large.