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In this paper, we construct an adaptive entropy coded trellis lossy source coding coding scheme for memoryless sources. The scheme utilizes a fully connected trellis with a set of fixed-rate state codebooks and adaptive probability mass function for the codewords. Since the usage frequency of each codeword is known at the decoder, no overhead is necessary for adapting the probabilities mass functions of the codebooks. Thus, using recent redundancy result in trellis coding [G.-C. Zhou et al., 2002] and universal random block codes designed in [E.-H. Yang et al., 2001], we show that the proposed coding scheme is universal and its redundancy is also of the form O(1/n), which is the same order as that of trellis source coding with known statistics.