MathDB

Call a subset

S

\{1,2,\dots,n\}

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)\equal{}7.] Find all positive integers

n

such that A(n\plus{}2)\minus{}2A(n\plus{}1)\plus{}A(n)\equal{}1.

B3