You can find old bash source here. In particular I downloaded bash-1.14.7.tar.gz. In the documentation/bash.txt you will find:
Arithmetic Expansion
Arithmetic expansion allows the evaluation of an arithmetic expression and the substitution of the result. There are two formats for arithmetic expansion:
$[expression]
$((expression))
The references to $[ are gone in doc/bash.html from the bash-doc-2.0.tar.gz download and the NEWS file mentions that:
The $[...] arithmetic expansion syntax is no longer supported, in favor of $((...)).
$((...)) is also the standard syntax for an arithmetic expansion, but may have been added to the standard later than the original Bash implementation.
However, $[...] does still seem to work in Bash 5.0, so it's not completely removed.
bashspecific$[...]syntax is non-standard. The$((...))(that stems fromksh) is POSIX standard. – Janis Jun 15 '15 at 16:20man bash | grep -E '\$\['. Output: The old format $[expression] is deprecated and will be removed in upcoming versions – Cyrus Jun 15 '15 at 17:17