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Sample-based (or adaptive) block codes represent a transitional case: while the exact parameters of a source are not known, one can obtain finite length sequences of symbols (or samples) produced by this source in the past. It turns out, that using such samples it is possible to construct codes that are more efficient in encoding this particular source than pure universal codes. In order to be able to compare adaptive codes and codes for a known source more precisely one clearly needs to know the exact expression of an O (1/n). Obtaining the exact expression for this and subsequent terms describing the average behavior of adaptive block codes is the main goal of this paper.