Seemingly Tough sequences

Let $a_0, a_1, a_2, \ldots$ be a sequence of real numbers such that $a_0 = 0, a_1 = 1$ and for every $n \ge 2$ there exists $1 \le k \le n$ satisfying

$\ { a }_{ n } = \displaystyle \frac { 1 }{ k } \sum _{ i=1 }^{ k }{ { a }_{ n - i } } .$

The maximal possible value of $\ { a }_{ 2018 } - { a }_{ 2017 }$ can be represented as $\dfrac a{b^c}$ , where $a, b, c$ are natural numbers with $\gcd(a, b) = 1$ .

Find $b - a + c.$