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Counting nine-term sequences with ratio constraints

High school · ●●●○○ 50 pts

The sequence {an}\{a_n\} has nine terms, a1=a9=1a_1=a_9=1, and for each i∈{1,2,…,8}i\in\{1,2,\ldots,8\} we have ai+1ai∈{2,1,−12}\frac{a_{i+1}}{a_i}\in\left\{2,1,-\frac{1}{2}\right\}. How many such sequences are there?

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countingcombinatoricsbinomial coefficientssequences