Another mathematical number sequence

Can you work out the mathematical rule for this number sequence?

354, 72, 345, 132, 6, 216, 9, 729, 354, 72, 345........

Hint 1

this is a mathematical puzzle

From time to time hints will be updated until solved.

Note: this sequence cannot be found in The On-Line Encyclopedia of Integer Sequences

Answer

Take

the $n$-th digit, where $n$ is the first digit (if $n$ is more than the number of the digits then "modulo" it -- wrap around)

and

cube it

then

sum with other digits

so you will get

$354$ -> $3+5+4^3$ (3rd digit) = $72$
$72$ -> $7^3+2$ (7th digit) = $345$

$345$ -> $3+4+5^3$ (3rd digit) = $132$
$132$ -> $1^3+3+2$ (1st digit) = $6$
$6$ -> $6^3$ (6th digit) = $216$
$216$ -> $2+1^3+6$ (2nd digit) = $9$
$9$ -> $9^3$ (9th digit) = $729$
$729$ -> $7^3+2+9$ (7th digit) = $354$
...