On periodic sequences for algebraic numbers

For each positive integer n ≥ 2, a new approach to expressing real numbers as sequences of nonnegative integers is given. The n = 2 case is equivalent to the standard continued fraction algorithm. For n = 3, it reduces to a new iteration of the triangle. Cubic irrationals that are roots of x3 +kx2 +x−1 are shown to be precisely those numbers with purely periodic expansions of period length one. For general positive integers n, it reduces to a new iteration of an n dimensional simplex. Algebraic numbers that are roots of x + kxn−1 + xn−2 − x − 1 are precisely those with purely periodic expansions of period length one.