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Omni-MATH / Call a subset S of {1,2,…,n} …a and b are elements of S whose average is an integer, that average is also an element of S. Let A(n) be the number of mediocre subsets of {1,2,…,n}. …

Problem

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problem

Call a subset SS of {1,2,…,n}\{1, 2, \dots, n\} \emph{mediocre} if it has the following property: Whenever aa and bb are elements of SS whose average is an integer, that average is also an element of SS. Let A(n)A(n) be the number of mediocre subsets of {1,2,…,n}\{1,2,\dots,n\}. [For instance, every subset of {1,2,3}\{1,2,3\} except {1,3}\{1,3\} is mediocre, so A(3)=7A(3) =7.] Find all positive integers nn such that A(n+2)−2A(n+1)+A(n)=1A(n+2) - 2A(n+1) + A(n) = 1.
Plain-text mathematical notation (without MathML)
Call a subset S of {1,2,…,n} \emph{mediocre} if it has the following property:
Whenever a and b are elements of S whose average is an integer, that average is also
an element of S. Let A(n) be the number of mediocre subsets of {1,2,…,n}.
[For instance, every subset of {1,2,3} except {1,3} is mediocre, so A(3)=7.]
Find all positive integers n such that A(n+2)−2A(n+1)+A(n)=1.
Original LaTeX notation
Call a subset $S$ of $\{1, 2, \dots, n\}$ \emph{mediocre} if it has the following property:
Whenever $a$ and $b$ are elements of $S$ whose average is an integer, that average is also
an element of $S$. Let $A(n)$ be the number of mediocre subsets of $\{1,2,\dots,n\}$.
[For instance, every subset of $\{1,2,3\}$ except $\{1,3\}$ is mediocre, so $A(3) =7$.]
Find all positive integers $n$ such that $A(n+2) - 2A(n+1) + A(n) = 1$.

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Source and history

task_id
a59ecf7d-49f7-5302-a7cb-8d24036f72a4
task_key
test--a59ecf7d-49f7-5302-a7cb-8d24036f72a4
task_revision_id
1
config / split
/ test

Official source

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