Statistical Span Property of Binary Run Sequences

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.