Mock AIME 6 2006-2007 Problems/Problem 15

Problem

For any finite sequence of positive integers $A=(a_1,a_2,\cdots,a_n)$ , let $f(A)$ be the sequence of the differences between consecutive terms of $A$ . i.e. $f(A)=(a_2-a_1,a_3-a_2,\cdots,a_n-a_{n-1})$ . Let $F^k(A)$ denote $F$ applied $k$ times to $A$ . If all of the sequences $A, f(A), f^2(A),\cdots, f^{n-2}(A)$ are strictly increasing and the only term of $f^{n01}(A)$ is $1$ , we call the sequence $A$ $\textit{superpositive}$ . How many sequences $A$ with at least two terms and no terms exceeding $18$ are $\textit{superpositive}$ ?

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Mock AIME 6 2006-2007 Problems/Problem 15

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