The perfect Gaussian integer sequences, which satisfy the ideal periodic autocorrelation functions, are of practical interest in both orthogonal frequency division multiplexing (OFDM) systems and code division multiple access (CDMA) systems. In this work, we propose an infinity family of the perfect Gaussian integer sequences constructed from a multiple of the original Payne's o-polynomial. Experimental results show that the obtained sequences are balanced and have odd period. The energy efficiency of the resulting sequences is slightly increased when compared with those of the already known sequences.