Let $W = 01001010010010 \ldots$ be the infinite Fibonacci word, A003849 in the OEIS.  Let $B(m)$ be the set of $m+1$ subwords of $W$ that have length $m$, and for each such subword $u$, let $p(u)$ be the sequence of positions in $W$ where $u$ begins. Then $p(u)$ is a composite of the famous Wythoff sequences $A=A000201$ and $B=A001950$. Can someone figure out (or cite a reference) exactly which composites represent the subwords in $B(m)$?

Here's how it looks for $m=4$:


\begin{array}{|c|c|c|c|}
\text{subword, } u & \text{positions, } p(u) & \text{composite} & \text{OEIS} \\
0100 & 1,6,9,14,\ldots & AAA & A134859 \\
1001 & 2,7,10,15,\ldots & BA & A035336 \\
0010 & 3,8,11,16,\ldots & AB & A003623 \\
0101 & 4,12,17,25,\ldots & AAB & A134860 \\
1010 & 5,13,18,26,\ldots & BB & A101864
\end{array}




(Note that for every $m$, the difference sequence of every $p(u)$ consists of Fibonacci numbers.)