  The classical Fibonacci sequence is known to exhibit many fascinating
properties. In this paper, we explore the Fibonacci sequence and integer
sequences generated by second order linear recurrence relations with positive
integer coefficients from the point of view of probability distributions that
they induce. We obtain the generalizations of some of the known limiting
properties of these probability distributions and present certain optimal
properties of the classical Fibonacci sequence in this context. In addition, we
also look at the self linear convolution of linear recurrence relations with
positive integer coefficients. Analysis of self linear convolution is focused
towards locating the maximum in the resulting sequence. This analysis, also
highlights the influence that the largest positive real root, of the
"characteristic equation" of the linear recurrence relations with positive
integer coefficients, has on the location of the maximum. In particular, when
the largest positive real root is 2,the location of the maximum is shown to
depend on whether the sequence length is odd or even.
