Beatty sequences

There is a beautiful theorem by Samuel Beatty which states:

If $\alpha$ and $\beta$ are positive irrational numbers such as $\frac{1}{\alpha }+\frac{1}{\beta }=1$
then $A=\{n\in N,\lfloor n\times \alpha \rfloor \}$ and $B=\{n\in N,\lfloor n\times \beta \rfloor \}$ are a partition of N:
$N=A\cup B$ and $A\cap B=\mathrm{\varnothing }$

Let's $A$ the vector contained in this file data389.txt

If $A$ is a Beatty sequence, can you find the corresponding value $\alpha$

Answer format: $\alpha$ // truncated at 6 positions after the decimal point.

[My timing: instant]