Show category children, one level

Why are you using the parameter parent=1?
If you remove this parameter it should work.

If you want children and grandchildren, you should use:

$categories =  get_categories('hide_empty=0&child_of=".$cat);

If you want only direct children, you should use:

$categories=  get_categories("hide_empty=0&parent=".$cat);

Also note that you are echoing $mycat inside the loop, and you are joining the multiple categories in this var, so the categories should appear repeated…