Sequences

Number Theory Level pending

Let \(S\) be the set of all ordered triplets of integers \((a_1,a_2,a_3)\) with \(1\le { a }_{ 1 },{ a }_{ 2 },{ a }_{ 3 }\le 10\). Each ordered triple in \(S\) generates a sequence according to the rule \({ a }_{ n }={ a }_{ n-1 }\cdot \left| { a }_{ n-2 }-{ a }_{ n-3 } \right|\) for all \(n\ge 4\). Find the number of such sequences for which \(a_n=0\) for some value of \(n\).

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