# Residue of a sequence

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 $$a+b$$.

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