Wikipedia's page on AIC gives a formula for the AICc, a "corrected" version of the AIC that helps to avoid overfitting when the sample size is small relative to the number of parameters in the models being considered.
Wikipedia's formula is consistent with Burnham & Anderson (2002), with the correction term $2K(K+1)/(n-K-1)$, where $K$ is the number of parameters and $n$ is the sample size.
However, in every source I have found prior to Burnham & Anderson (2002), the correction term is stated as $2(K+1)(K+2)/(n-K-2)$. For example, see Anderson et. al. (1994).
Why are the two formulas different? Was there a mistake in the original derivation that was later corrected, or did the set of assumptions change at some point? I have not been able to find any explicit mention of the difference between these two formulas.