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An algorithm is presented which requires computations for solving a set of linear Toeplitz equations for n = 2k, where k is a positive integer. The previously known algorithms require a minimum of 3n2computations. The major advantage of the algorithm is realized for those applications where the set of n Toeplitz equations Tx = c is to be solved for a different c and the same T for the unknown x. Each additional solution requires storing only (4n + 1) elements from the first solution and computations.