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In this paper, we give sufficient conditions for the construction of periodic and generalized periodic Jacobi matrices from particular spectral data. For the construction of a periodic Jacobi matrix, the smallest and largest eigenvalues of all its leading principal submatrices, and a prescribed entry are required. For the case of the generalized periodic Jacobi matrix, an eigenvector associated with the largest eigenvalue is additionally considered. The results obtained provide algorithmic procedures that are illustrated in some numerical experiments. [ABSTRACT FROM AUTHOR]