Question

what is the name and syntax of the construction ((..)) in example below?

for ((i=1;i<10;i++)) do echo $i; done

it has strange variable i
where are other constructions for numeric cycling in shells?

Answer 1

In section §3.2.4.2 of the ‘Bash Reference Manual’ (4.0), the ‘((...))‘ notation is classified as as an arithmetic expression. It is closely related to the ‘$((...))‘ notation described in §3.5.5 as Arithmetic Expansion. And §3.2.4.1 ‘Looping Constructs’ says:

An alternate form of the for command is also supported:

    for (( expr1 ; expr2 ; expr3 )) ; do commands ; done

First, the arithmetic expression expr1 is evaluated according to the rules described
below (see Section 6.5 [Shell Arithmetic], page 78). The arithmetic
expression expr2 is then evaluated repeatedly until it evaluates to zero. Each
time expr2 evaluates to a non-zero value, commands are executed and the arithmetic
expression expr3 is evaluated. If any expression is omitted, it behaves as
if it evaluates to 1. The return value is the exit status of the last command in
list that is executed, or false if any of the expressions is invalid.