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Difference between revisions of "Mock AIME 6 2006-2007 Problems/Problem 15"

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

Revision as of 20:36, 29 April 2025

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 $f^{n-1}(A)$ is $1$, we call the sequence $A$ $\textit{superpositive}$. How many sequences $A$ with at least two terms and no terms exceeding $18$ are $\textit{superpositive}$?

Solution

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