We define run sequences of period $2^{n}-1$ as the binary sequences where the distribution of runs of 0’s and runs of 1’s is exactly same as that for the maximal length linear shift resister sequences of period $2^{n}-1$ . We first count the number of all the cyclically distinct run sequences of period $2^{n}-1$ . For each $n$ -tuple, we consider the average number of occurrences over all the run sequences of period $2^{n}-1$ . We identify the $n$ -tuples with average number 1 and, in particular, those that occur exactly once in every run sequence of period $2^{n}-1$ . We finally prove that, as $n$ increases, the average number of every non-zero $n$ -tuple approaches to 1.
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