Residue of a sequence

Probability Level 2

Consider a sequence $\{a_i\}$ of positive integers defined by $a_1= 1, a_2= 2$ , and for all integers $n>2$ , $a_n= 3a_{n-1} + 5a_{n-2}$ Consider the set $S= \{a_1, a_2, \cdots , a_{1200} \}$ Sam randomly picks an element from this set. The probability that this element is a multiple of $8$ can be expressed as $\dfrac{a}{b}$ , where $a, b$ are coprime positive integers. Find $a+b$ .