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Recursions going to zero

Source: 2015 AIME I Problem 9

March 20, 2015
AMCAIMEAIME I

Problem Statement

Let SS be the set of all ordered triples of integers (a1,a2,a3)(a_1,a_2,a_3) with 1≤a1,a2,a3≤101 \le a_1,a_2,a_3 \le 10. Each ordered triple in SS generates a sequence according to the rule an=an−1⋅∣an−2−an−3∣a_n=a_{n-1}\cdot | a_{n-2}-a_{n-3} | for all n≥4n\ge 4. Find the number of such sequences for which an=0a_n=0 for some nn.
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