Regular meetings of the council shall be held on the first Monday after the election of councilmen, and thereafter at least once each month. The council shall provide by ordinance for the times of holding regular meetings, and special meetings may be called from time to time by the mayor or two councilmen. All meetings of the council, whether regular or special, at which any person not a city officer is admitted, shall be open to the public. The mayor shall be president of the council and preside at its meetings, and shall supervise all departments and report to the council for its action all matters requiring attention in either. The superintendent of the department of accounts and finances shall be vice president of the council, and in case of vacancy in the office of mayor, or the absence or inability of the mayor, shall perform the duties of mayor.