Makri and Psillakis  derived the same formula for it and
they also established an explicit expression of
. In both
papers ( and ) the approach was relied on the definition
, . Recently, Sinha and Sinha 
solving explicitly a recursive generation
scheme for it.
The proposed, in the present note, approach is a new one
and treats under the same frame all the numbers
simple, unified and systematic way. Accordingly, by (7) we
effortless get a recursive scheme for
, . It gives a
way to generate
and it offers further insight in
understanding the interdependencies among the studied
numbers. Specifically, it holds
. We note that
for the particular case we capture by the relevant entries
of (9) and (7), Theorems 2 and 3 of , respectively.
TABLE I. NUMBERS OF OCCURRENCES OF 1-RUNS,
NUMBER OF 1S,
, IN BINARY STRINGS OF LENGTH
In this note we stated three run statistics which are
important in many areas of applied probability. We defined
them on a binary (0-1) sequence, and we then provided
explicitly their mean values for a Bernoulli sequence. After that,
we considered binary strings (symmetric Bernoulli sequences)
and we showed how the analytic expressions of the means of
these RVs provide eventually the respective explicit
expressions of three numbers studied recently by different
methods. Finally, as a byproduct of our approach, we proposed
a unified recursive scheme which clarifies further the
interdependencies among these numbers. The examined
numbers are potential useful in many engineering applications
like the ones mentioned briefly in the Introduction. Early
results are encouraging in this direction.
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