N.H. Rev. Stat. Ann. § 45:4

If no person is elected mayor, or if the person elected declines, the convention shall make a record of the fact and appoint a day for another election; and the mayor and aldermen shall issue their precept to the selectmen of the several wards, requiring them to call meetings of their wards on the day so appointed; and the same proceedings shall be had from time to time until a mayor is chosen.