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.)

Following Sam Hopkins's note, here's a definition. Start with $0$ and apply the substitutions $0 \rightarrow 01$ and $1 \rightarrow 0$ repeatedly, like this: $$0,01,010,01001,01001010,0100101001001,\ldots.$$ The limiting word is $A003849$, one of several called the infinite Fibonacci word, but this one is regarded as the standard form, according to the Crossrefs section of $A014675$.

Writing those words as $w_0,w_1,w_2,\ldots$, respectively, note that $w_n$ is, for $n \geq 2$, the concatention indicated by $w_n=w_{n-1}w_{n-2}$, so that the length of $w_n$ is a Fibonacci number.