I have a question pertaining to whether the following way of writing the Wythoff array obtained from the Fibonacci sequence / infinite Fibonacci word is known or obvious.
Let $n$ be an integer.
Let $F(n) = F(n-1) + F(n-2)$ denote the $n$-th Fibonacci number such that $F(0) = 0$ and $F(1) = 1$.
Let $S_{\infty}$ denote the infinite Fibonacci word function (A014675) and let $N_{\infty}$ be the function that maps the occurrences of the natural numbers $\{0, 1, 2, 3, \ldots\}$ to the digits of the Fibonacci word, so for instance,
$$ \begin{equation} \begin{aligned} S_0 = 2 &: N_0 = 00 \\ S_1 = 21 &: N_1 = 001 \\ S_2 = 212 &: N_2 = 00122 \\ S_3 = 21221 &: N_3 = 00122334 \\ \vdots \end{aligned} \end{equation} $$
We can treat $N_{\infty}$ as a vector such that $N_{\infty}(i)$ is the $i$-th indexed element.
For some non-negative integers $i$ and $j$, let $W(i,j)$ denote the $i,j$-th element of the Wythoff array (A035513)
Question: Is it obvious that
$$ W(i,j) = 2 F_j + \alpha F_{j+1} + \beta F_{j+2} $$
for all $i \geq 2$ and for all $j \geq 1$ where
$$ \alpha = - N_{\infty}(i-2)) + i - 1 \quad \text{and} \quad \beta = N_{\infty}(i-2) $$
Notes:
- This might be obvious and/or I may just be writing something in a convoluted way.
- I am a computer scientist and not a number theorist by training (see point 1, and my apologies if this is trivial).
- I have code that verifies this pattern for several hundred rows/terms of the Wythoff sequence (see point 2) and would be happy to share it if helpful.